# Basic non-Archimedean functional analysis over a non-Archimedean field $\star_{\_c^{\#}}$.

jaykov foukzon

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Basic non-Archimedean functional analysis over a
non-Archimedean field $\star_{\_c^{\#}}$.

jaykov foukzon

► To cite this version:

jaykov foukzon. Basic non-Archimedean functional analysis over a non-Archimedean field $\star_{\_c^{\#}}$.2024.hal-04583394v2

HAL Id: hal-04583394

https://hal.science/hal-04583394v2

Preprint submitted on 16 Oct 2024

**HAL** is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
Basic non-Archimedean functional analysis over non-Archimedean field *$\mathbb{R}_c^#$. Applications to constructive quantum field theory.

Jaykov Foukzon

Center for Mathematical Sciences, Israel Institution of Technology, Haifa, Israel.

E-mail: jaykovfoukzon@list.ru

E-mail: jaikovfoukzon@gmail.com

https://orcid.org/0000-0002-6837-4004

**Abstract.** Functional analysis works with TVS (Topological Vector Spaces), classically over archimedean fields like $\mathbb{R}$ and $\mathbb{C}$. Canonical non-Archimedean functional analysis, where alternative but equally valid number systems such as p-adic numbers $\mathbb{Q}_p$ etc. are fundamental, is a fast-growing discipline.

This paper deals with TVS over non-classical non-Archimedean fields *$\mathbb{R}_c^#$, $\widetilde{\mathbb{R}_c^#}$ and *$\mathbb{C}_c^#$, $\widetilde{\mathbb{C}_c^#}$.

Definitions and theorems related to non-Archimedean functional analysis on non-Archemedean field $\widetilde{\mathbb{R}_c^#}$ and on complex field $\widetilde{\mathbb{C}_c^#} = \widetilde{\mathbb{R}_c^#} + i\widetilde{\mathbb{R}_c^#}$ are considered.

Applications to constructive quantum field theory also are considered
[6] https://doi.org/10.1063/5.0162832

[12] https://iopscience.iop.org/article/10.1088/1742-6596/2701/1/012113

[notice in [6] and [12] we abbreviate *$\mathbb{R}_c^#$ instead $\widetilde{\mathbb{R}_c^#}$ for the sake of brevity].

Definitions and theorems appropriate to analysis on non-Archemedean
field *$\mathbb{R}_c^#$ and on complex field *$\mathbb{C}_c^# = \mathbb{R}_c^# + i\mathbb{R}_c^#$ are given in [1]-[2].
Content.

**Chapter I. Basic analysis on external non-Archimedean field $\mathbb{R}_c^#$ and $\mathbb{R}_c^{\#}$**

§1. Bivalent hyper Infinitary first-order logic $^2L_{\infty}^{\#}$ with restricted rules of conclusion. Generalized Deduction Theorem.

§2. Set theory $NC_{\infty}^{\#}$.

§3. Nonconservative extension of the model theoretical NSA based on bivalent hyper Infinitary first-order logic $^2L_{\infty}^{\#}$ with restricted canonical rules of conclusion.

§4. Internal Set Theory IST.

§5. Internal Set Theory IST#.

§6. Hypernaturals $N^#.

§7. Axioms of the nonstandard arithmetic $A^#.

§8. The Generalized Recursion Theorem.

§9. General associative and commutative laws.

§10. Hyperrationals $\mathbb{Q}^#.

§11. External Cauchy hyperreals $\mathbb{R}_c^#$ via Cauchy completion of $\mathbb{Q}^#.
Basic non-Archimedean functional analysis over a non-Archimedean field $\mathbb{R}^\#_c$ and $\mathbb{R}^\#_c$. Applications to constructive quantum field theory.

Jaykov Foukzon

Center for Mathematical Sciences, Israel Institution of Technology, Haifa, Israel.

E-mail: jaykovfoukzon@list.ru

E-mail: jaikovfoukzon@gmail.com

https://orcid.org/0000-0002-6837-4004

**Abstract.** Functional analysis works with TVS (Topological Vector Spaces), classically over archimedean fields like $\mathbb{R}$ and $\mathbb{C}$. Canonical non-Archimedean functional analysis, where alternative but equally valid number systems such as p-adic numbers $\mathbb{Q}_p$ etc. are fundamental, is a fast-growing discipline.
This paper deals with TVS over non-classical non-Archimedean fields $\mathbb{R}^\#_c, \mathbb{R}^{\tilde{\#}}_c$ and $\mathbb{C}^\#_c, \mathbb{C}^{\tilde{\#}}_c$.

Definitions and theorems related to non-Archimedean functional analysis on non-Archemedean field $\mathbb{R}^\#_c$ and on complex field $\mathbb{C}^\#_c = \mathbb{R}^\#_c + i\mathbb{R}^\#_c$ are considered.

Applications to constructive quantum field theory also are considered
[6] https://doi.org/10.1063/5.0162832

[12] https://iopscience.iop.org/article/10.1088/1742-6596/2701/1/012113
[notice in [6] and [12] we abbreviate $\mathbb{R}^\#_c$ instead $\mathbb{R}^{\tilde{\#}}_c$ for the sake of brevity].
Definitions and theorems appropriate to analysis on non-Archemedean
field $\mathbb{R}^\#_c$ and on complex field $\mathbb{C}^\#_c = \mathbb{R}^\#_c + i\mathbb{R}^\#_c$ are given in [1]-[2].

Content.

**Chapter I. Basic analysis on external non-Archimedean field $\mathbb{R}^\#_c$ and $\mathbb{R}^{\tilde{\#}}_c$**

§1. Bivalent hyper Infinitary first-order logic $^2L^\#_{\infty}$ with restricted rules of conclusion. Generalized Deduction Theorem.

§2. Set theory $\mathbb{N}^\#_{\infty}$.

§3. Nonconservative extension of the model theoretical NSA based on bivalent hyper Infinitary first-order logic $^2L^\#_{\infty}$ with restricted canonical rules of conclusion.

§4. Internal Set Theory IST.

§5. Internal Set Theory IST\#.

§6. Hypernaturals $\mathbb{N}^\#$.

§7. Axioms of the nonstandard arithmetic $\mathbb{A}^\#$.

§8. The Generalized Recursion Theorem.

§9. General associative and commutative laws.

§10. Hyperrationals $\mathbb{Q}^\#$.

§11. External Cauchy hyperreals $\mathbb{R}^\#_c$ via Cauchy completion of $\mathbb{Q}^\#$.
§12. The Extended Hyperreal Number System $\hat{\mathbb{R}}^#$.

§13. External non-Archimedean field $*\mathbb{R}^#$. via Cauchy completion of $*\mathbb{R}$.

§13.1. The Extended Hyperreal Number System $*\hat{\mathbb{R}}^#$.

§13.2. #-Open and #-Closed Sets on $*\hat{\mathbb{R}}^#$.

§13.3. #-Open Coverings.

§13.4. External Cauchy hyperreals $\mathbb{R}^#$ and $*\mathbb{R}^#$ axiomatically.

§14.1. External non-Archimedean field $\mathbb{R}^\sim_c$ via special extension of non-Archimedean field $\mathbb{R}^#_c$

§14.2. External non-Archimedean field $*\mathbb{R}^\sim_c$ via non-Archimedean field special extension of field $*\mathbb{R}^#_c$.

§15. Basic analysis on external non-Archimedean field $\mathbb{R}^#_c$.

§15.1. The #-limit of a function $f: \mathbb{R}^#_c \to \mathbb{R}^#_c$

§15.2. Monotonic Functions $f: \mathbb{R}^#_c \to \mathbb{R}^#_c$.

§15.3. #-Limits Inferior and Superior.

§15.4. The #-continuity of a function $f: \mathbb{R}^#_c \to \mathbb{R}^#_c$.

§15.5. Removable #-discontinuities.

§15.6. Composite Functions $f: \mathbb{R}^#_c \to \mathbb{R}^#_c$.

§15.7. Bounded and Hyperbounded Functions $f: \mathbb{R}^#_c \to \mathbb{R}^#_c$.

§15.8. Generalized Intermediate Value Theorem.

§15.9. Uniform #-Continuity.

§15.10. Monotonic External Functions $f: \mathbb{R}^#_c \to \mathbb{R}^#_c$.

§15.11. The #-derivative of a $\mathbb{R}^#_c$-valued function $f: D \to \mathbb{R}^#_c$.

§15.12. The Riemann integral of a $\mathbb{R}^#_c$-valued external function $f(x)$.

§15.13. Existence of the integral of a $\mathbb{R}^#_c$-valued external function $f(x)$.

§15.14. Improper integrals.

§15.15. Improper integrals of nonnegative functions $f: D \to \mathbb{R}^#_c$. Absolute Integrability.

§15.16. Change of Variable in an Improper Integral.

§15.17. Generalized integrability criterion due to Lebesgue.

16. Hyper infinite external sequences and series.

16.1. Hyper infinite external sequences.

§16.2. Hyper infinite external series of constant.

§16.4. Absolute and Conditional #-Convergence.

§16.5. Grouping Terms in a Hyper Infinite Series.

§16.6. Rearrangement of hyper infinite series.

§16.7. Multiplication of hyper infinite Series.

§16.8. Double Hyper Infinite Sequences.

§16.9. External Double Hyper Infinite Series.

§16.10. Interchanging the order of summation of hyper infinite sum.

§17. Hyper infinite sequences and series of $\mathbb{R}^#_c$-valued functions.

§17.1. Uniform #-Convergence.

§17.2. Hyper Infinite Series of Functions.

§17.3. #-Continuity, #-Differentiability, and Integrability of hyper infinite Series.

§18. Hyper Infinite Power Series.

§18.1. The #-convergence properties of hyper infinite power series.

§18.2. The $\mathbb{R}^#_c$-valued #-exponential $\text{Ext}-\exp(x)$

§18.3. The $\mathbb{R}^#_c$-valued Trigonometric Functions $\text{Ext}-\sin(x)$ and $\text{Ext}-\cos(x)$.
§12. The Extended Hyperreal Number System $\mathbb{R}^#.$

§13. External non-Archimedean field $*\mathbb{R}^#.$ via Cauchy completion of $*\mathbb{R}$.

§13.1. The Extended Hyperreal Number System $*\mathbb{R}^#.$

§13.2. #-Open and #-Closed Sets on $*\mathbb{R}^#.$

§13.3. #-Open Coverings.

§13.4. External Cauchy hyperreals $\mathbb{R}^#.$ and $*\mathbb{R}^#.$ axiomatically.

§14.1. External non-Archimedean field $\tilde{\mathbb{R}}^#.$ via special extension of non-Archimedean field $\mathbb{R}^#.$

§14.2. External non-Archimedean field $*\tilde{\mathbb{R}}^#.$ via special extension of field $*\mathbb{R}^#.$

§15. Basic analysis on external non-Archimedean field $\mathbb{R}^#.$

§15.1. The #-limit of a function $f: \mathbb{R}^# \to \mathbb{R}^#.$

§15.2. Monotonic Functions $f: \mathbb{R}^# \to \mathbb{R}^#.$

§15.3. #-Limits Inferior and Superior.

§15.4. The #-continuity of a function $f: \mathbb{R}^# \to \mathbb{R}^#.$

§15.5. Removable #-discontinuities.

§15.6. Composite Functions $f: \mathbb{R}^# \to \mathbb{R}^#.$

§15.7. Bounded and Hyperbounded Functions $f: \mathbb{R}^# \to \mathbb{R}^#.$

§15.8. Generalized Intermediate Value Theorem.

§15.9. Uniform #-Continuity.

§15.10. Monotonic External Functions $f: \mathbb{R}^# \to \mathbb{R}^#.$

§15.11. The #-derivative of a $\mathbb{R}^#$-valued function $f: D \to \mathbb{R}^#.$

§15.12. The Riemann integral of a $\mathbb{R}^#$-valued external function $f(x)$.

§15.13. Existence of the integral of a $\mathbb{R}^#$-valued external function $f(x)$.

§15.14. Improper integrals.

§15.15. Improper integrals of nonnegative functions $f: D \to \mathbb{R}^#.$
Absolute Integrability.

§15.16. Change of Variable in an Improper Integral.

§15.17. Generalized integrability criterion due to Lebesgue.

16. Hyper infinite external sequences and series.

16.1. Hyper infinite external sequences.

§16.2. Hyper infinite external series of constant.

§16.4. Absolute and Conditional #-Convergence.

§16.5. Grouping Terms in a Hyper Infinite Series.

§16.6. Rearrangement of hyper infinite series.

§16.7. Multiplication of hyper infinite Series.

§16.8. Double Hyper Infinite Sequences.

§16.9. External Double Hyper Infinite Series.

§16.10. Interchanging the order of summation of hyper infinite sum.

§17. Hyper infinite sequences and series of $\mathbb{R}^#$-valued functions.

§17.1. Uniform #-Convergence.

§17.2. Hyper Infinite Series of Functions.

§17.3. #-Continuity, #-Differentiability, and Integrability of hyper infinite Series.

§18. Hyper Infinite Power Series.

§18.1. The #-convergence properties of hyper infinite power series.

§18.2. The $\mathbb{R}^#$-valued #-exponential $\text{Ext}\text{-exp}(x)$

§18.3. The $\mathbb{R}^#$-valued Trigonometric Functions $\text{Ext}\text{-sin}(x)$ and $\text{Ext}\text{-cos}(x)$.
§18.4. $\mathbb{R}_c^\#$-valued functions of several variables.

§18.5. Line Segments in $\mathbb{R}_c^\#$, $n \in \mathbb{N}^\#$.

§18.6. #-Neighborhoods and #-open sets in $\mathbb{R}_c^\#$, $n \in \mathbb{N}^\#$. Connected Sets and Regions in $\mathbb{R}_c^\#$.

§18.7. The #-limits and #-continuity $\mathbb{R}_c^\#$-valued functions of $n \in \mathbb{N}^\#$ variables.

§18.8. Partial #-Derivatives and the #-Differential.

§19. #-Analytic functions $f: \mathbb{C}_c^\# \to \mathbb{C}_c^\#$.

§19.1. $\mathbb{C}_c^\#$-valued #-analytic functions $f: \mathbb{C}_c^\# \to \mathbb{C}_c^\#$.

§19.2. The $\mathbb{C}_c^\#$-valued #-Exponential $\mathrm{Ext}\text{-exp}(z)$.

§19.3. The $\mathbb{C}_c^\#$-valued Trigonometric Functions $\mathrm{Ext}\text{-sin}(z)$, $\mathrm{Ext}\text{-cos}(z)$.

§19.4. The periodicity of the #-exponential $\mathrm{Ext}\text{-exp}(iz)$.

§19.5. The $\mathbb{C}_c^\#$-valued Logarithm.

§ 20. Complex Integration of the $\mathbb{C}_c^\#$-valued function $f(t)$.

§ 20.1. Definition and basic properties of the complex integral.

§ 20.2. Line Integrals as Functions of Arcs.

§ 20.3. Generalized Cauchy's Theorem for a Rectangle.

§ 20.4. Generalized Cauchy's Theorem in a Disk.

§ 20.5. Generalized Cauchy's integral formula.

§ 20.6. Generalized Liouville's theorem.

§ 20.7. Generalized fundamental theorem of algebra.

§ 21. The local properties of #-analytic function.

§ 21.1. Removable Singnlarities. Taylor's Theorem.

**Chapter II.** $\mathbb{R}_c^\#$-Valued Lebesgue Integral.

§1. External $\mathbb{R}_c^\#$-Valued Lebesgue Measure.

§ 2. External $\mathbb{R}_c^\#$-Valued Lebesgue outer measure.

§ 3 Lebesgue Integral of a $\mathbb{R}_c^\#$-valued external function $f(x)$.

§ 4. General Definition and Basic Properties of the external Lebesgue Integral.

**Chapter III.** $*\mathbb{R}_c^\#$-Valued abstract measures and integration.

§ 1. $\sigma^\#$-Algebras

§ 2. $*\mathbb{R}_c^\#$-Valued #-measures

§ 2.1. #-Convergence of functions and the generalized Egoroff theorem.

§ 2.2. Vector-valued #-measures

§ 3. The Lebesgue #-Integral

§ 4. #-Convergence in #-measure.

§ 5. The Extension of #-Measure

§ 5.1. Outer #-measures.

§ 5.2. The Lebesgue and Lebesgue-Stieltjes #-measure on $*\mathbb{R}_c^\#$.

§ 5.3. Product #-measures.

§ 5.4. Lebesgue #-measure and Lebesgue #-integral in $*\mathbb{R}_c^{\#n}$.

§ 5.5. Lebesgue #-integrable functions on $*\mathbb{R}_c^{\#n}$.

**Chapter IV.** $*\mathbb{R}_c^\#$-valued distribution.

§1. $*\mathbb{R}_c^\#$-valued test functions and distributions

§ 2. The non-Archimedean external $*\mathbb{C}_c^\#$-Valued Schwartz distributions.

§ 2.1. Schwartz space $S^#(*\mathbb{R}_c^{\#n})$.

§ 2.2. Schwartz space $S_{\mathrm{fin}}^#(*\mathbb{R}_c^{\#n})$

§ 2.3. $*\mathbb{C}_c^\#$-Valued tempered distributions.
§18.4. $\mathbb{R}_c^\#$-valued functions of several variables.

§18.5. Line Segments in $\mathbb{R}_c^\#$, $n \in \mathbb{N}^\#$.

§18.6. #-Neighborhoods and #-open sets in $\mathbb{R}_c^\#$, $n \in \mathbb{N}^#$. Connected Sets and Regions in $\mathbb{R}_c^\#$.

§18.7. The #-limits and #-continuity $\mathbb{R}_c^\#$-valued functions of $n \in \mathbb{N}^#$ variables.

§18.8. Partial #-Derivatives and the #-Differential.

§19. #-Analytic functions $f: \mathbb{C}^\# \to \mathbb{C}^\#$.

§19.1. $\mathbb{C}_c^\#$-valued #-analytic functions $f: \mathbb{C}_c^\# \to \mathbb{C}_c^\#$.

§19.2. The $\mathbb{C}_c^\#$-valued #-Exponential $\mathrm{Ext}\text{-exp}(z)$.

§19.3. The $\mathbb{C}_c^\#$-valued Trigonometric Functions $\mathrm{Ext}\text{-sin}(z)$, $\mathrm{Ext}\text{-cos}(z)$.

§19.4. The periodicity of the #-exponential $\mathrm{Ext}\text{-exp}(iz)$.

§19.5. The $\mathbb{C}_c^\#$-valued Logarithm.

§ 20. Complex Integration of the $\mathbb{C}_c^\#$-valued function $f(t)$.

§ 20.1. Definition and basic properties of the complex integral.

§ 20.2. Line Integrals as Functions of Arcs.

§ 20.3. Generalized Cauchy's Theorem for a Rectangle.

§ 20.4. Generalized Cauchy's Theorem in a Disk.

§ 20.5. Generalized Cauchy's integral formula.

§ 20.6. Generalized Liouville's theorem.

§ 20.7. Generalized fundamental theorem of algebra.

§ 21. The local properties of #-analytic function.

§ 21.1. Removable Singnlarities. Taylor's Theorem.

**Chapter II. Basic analysis on external non-Archimedean field $\tilde{\mathbb{R}}_c^\#$**

§ 1. The #-limit of a function $f: \tilde{\mathbb{R}}_c^\# \to \tilde{\mathbb{R}}_c^\#$.

§ 2. Monotonic Functions $f: \tilde{\mathbb{R}}_c^\# \to \tilde{\mathbb{R}}_c^\#$.

§ 3. #-Limits Inferior and Superior.

§ 4. The #-continuity of a function $f: \tilde{\mathbb{R}}_c^\# \to \tilde{\mathbb{R}}_c^\#$.

§ 5. Removable #-discontinuities.

§ 6. Composite Functions $f: \tilde{\mathbb{R}}_c^\# \to \tilde{\mathbb{R}}_c^\#$.

§ 7. Bounded and Hyperbounded Functions $f: \tilde{\mathbb{R}}_c^\# \to \tilde{\mathbb{R}}_c^\#$.

§ 8. Generalized Intermediate Value Theorem.

**Chapter III.** $\mathbb{R}_c^\#$-Valued Lebesgue Integral.

§1. External $\mathbb{R}_c^\#$-Valued Lebesgue Measure.

§ 2. External $\mathbb{R}_c^\#$-Valued Lebesgue outer measure.

§ 3. Lebesgue Integral of a $\mathbb{R}_c^\#$-valued external function $f(x)$.

§ 4. General Definition and Basic Properties of the external Lebesgue Integral.

**Chapter IV.** $*\mathbb{R}_c^\#$-Valued abstract measures and integration.

§ 1. $\sigma^\#$-Algebras

§ 2. $*\mathbb{R}_c^\#$-Valued #-measures

§ 2.1. #-Convergence of functions and the generalized Egoroff theorem.

§ 2.2. Vector-valued #-measures

§ 3. The $*\mathbb{R}_c^\#$-Valued Lebesgue #-Integral

§ 4. #-Convergence in $*\mathbb{R}_c^\#$-Valued #-measure.

§ 5. The Extension of $*\mathbb{R}_c^\#$-Valued #-Measure.

§ 5.1. Outer #-measures.
§ 3. The Fourier transform on $S^{#}(*\mathbb{R}_c^{\#n})$, $S_{\text{fin}}^{#}(*\mathbb{R}_c^{\#n})$

**Chapter V. Non-Archimedean Hilbert Spaces over field $\widehat{*\mathbb{C}}_c^\# = \widehat{*\mathbb{R}}_c + i\widehat{\mathbb{R}}_c$**

§ 1. Non-Archimedean Hilbert Spaces over field $\widehat{*\mathbb{C}}_c^\# = \widehat{*\mathbb{R}}_c + i\widehat{\mathbb{R}}_c$. Basics.

§ 2. Non-Archimedean Hilbert Space Basis.

§ 3.1. Weak #-Convergence.

§ 3.2. #-Analytic vectors. Generalized Nelson's #-analytic vector theorem.

§ 4. The generalized Spectral Theorem.

§ 4.1. The #-continuous functional calculus related to bounded in $\ast\mathbb{R}_c^\#$ self-#-adjoint operators.

§ 4.2. The spectral #-measures.

§ 4.2.1. Multiplicity free operators.

§ 4.2.2. Operators of uniform multiplicity.

§ 4.2.3. Disjoint #-measure classes.

§ 4.3. Spectral projections.

§ 4.4. The #-continuous functional calculus related to unbounded in $\ast\mathbb{R}_c^\#$ self-#-adjoint operators.

§ 4.5. Nearstandard $C_\#^*$ algebras generated by spectral projections related to unbounded in $\ast\mathbb{R}_c^\#$ self-#-adjoint operators.

§ 4.6. $\widehat{*\mathbb{C}}_c^\#$-valued quadratic forms.

§ 4.7. #-Convergence of unbounded in $\ast\mathbb{R}_c^\#$ operators.

§ 4.8. Graph #-limits.

§ 4.9. The polar decomposition.

§ 4.10. Generalized Trotter product formula.

§ 5. Tensor products and second quantization.

§ 5.1. Tensor products.

§ 5.2. Non-Archimedean Fock spaces.

§ 5.3. Second quantization of the free Hamiltonian.

**Chapter VI.** Non-Archimedean Banach spaces endroved with $\widehat{\ast\mathbb{R}}_c^\#$-valued norm.

§1. Definitions and examples.

§2. Linear operators, isomorphisms.

**Chapter VII.** Semigroups of operators on a non-Archimedean Banach spaces.

§1. Semigroups on non-Archimedean Banach spaces and their generators.

§2. Hypercontractive semigroups.

**Chapter VIII.** Singular Perturbations of Selfadjoint Operators on a non-Archimedean Hilbert space.

§1. Introduction.

§2. Strong #-Convergence of Operators.

§3. Estimates on a $G$ #-Convergent hyper infinite Sequence.

§ 4. Estimates for singular perturbations.

**Chapter IX.**

§1. Free scalar field.

§2. $Q^#$-space representation of the non-Archimedean Fock space structures.

**Chapter X.** A non-Archemedean Banach algebras and $C_\#^*$-Algebras

§1. A non-Archemedean Banach algebra $B(H^*)$

§1.1. Basic Properties.
§ 5.2. The Lebesgue and Lebesgue-Stieltjes \#-measure on $\mathbb{R}^n$.

§ 5.3. Product \#-measures.

§ 5.4. Lebesgue \#-measure and Lebesgue \#-integral in $\mathbb{R}^n$.

§ 5.5. Lebesgue \#-integrable functions on $\mathbb{R}^n$.

§ 6. \#-Valued measurable functions

§ 7. \#-Valued \#-measures.

§ 8. \#-Convergence of functions and the generalized Egoroff theorem.

§ 9. Vector-valued \#-measures.

§ 10. The \#-valued Lebesgue \#-Integral.

§ 11. \#-Convergence in \#-Valued \#-measure.

**Chapter V. \#-valued distributions.**

§ 1. \#-valued test functions and distributions

§ 2. The non-Archimedean external \#-Valued Schwartz distributions.

§ 2.1. Schwartz space $S^#(*\mathbb{R}^n)$.

§ 2.2. Schwartz space $S_{\text{fin}}^#(*\mathbb{R}^n_{c,\text{fin}})$

§ 2.3. \#-Valued tempered distributions.

§ 3. The Fourier transform on $S^#(*\mathbb{R}^n)$, $S_{\text{fin}}^#(*\mathbb{R}^n)$.

§ 4. Schwartz space $S^#(*\tilde{\mathbb{R}}^n)$.

§ 5. Non-Archimedean tempered distributions $S^#(*\tilde{\mathbb{R}}^n)$.

§ 5.1. The Fourier transform on $S^#(*\tilde{\mathbb{R}}^n)$, $S_{\text{fin}}^#(*\tilde{\mathbb{R}}^n)$.

**Chapter VI. Non-Archimedean Hilbert Spaces over field $\tilde{\mathbb{C}}^n = *\tilde{\mathbb{R}}_c + i*\tilde{\mathbb{R}}_c$**

§ 1. Non-Archimedean Hilbert Spaces over field $\tilde{\mathbb{C}}^n = *\tilde{\mathbb{R}}_c + i*\tilde{\mathbb{R}}_c$. Basics.

§ 2. Non-Archimedean Hilbert Space Basis.

§ 3. Weak \#-Convergence.

§ 4. \#-Analytic vectors. Generalized Nelson's \#-analytic vector theorem.

§ 5. The generalized Spectral Theorem.

§ 5.1. The \#-continuous functional calculus related to bounded in $\tilde{\mathbb{R}}^n$ self-\#-adjoint operators.

§ 5.2. The spectral \#-measures.

§ 5.2.1. Multiplicity free operators.

§ 5.2.2. Operators of uniform multiplicity.

§ 5.2.3. Disjoint \#-measure classes.

§ 5.3. Spectral projections.

§ 5.4. The \#-continuous functional calculus related to unbounded in $\tilde{\mathbb{R}}^n$ self-\#-adjoint operators.

§ 5.5. Nearstandard $C^*_\#$ algebras generated by spectral projections related to unbounded in $\mathbb{R}^n$ self-\#-adjoint operators.

§ 5.6. \#-valued quadratic forms.

§ 5.7. \#-Convergence of unbounded in $\mathbb{R}^n$ operators.

§ 5.8. Graph \#-limits.

§ 5.9. The polar decomposition.

§ 5.10. Generalized Trotter product formula.

§ 5.11. Tensor products and second quantization.

§ 5.12. Tensor products.
§1.2. Types of Operators.

§1.3. Basic Spectral Theory.

§1.4. $l_2^#(G)$ and $B(l_2^#(G))$

§1.5. Topologies on $B(H^#)$.

§2. Non-Archemedean Banach algebras and $C_\#^*$-Algebras.

§2.1. Initial Definitions and #-Continuous Functional Calculus.

§2.2. $*C_c^\#$-valued States.

§2.3. Representations and the generalized Gelfand-Naimark-Segal Construction.

## Introduction

The incompleteness of set theory ZFC leads one to look for natural nonconservative extensions of ZFC in which one can prove statements independent of ZFC which appear to be “true”. One approach has been to add large cardinal axioms.

Or, one can investigate second-order expansions like Kelley-Morse class theory, KM or Tarski-Grothendieck set theory TG or It is a nonconservative extension of ZFC and is obtained from other axiomatic set theories by the inclusion of Tarski's axiom which implies the existence of inaccessible cardinals. See also related set theory with a filter quantifier $ZF(aa)$. In this paper we look at a set theory $NC_\infty^\#$ [18], based on bivalent hyper infinitary logic with restricted Modus Ponens Rule [18]. Nonconservative extension namely IST# of the canonical internal set theory IST was presented in [18].

## §1. Bivalent hyper Infinitary first-order logic $^{2}L_\infty^\#$ with restricted rules of conclusion. Generalized Deduction Theorem.

Hyper infinitary language $L_\infty^\#$ are defined according to the length of hyper infinitary conjunctions/disjunctions as well as quantification it allows. In that way, assuming a supply of $κ < ℵ_0^\# = card(\mathbb{N}^\#)$ variables to be interpreted as ranging over a nonempty domain, one includes in the inductive definition of formulas an infinitary clause for conjunctions and disjunctions, namely, whenever the hypernaturals indexed hyper infinite sequence $\{A_\delta\}_{\delta \in \mathbb{N}^\#}$ of formulas has length less than $κ$, one can form the hyperfinite conjunction/disjunction of them to produce a formula. Analogously, whenever an hypernaturals indexed sequence of variables has length less than $λ$, one can introduce one of the quantifiers $∀$ or $∃$ together with the sequence of variables in front of a formula to produce a new formula. One also stipulates that the length of any well-formed formula is less than $ℓ_0^\#$ itself.

The syntax of bivalent hyper infinitary first-order logics $^{2}L_\infty^\#$ consists of a (ordered) set of

sorts and a set of function and relation symbols, these latter together with the corresponding type, which is a subset with less than $ℓ_0^\# = card(\mathbb{N}^\#)$ many sorts. Therefore, we assume that our signature may contain relation and function symbols on $γ < ℵ_0^\#$ many variables, and we suppose there is a supply of $κ < ℵ_0^\#$ many fresh variables of each sort. Terms and atomic formulas are defined as usual, and general
§ 5.13. Non-Archimedean Fock spaces.

§ 5.14. Second quantization of the free Hamiltonian.

§ 6. #-Convergence of unbounded in*R̃# operators.

§ 6.1. Graph #-limits unbounded in *R̃# operators.

**Chapter VII.** Non-Archimedean Banach spaces endroved with \*\~R\#-valued norm.

§1. Definitions and examples.

§2. Linear operators, isomorphisms.

§3. Generalized #-convergence of #-closed operators

§3.1. The gap between subspaces

§3.2. The gap and the dimension

§3.3. Duality

§3.4. The gap between closed operators

§3.5. Further results on the stability of bounded invertibility

§3.6. Generalized #-convergence

**Chapter VIII.** Semigroups of operators on a non-Archimedean Banach spaces.

§1. Semigroups on non-Archimedean Banach spaces and their generators.

§2. Hypercontractive semigroups.

**Chapter IX.** Singular Perturbations of Selfadjoint Operators on a non-Archimedean Hilbert space.

§1. Introduction.

§2. Strong #-Convergence of Operators.

§3. Estimates on a G #-Convergent hyper infinite Sequence.

§ 4. Estimates for singular perturbations.

**Chapter X.**

§1. Free scalar field.

§2. Q#-space representation of the non-Archimedean Fock space structures.

**Chapter XI.** A non-Archemedean Banach algebras and C#*-Algebras

§1. A non-Archemedean Banach algebra B(H#)

§1.1. Basic Properties.

§1.2. Types of Operators.

§1.3. Basic Spectral Theory.

§1.4. $I_2^#(G)$ and $B(I_2^#(G))$

§1.5. Topologies on B(H#).

§2. Non-Archemedean Banach algebras and C#*-Algebras.

§2.1. Initial Definitions and #-Continious Functional Calculus.

§2.2. \*C#-valued States.

§2.3. Representations and the generalized Gelfand-Naimark-Segal Construction.

Introduction

The incompleteness of set theory ZFC leads one to look for natural nonconservative extensions of ZFC in which one can prove statements independent of ZFC which appear to be “true”. One approach has been to add large cardinal axioms.

Or, one can investigate second-order expansions like Kelley-Morse class theory, KM or Tarski-Grothendieck set theory TG or It is a nonconservative extension of ZFC and is obtained from other axiomatic set theories by the inclusion of Tarski’s
formulas are defined inductively according to the following rules.

If $\phi, \psi, \{\phi_\alpha : \alpha < \gamma\}$ (for each $\gamma < \kappa$) are formulas of $L_\infty^\#$, the following are also formulas:

(i) $\bigwedge_{\alpha<\gamma} \phi_\alpha, \bigwedge_{\alpha\le\gamma} \phi_\alpha,$

(ii) $\bigvee_{\alpha<\gamma} \phi_\alpha, \bigvee_{\alpha\le\gamma} \phi_\alpha,$

(iii) $\phi \rightarrow \psi, \phi \wedge \psi, \phi \vee \psi, \neg \phi$

(iv) $\forall \alpha<\gamma x_\alpha \phi$ (also written $\forall \mathbf{x}_\gamma \phi$ if $\mathbf{x}_\gamma = \{x_\alpha : \alpha < \gamma\}$),

(v) $\exists \alpha<\gamma x_\alpha \phi$ (also written $\exists \mathbf{x}_\gamma \phi$ if $\mathbf{x}_\gamma = \{x_\alpha : \alpha < \gamma\}$),

(vi) the statement $\bigwedge_{\alpha<\gamma} \phi_\alpha$ holds if and only if for any $\alpha$ such that $\alpha < \gamma$

the statement holds $\phi_\alpha$,

(vii) the statement $\bigvee_{\alpha<\gamma} \phi_\alpha$ holds if and only if there exist $\alpha$ such that $\alpha < \gamma$

the statement holds $\phi_\alpha$.

**Definition 1.1.** A valuation of a syntactic system is a function that as signs $\vdash$ (true) to some of its sentences, and/or $\perp$ (false) to some of its sentences. Precisely, a valuation maps a nonempty subset of the set of sentences into the set $\{\vdash, \perp\}$. We call a valuation bivalent iff it maps all the sentences into $\{\vdash, \perp\}$.

**Definition 1.2.** Let $L$ be a propositional language. $L$ is a classical bivalent propositional language iff its admissible valuations are the functions $v$ such that for all sentences $A, B$

of $L$ the following properties hold

(a) $v(A) \in \{\vdash, \perp\}$

(b) $v(\neg A) = \vdash$ iff $v(A) = \perp$

(c) $v(A \wedge B) = \vdash$ iff $v(A) = v(B) = \vdash$.

(d) by definition of the classical implication $A \Rightarrow B$ the following truth table holds

$$v(A) \quad v(B) \quad v(A \Rightarrow B)$$

(1) $\vdash$ $\vdash$ $\vdash$

(2) $\vdash$ $\perp$ $\perp$

(3) $\perp$ $\vdash$ $\vdash$

(4) $\perp$ $\perp$ $\vdash$

Truth table 1.

(e) $v^*(A) \in \{\vdash, \perp\}$

(f) $v^*(\neg A) = \vdash$ iff $v^*(A) = \perp$

(g) $v^*(A \wedge B) = \vdash$ iff $v^*(A) = v^*(B) = \vdash$.

(h) by definition of the nonclassical implication $A \Rightarrow B$ the following truth table holds

$$v^*(A) \quad v^*(B) \quad v^*(A \Rightarrow B)$$

(1) $\vdash$ $\vdash$ $\vdash$

(2) $\vdash$ $\perp$ $\vdash$

(3) $\perp$ $\vdash$ $\vdash$

(4) $\perp$ $\perp$ $\vdash$

Truth table 2.

**Remark 1.1.** Note that in the case (2) of the truth table 2

$$\vdash = v^*(A \Rightarrow B) \neq v(A \Rightarrow B) = \perp.$$

In this case we call implication $A \Rightarrow B$ a weak implication and abbreviate
axiom which implies the existence of inaccessible cardinals. See also related set
theory with a filter quantifier ZF(aa). In this paper we look at a set theory $N\mathcal{C}_{\infty}^{\#}$ [18],
based on bivalent hyper infinitary logic with restricted Modus Ponens Rule [18].
Nonconservative extension namely IST# of the canonical internal set theory IST was
presented in [18].

§1. Bivalent hyper Infinitary first-order logic $^2L_{\infty}^{\#}$ with
restricted rules of conclusion. Generalized Deduction
Theorem.

Hyper infinitary language $L_{\infty}^{\#}$ are defined according to the length of hyper infinitary
conjunctions/disjunctions as well as quantification it allows. In that way, assuming a
supply of $\kappa < \aleph_0 = \text{card}(\mathbb{N}^{\#})$ variables to be interpreted as ranging over a nonempty
domain, one includes in the inductive definition of formulas an infinitary clause for
conjunctions and disjunctions, namely, whenever the hypernaturals indexed hyper
infinite sequence $\{A_\delta\}_{\delta\in\mathbb{N}^\#}$ of formulas has length less than $\kappa$, one can form the
hyperfinite conjunction/disjunction of them to produce a formula. Analogously, whenever
an hypernaturals indexed sequence of variables has length less than $\lambda$, one can
introduce one of the quantifiers $\forall$ or $\exists$ together with the sequence of variables in front of
a formula to produce a new formula. One also stipulates that the length of any
well-formed formula is less than $\aleph_0$ itself.

The syntax of bivalent hyper infinitary first-order logics $^2L_{\infty}^{\#}$ consists of a (ordered) set of sorts and a set of function and relation symbols, these latter together with the

corresponding type, which is a subset with less than $\aleph_0 = \text{card}(\mathbb{N}^\#)$ many sorts.
Therefore, we assume that our signature may contain relation and function symbols on
$\gamma < \aleph_0$ many variables, and we suppose there is a supply of $\kappa < \aleph_0$ many fresh
variables of each sort. Terms and atomic formulas are defined as usual, and general
formulas are defined inductively according to the following rules.

If $\phi, \psi, \{\phi_\alpha : \alpha < \gamma\}$ (for each $\gamma < \kappa$) are formulas of $L_{\infty}^{\#}$, the following are also
formulas:

(i) $\bigwedge_{\alpha<\gamma} \phi_{\alpha}, \bigwedge_{\alpha\leq\gamma} \phi_{\alpha},$

(ii) $\bigvee_{\alpha<\gamma} \phi_{\alpha}, \bigvee_{\alpha\leq\gamma} \phi_{\alpha},$

(iii) $\phi \rightarrow \psi, \phi \wedge \psi, \phi \vee \psi, \neg \phi$

(iv) $\forall_{\alpha<\gamma} x_{\alpha} \phi$ (also written $\forall \mathbf{x}_{\gamma} \phi$ if $\mathbf{x}_{\gamma} = \{x_{\alpha} : \alpha < \gamma\}$),

(v) $\exists_{\alpha<\gamma} x_{\alpha} \phi$ (also written $\exists \mathbf{x}_{\gamma} \phi$ if $\mathbf{x}_{\gamma} = \{x_{\alpha} : \alpha < \gamma\}$),

(vi) the statement $\bigwedge_{\alpha<\gamma} \phi_{\alpha}$ holds if and only if for any $\alpha$ such that $\alpha < \gamma$

the statement holds $\phi_a$,

(vii) the statement $\bigvee_{\alpha<\gamma} \phi_{\alpha}$ holds if and only if there exist $\alpha$ such that $\alpha < \gamma$

the statement holds $\phi_a$.

**Definition 1.1.** A valuation of a syntactic system is a function that as signs $\uparrow$ (true)
to some of its sentences, and/or $\perp$ (false) to some of its sentences. Precisely, a
valuation maps a nonempty subset of the set of sentences into the set $\{\uparrow, \perp\}$.
We call a valuation bivalent iff it maps all the sentences into $\{\uparrow, \perp\}$.
$$A \Rightarrow_B^w C \qquad (1.1)$$

We call a statement (1.1) as a weak statement and often abbreviate $v(A \Rightarrow B) = \tau_w$ instead (1).

**Definition 1.3.** [7-8]. A is a valid (logically valid) sentence (in symbols, $\vdash A$) in $L$ iff every admissible valuation of $L$ satisfies A.

The axioms of hyper infinitary first-order logic $^2L_{\infty}^{\#}$ consist of the following schemata:

**I. Logical axiom**

A 1. $A \to [B \to A]$

A 2. $[A \to [B \to C]] \to [[A \to B] \to [A \to C]]$

A 3. $[\neg B \to \neg A] \to [A \to B]$

A 4. $[\bigwedge_{i<\alpha} [A \to A_i]] \to [A \to \bigwedge_{i<\alpha} A_i], \alpha \in \mathbb{N}^{\#}$

A 5. $[\bigwedge_{i<\alpha} A_i] \to A_j, \alpha \in \mathbb{N}^{\#}$

A 6. $[\forall \mathbf{x}[A \to B] \to [A \to \forall \mathbf{x}B]]$ provided no variable in $\mathbf{x}$ occurs free in A;

A 7. $\forall \mathbf{x}A(\mathbf{x}) \to S_f(A)$,

where $S_f(A)$ is a substitution based on a function $f$ from $\mathbf{x}$ to the terms of the language; in particular:

A 7'. $\forall x_i [A(x_i)] \Rightarrow A(t)$ is a wff of $^2L_{\infty}^{\#}$ and $t$ is a term of $^2L_{\infty}^{\#}$ that is free for $x_i$ in $A(x_i)$. Note here that $t$ may be identical with $x_i$; so that all wffs $\forall x_i A \Rightarrow A$ are axioms by virtue of axiom (7), see [8].

A 8. Gen (Generalization).

$\forall x_i B$ follows from B.

**II. Restricted rules of conclusion.**

Let $\mathcal{F}_{\text{wff}}$ be a set of the all closed wffs of $L_{\infty}^{\#}$.

**R1.** RMP (Restricted Modus Ponens).

There exist subsets $\Delta_1, \Delta_2 \subset \mathcal{F}_{\text{wff}}$ such that the following rules are satisfied.

From A and $A \Rightarrow B$, conclude $B$ iff $A \notin \Delta_1$ and $(A \Rightarrow B) \notin \Delta_2$, where $\Delta_1, \Delta_2 \subset \mathcal{F}_{\text{wff}}$.

In particular for any $A, B \in \mathcal{F}_{\text{wff}}: A \Rightarrow_B B \in \Delta_2$.

If $A \notin \Delta_1$ and $(A \Rightarrow B) \notin \Delta_2$ we also abbreviate by $A, A \Rightarrow_B B \vdash_{RMP} B$.

**R2.** RMT (Restricted Modus Tollens)

There exist subsets $\Delta'_1, \Delta'_2 \subset \mathcal{F}_{\text{wff}}$ such that the following rules are satisfied.

$P \Rightarrow Q, \neg Q \vdash_{RMT} \neg P$ iff $P \notin \Delta'_1$ and $(P \Rightarrow Q) \notin \Delta'_2$, where $\Delta'_1, \Delta'_2 \subset \mathcal{F}_{\text{wff}}$.

**Remark 1.2.** Note that RMP and RMT easily prevent any paradoxes of naive Cantor set theory (NC), see [1],[9].

**III. Additional derived rule of conclusion.**

**Particularization rule (RPR)**

Remind that canonical unrestricted particularization rule (UPR) reads

**UPR:** If $t$ is free for $x$ in $B(x)$, then $\forall x[B(x)] \vdash B(t)$, see [8].

**Proof.** From $\forall x[B(x)]$ and the instance $\forall x[B(x)] \Rightarrow B(t)$ of axiom (A7), we obtain $B(t)$ by unrestricted modus ponens rule. Since $x$ is free for $x$ in $B(x)$, a special case of unrestricted particularization rule is: $\forall xB \vdash B$.

**Definition 1.4.** Any formal theory $L$ with a hyper infinitary language $L_{\infty}^{\#}$ is defined when the following conditions are satisfied:

1. A hyper infinite set of symbols is given as the symbols of $L$. A finite or hyperfinite sequence of symbols of $L$ is called an expression of $L$.
**Definition 1.2.** Let $L$ be a propositional language. $L$ is a classical bivalent propositional language iff its admissible valuations are the functions $v$ such that for all sentences $A, B$

of $L$ the following properties hold

(a) $v(A) \in \{\top, \bot\}$

(b) $v(\neg A) = \top$ iff $v(A) = \bot$

(c) $v(A \land B) = \top$ iff $v(A) = v(B) = \top$.

(d) by definition of the classical implication $A \Rightarrow B$ the following truth table holds

$$v(A) \quad v(B) \quad v(A \Rightarrow B)$$

(1) $\top$ $\top$ $\top$

(2) $\top$ $\bot$ $\bot$

(3) $\bot$ $\top$ $\top$

(4) $\bot$ $\bot$ $\top$

Truth table 1.

(e) $v^*(A) \in \{\top, \bot\}$

(f) $v^*(\neg A) = \top$ iff $v^*(A) = \bot$

(g) $v^*(A \land B) = \top$ iff $v^*(A) = v^*(B) = \top$.

(h) by definition of the nonclassical implication $A \Rightarrow B$ the following truth table holds

$$v^*(A) \quad v^*(B) \quad v^*(A \Rightarrow B)$$

(1) $\top$ $\top$ $\top$

(2) $\top$ $\bot$ $\top$

(3) $\bot$ $\top$ $\top$

(4) $\bot$ $\bot$ $\top$

Truth table 2.

**Remark 1.1.** Note that in the case (2) of the truth table 2

$$\top = v^*(A \Rightarrow B) \neq v(A \Rightarrow B) = \bot.$$

In this case we call implication $A \Rightarrow B$ a weak implication and abbreviate

$$A \Rightarrow_B w \quad (1.1)$$

We call a statement (1.1) as a weak statement and often abbreviate $v(A \Rightarrow B) = \top_w$ instead (1).

**Definition 1.3.** [7-8]. $A$ is a valid (logically valid) sentence (in symbols, $\models A$) in $L$ iff every admissible valuation of $L$ satisfies $A$.

The axioms of hyper infinitary first-order logic $^{2}L_{\infty}^{\#}$ consist of the following schemata:

**I. Logical axiom**

A 1. $A \rightarrow [B \rightarrow A]$

A 2. $[A \rightarrow [B \rightarrow C]] \rightarrow [[A \rightarrow B] \rightarrow [A \rightarrow C]]$

A 3. $[\neg B \rightarrow \neg A] \rightarrow [A \rightarrow B]$

A 4. $[\bigwedge_{i<\alpha} [A \rightarrow A_i]] \rightarrow [A \rightarrow \bigwedge_{i<\alpha} A_i], \alpha \in \mathbb{N}^{\#}$

A 5. $[\bigwedge_{i<\alpha} A_i] \rightarrow A_j, \alpha \in \mathbb{N}^{\#}$

A 6. $[\forall x[A \rightarrow B] \rightarrow [A \rightarrow \forall xB]]$ provided no variable in $x$ occurs free in $A$;

A 7. $\forall x A(x) \rightarrow S_f(A)$,

where $S_f(A)$ is a substitution based on a function $f$ from $x$ to the terms of the
2. There is a subset of the set of expressions of $L$ called the set of well formed formulas (wffs) of $L$. There is usually an effective procedure to determine whether a given expression is a wff.

3. There is a set of wfs called the set of axioms of $L$. Most often, one can effectively decide whether a given wff is an axiom; in such a case, $L$ is called an axiomatic theory.

4. There is a *finite* set $R_1, \dots, R_n$, of relations among wffs, called rules of conclusion. For each $R_i$, there is a unique positive integer $j$ such that, for every set of $j$ wfs and each wff $B$, one can effectively decide whether the given $j$ wffs are in the relation $R_i$ to $B$, and, if so, $B$ is said to follow from or to be a direct consequence of the given wffs by virtue of $R_j$.

**Definition 1.5.** A proof in $L$ is a finite or *hyperfinite* sequence $B_1, \dots, B_k, k \in \mathbb{N}^$
of wffs such that for each $i$, either $B_i$ is an axiom of $L$ or $B_i$ is a direct consequence of some of the preceding wffs in the sequence by virtue of one of the rules of inference of $L$.

**Definition 1.6.** A theorem of $L$ is a wff $B$ of $Y$ such that $B$ is the last wff of some proof in $L$. Such a proof is called a proof of $B$ in $L$.

**Definition 1.7.** A wff $E$ is said to be a consequence in $L$ of a set of $\Gamma$ of wffs if and only if there is a finite or *hyperfinite* sequence $B_1, \dots, B_k, k \in \mathbb{N}^\#$ of wffs such that $E$ is $B_k$ and, for each $i$, either $B_i$ is an axiom or $B_i$ is in $\Gamma$, or $B_i$ is a direct consequence by some rule of inference of some of the preceding wffs in the sequence. Such a sequence is called a proof (or deduction) $E$ from $\Gamma$. The members of $\Gamma$ are called the hypotheses or premisses of the proof.
We use $\Gamma \vdash E$ as an abbreviation for $E$ as a consequence of $\Gamma$.
In order to avoid confusion when dealing with more than one theory, we write $\Gamma \vdash_L E$, adding the subscript $L$ to indicate the theory in question.

If $\Gamma$ is a finite or *hyperfinite* set $\{H_i\}_{1\le i\le m}, m \in \mathbb{N}^\#$ we write $H_1, \dots, H_m \vdash E$ instead of $\{H_i\}_{1\le i\le m} \vdash E$.

**Lemma 1.1.** [18]. $\vdash B \Rightarrow B$ for all wffs $B$.

**Theorem 1.1.** (Generalized Deduction Theorem 1). If $\Gamma$ is a set of wffs and $B$ and $E$ are wffs, and $\Gamma, B \vdash E$, then $\Gamma \vdash B \Rightarrow_s E$. In particular, if $B \vdash E$ then $\Gamma \vdash B \Rightarrow E$.

**Proof.** Let $E_1, \dots, E_n, n \in \mathbb{N}^\#$ be a proof of $E$ form $\Gamma \cup \{B\}$, where $E_n$ is $E$.
Let us prove, by *hyperfinite induction* on $j$, that $\Gamma \vdash B \Rightarrow_s E_j$ for $1 \le j \le n$.
First of all, $E_1$ must be either in $\Gamma$ or an axiom of $L$ or $B$ itself.

By axiom schema A1, $E_1 \Rightarrow_s (B \Rightarrow_s E_1)$ is an axiom. Hence, in the first two cases, by MP, $\Gamma \vdash B \Rightarrow_s E_1$. For the third case, when $E_1$ is $B$, we have $\vdash B \Rightarrow_s E_1$ by Lemma 1, and, therefore, $\Gamma \vdash B \Rightarrow_s E_1$. This takes care of the case $j = 1$.

Assume now that: $\vdash B \Rightarrow_s E_k$ for all $k < j, j \in \mathbb{N}^\#$. Either $E_j$ is an axiom, or $E_j$ is in $\Gamma$, or $E_j$ is $B$, or $E_j$ follows by modus ponens from some $E_l$ and $E_m$ where $l < j$, $m < j$, and $E_m$ has the form $E_l \Rightarrow_s E_j$. In the first three cases, $\Gamma \vdash B \Rightarrow_s E_j$ as in the case $j = 1$ above. In the last case, we have, by inductive hypothesis, $\Gamma \vdash B \Rightarrow_s E_l$ and $\Gamma \vdash B \Rightarrow_s (E_l \Rightarrow_s E_j)$. But, by axiom schema (A2), $\vdash B \Rightarrow_s (E_l \Rightarrow_s E_j) \Rightarrow_s ((B \Rightarrow_s E_l) \Rightarrow_s (B \Rightarrow_s E_j))$

Hence, by MP, $\Gamma \vdash (B \Rightarrow_s E_l) \Rightarrow_s (B \Rightarrow_s E_j)$ and, again by MP, $\Gamma \vdash B \Rightarrow_s E_j$.
Thus, the proof by hyperfinite induction is complete.
language; in particular:

A $7'$. $\forall x_i[A(x_i)] \Rightarrow A(t)$ is a wff of $^2L_{\infty}^{\#}$ and $t$ is a term of $^2L_{\infty}^{\#}$ that is free for $x_i$ in $A(x_i)$. Note here that $t$ may be identical with $x_i$; so that all wffs $\forall x_i A \Rightarrow A$ are axioms by virtue of axiom (7), see [8].

A 8.Gen (Generalization).

$\forall x_i B$ follows from $B$.

## II. Restricted rules of conclusion.

Let $\mathcal{F}_{wff}$ be a set of the all closed wffs of $L_{\infty}^{\#}$.

### R1. RMP (Restricted Modus Ponens).

There exist subsets $\Delta_1, \Delta_2 \subset \mathcal{F}_{wff}$ such that the following rules are satisfied.

From $A$ and $A \Rightarrow B$, conclude $B$ iff $A \notin \Delta_1$ and $(A \Rightarrow B) \notin \Delta_2$, where $\Delta_1, \Delta_2 \subset \mathcal{F}_{wff}$.

In particular for any $A, B \in \mathcal{F}_{wff}: A \Rightarrow_w B \in \Delta_2$.

If $A \notin \Delta_1$ and $(A \Rightarrow B) \notin \Delta_2$ we also abbreviate by $A, A \Rightarrow_S B \vdash_{RMP} B$.

### R2. RMT (Restricted Modus Tollens)

There exist subsets $\Delta'_1, \Delta'_2 \subset \mathcal{F}_{wff}$ such that the following rules are satisfied.

$P \Rightarrow Q, \neg Q \vdash_{RMT} \neg P$ iff $P \notin \Delta'_1$ and $(P \Rightarrow Q) \notin \Delta'_2$, where $\Delta'_1, \Delta'_2 \subset \mathcal{F}_{wff}$.

**Remark 1.2.** Note that RMP and RMT easily prevent any paradoxes of naive Cantor set theory (NC), see [1],[9].

## III. Additional derived rule of conclusion.

### Particularization rule (RPR)

Remind that canonical unrestricted particularization rule (UPR) reads

**UPR:** If $t$ is free for $x$ in $B(x)$, then $\forall x[B(x)] \vdash B(t)$, see [8].

**Proof.** From $\forall x[B(x)]$ and the instance $\forall x[B(x)] \Rightarrow B(t)$ of axiom (A7), we obtain $B(t)$ by unrestricted modus ponens rule. Since $x$ is free for $x$ in $B(x)$, a special case of unrestricted particularization rule is: $\forall x B \vdash B$.

**Definition 1.4.** Any formal theory $L$ with a hyper infinitary language $L_{\infty}^{\#}$ is defined when the following conditions are satisfied:

1. A hyper infinite set of symbols is given as the symbols of $L$. A finite or hyperfinite sequence of symbols of $L$ is called an expression of $L$.

2. There is a subset of the set of expressions of $L$ called the set of *well formed formulas* (wffs) of $L$. There is usually an effective procedure to determine whether a given expression is a wff.

3. There is a set of wfs called the set of axioms of $L$. Most often, one can effectively decide whether a given wff is an axiom; in such a case, $L$ is called an axiomatic theory.

4. There is a *finite* set $R_1, \dots, R_n$, of relations among wffs, called rules of conclusion. For each $R_i$, there is a unique positive integer $j$ such that, for every set of $j$ wfs and each wff $B$, one can effectively decide whether the given $j$ wffs are in the relation $R_i$ to $B$, and, if so, $B$ is said to follow from or to be a direct consequence of the given wffs by virtue of $R_j$.

**Definition 1.5.** A proof in $L$ is a finite or *hyperfinite* sequence $B_1, \dots, B_k, k \in \mathbb{N}^$
of wffs such that for each $i$, either $B_i$ is an axiom of $L$ or $B_i$ is a direct consequence of some of the preceding wffs in the sequence by virtue of one of the rules of inference of $L$.

**Definition 1.6.** A theorem of $L$ is a wff $B$ of $Y$ such that $B$ is the last wff of some
The case $j = n \in \mathbb{N}^#$ is the desired result. Notice that, given a deduction of $E$ from $\Gamma$ and $B$, the proof just given enables us to construct a deduction of $B \Rightarrow_s E$ from $\Gamma$. Also note that axiom schema A3 was not used in proving the generalized deduction theorem.

**Remark 1.3.** For the remainder of the chapter, unless something is said to the contrary,
we shall omit the subscript $L$ in $\vdash_L$. In addition, we shall use $\Gamma, B \vdash E$ to stand for $\Gamma \cup \{B\} \vdash E$. In general, we let $\Gamma, B_1, \dots, B_n \vdash E$ stand for $\Gamma \cup \{B_i\}_{1 \le i \le n} \vdash E$.

**Remark 1.4.** We shall use the terminology proof, theorem, consequence, axiomatic, etc. and notation $\Gamma \vdash E$ introduced above.

**Proposition 1.1.** Every wff $B$ of $K$ that is an instance of a tautology is a theorem of $K$, and it may be proved using only axioms A1-A3 and MP.

**Proposition 1.2.** If $E$ does not depend upon $B$ in a deduction showing that $\Gamma, B \vdash E$, then $\Gamma \vdash E$.

**Proof.** Let $D_1, \dots, D_n$ be a deduction of $E$ from $\Gamma$ and $B$, in which $E$ does not depend upon $B$. In this deduction, $D_n$ is $E$. As an inductive hypothesis, let us assume that the proposition is true for all deductions of length less than $n \in \mathbb{N}^#$. If $E$ belongs to $\Gamma$ or is an axiom, then $\Gamma \vdash E$. If $E$ is a direct consequence of one or two preceding wffs by Gen or MP, then, since $E$ does not depend upon $B$, neither do these preceding wfs. By the inductive hypothesis, these preceding wfs are deducible from $\Gamma$ alone. Consequently, so is $E$.

**Theorem 1.2.** (Generalized Deduction Theorem 2). Assume that, in some deduction showing that $\Gamma, B \vdash E$, no application of Gen to a wff that depends upon $B$ has as its quantified variable a free variable of $B$. Then $\Gamma \vdash B \Rightarrow_s E$.

**Proof.** Let $D_1, \dots, D_n$ be a deduction of $E$ from $\Gamma$ and $B$ satisfying the assumption of this theorem. In this deduction, $D_n$ is $E$. Let us show by hyperfinite induction that $\Gamma \vdash B \Rightarrow_s D_i$ for each $i \le n \in \mathbb{N}^#$. If $D_i$ is an axiom or belongs to $\Gamma$, then $\Gamma \vdash B \Rightarrow_s D_i$, since $D_i \Rightarrow_s (B \Rightarrow_s D_i)$ is an axiom. If $D_i$ is $B$, then $\Gamma \vdash B \Rightarrow_s D_i$, since, by Proposition 1, $\vdash B \Rightarrow_s B$. If there exist $j$ and $k$ less than $i$ such that $D_k$ is $\vdash D_j \Rightarrow_s D_i$, then, by inductive hypothesis, $\Gamma \vdash B \Rightarrow_s D_j$ and $\Gamma \vdash B \Rightarrow_s (D_j \Rightarrow_s D_i)$. Now, by axiom A2, $\vdash B \Rightarrow_s (D_j \Rightarrow_s D_i) \Rightarrow_s ((B \Rightarrow_s D_j) \Rightarrow_s (B \Rightarrow_s D_i))$. Hence, by MP twice, $\Gamma \vdash B \Rightarrow_s D_i$. Finally, suppose that there is some $j < i$ such that $D_i$ is $\forall x_k D_j$. By the inductive hypothesis, $\Gamma \vdash B \Rightarrow_s D_j$, and, by the hypothesis of the theorem, either $D_j$ does not depend upon $B$ or $x_k$ is not a free variable of $B$. If $D_j$ does not depend upon $B$, then, by Proposition 2, $\Gamma \vdash D_j$ and, consequently, by Gen, $\Gamma \vdash \forall x_k D_j$. Thus, $\Gamma \vdash D_i$. Now, by axiom A1, $\vdash D_i \Rightarrow_s (B \Rightarrow_s D_i)$. So, $\Gamma \vdash B \Rightarrow_s D_i$ by MP. If, on the other hand, $x_k$ is not a free variable of $B$, then, by axiom A5, $\vdash \forall x_k (B \Rightarrow_s D_j) \Rightarrow_s (B \Rightarrow_s \forall x_k D_j)$. Since $\Gamma \vdash B \Rightarrow_s D_j$, we have, by Gen, $\Gamma \vdash \forall x_k (B \Rightarrow_s D_j)$, and so, by MP, $\Gamma \vdash B \Rightarrow_s \forall x_k D_j$ that is, $\Gamma \vdash B \Rightarrow_s D_i$. This completes the induction, and our proposition is just the special case $i = n$.

# §2. Set theory $\text{NC}^#_{\infty\#}$.

Set theory $\text{NC}^#_{\infty\#}$ is formulated as a system of axioms based on bivalent hyper infinitary logic $^2L^#_{\infty\#}$ with restricted modus ponens rule [1],[18]. The language of set
proof in L. Such a proof is called a proof of B in L.

**Definition 1.7.** A wff *E* is said to be a consequence in *L* of a set of Γ of wffs if and only if there is a finite or *hyperfinite* sequence *B*₁, ..., *B*ₖ, *k* ∈ N# of wffs such that *E* is *B*ₖ and, for each *i*, either *B*ᵢ is an axiom or *B*ᵢ is in Γ, or *B*ᵢ is a direct consequence by some rule of inference of some of the preceding wffs in the sequence. Such a sequence is called a proof (or deduction) *E* from Γ. The members of Γ are called the hypotheses or premisses of the proof.

We use Γ ⊢ *E* as an abbreviation for *E* as a consequence of Γ.

In order to avoid confusion when dealing with more than one theory, we write
Γ ⊢*L* *E*, adding the subscript *L* to indicate the theory in question.

If Γ is a finite or *hyperfinite* set {Hᵢ}₁≤i≤m, *m* ∈ N# we write H₁, ..., Hm ⊢ *E* instead
of {Hᵢ}₁≤i≤m ⊢ *E*.

**Lemma 1.1.** [18]. ⊢ *B* ⇒ *B* for all wffs *B*.

**Theorem 1.1.** (Generalized Deduction Theorem 1). If Γ is a set of wffs and B and E are wffs, and Γ, B ⊢ E, then Γ ⊢ B ⇒ₐ E. In particular, if B ⊢ E then Γ ⊢ B ⇒ E.

**Proof.** Let $E_1, \dots, E_n, n \in \mathbb{N}^#$ be a proof of E form $\Gamma \cup \{B\}$, where $E_n$ is E.

Let us prove, by *hyperfinite induction* on *j*, that $\Gamma \vdash B \Rightarrow_s E_j$ for $1 \le j \le n$.

First of all, $E_1$ must be either in $\Gamma$ or an axiom of $L$ or $B$ itself.

By axiom schema A1, $E_1 \Rightarrow_s (B \Rightarrow_s E_1)$ is an axiom. Hence, in the first two cases, by MP, $\Gamma \vdash B \Rightarrow_s E_1$. For the third case, when $E_1$ is $B$, we have $\vdash B \Rightarrow_s E_1$ by Lemma 1, and, therefore, $\Gamma \vdash B \Rightarrow_s E_1$. This takes care of the case $j = 1$.

Assume now that: $\vdash B \Rightarrow_s E_k$ for all $k < j, j \in \mathbb{N}^#$. Either $E_j$ is an axiom, or $E_j$ is in $\Gamma$, or $E_j$ is $B$, or $E_j$ follows by modus ponens from some $E_l$ and $E_m$ where $l < j$, $m < j$, and $E_m$ has the form $E_l \Rightarrow_s E_j$. In the first three cases, $\Gamma \vdash B \Rightarrow_s E_j$ as in the case $j = 1$ above. In the last case, we have, by inductive hypothesis, $\Gamma \vdash B \Rightarrow_s E_l$ and $\Gamma \vdash B \Rightarrow_s (E_l \Rightarrow_s E_j)$. But, by axiom schema (A2),

$$
\vdash B \Rightarrow_s (E_l \Rightarrow_s E_j) \Rightarrow_s ((B \Rightarrow_s E_l) \Rightarrow_s (B \Rightarrow_s E_j))
$$

Hence, by MP, $\Gamma \vdash (B \Rightarrow_s E_l) \Rightarrow_s (B \Rightarrow_s E_j)$ and, again by MP, $\Gamma \vdash B \Rightarrow_s E_j$.

Thus, the proof by hyperfinite induction is complete.

The case $j = n \in \mathbb{N}^#$ is the desired result. Notice that, given a deduction of *E* from
Γ and *B*, the proof just given enables us to construct a deduction of *B* ⇒*s* *E*
from Γ. Also note that axiom schema A3 was not used in proving the
generalized deduction theorem.

**Remark 1.3.** For the remainder of the chapter, unless something is said to the contrary,
we shall omit the subscript *L* in ⊢*L* . In addition, we shall use Γ, *B* ⊢ *E* to stand for
Γ ∪ {B} ⊢ *E*. In general, we let Γ, *B*₁, ..., *B*ₙ ⊢ *E* stand for Γ ∪ {*B*ᵢ}₁≤i≤n ⊢ *E*.

we shall omit the subscript *L* in ⊢*L*. In addition, we shall use Γ, *B* ⊢ *E* to stand for
Γ ∪ {B} ⊢ *E*. In general, we let Γ, *B*₁, ..., *B*ₙ ⊢ *E* stand for Γ ∪ {Bᵢ}₁≤i≤n ⊢ *E*.

**Remark 1.4.** We shall use the terminology proof, theorem, consequence, axiomatic, etc. and notation Γ ⊢ *E* introduced above.

**Proposition 1.1.** Every wff *B* of *K* that is an instance of a tautology is a theorem of *K*, and it may be proved using only axioms A1-A3 and MP.

**Proposition 1.2.** If *E* does not depend upon *B* in a deduction showing that
Γ, B ⊢ E, then Γ ⊢ E.

**Proof.** Let $D_1, \dots, D_n$ be a deduction of $E$ from $\Gamma$ and $B$, in which $E$ does not depend upon $B$. In this deduction, $D_n$ is $E$. As an inductive hypothesis, let
theory $\mathbf{NC}_{\infty}^{\#}$ is a first-order hyper infinitary language $L_{\infty}^{\#}$ with equality $=$, which includes a binary symbol $\in$. We write $x \neq y$ for $\neg (x = y)$ and $x \notin y$ for $\neg (x \in y)$. Individual variables $x, y, z, \dots$, and $x^{\text{CL}}, y^{\text{CL}}, z^{\text{CL}}, \dots$ of $L_{\infty}^{\#}$ will be understood as ranging over classical sets. The unique existential quantifier $\exists!$ is introduced by writing, for any formula $\varphi(x), \exists!x\varphi(x)$ as an abbreviation of the formula $\exists x[\varphi(x) \text{ & } \forall y(\varphi(y) \Rightarrow_s x = y)]$. The language $L_{\infty}^{\#}$ will also contains the formation of terms of the form $\{x|\varphi(x)\}^{\text{NCL}}$, for any formula $\varphi(x)$ containing the free variable $x$.

Such terms are called non-classical sets; we shall use upper case letters $A, B, \dots$, and $A^{\text{NCL}}, B^{\text{NCL}}, \dots$ for such sets. For each non-classical set $A = \{x|\varphi(x)\}^{\text{NCL}}$ the formulas

$\forall x[x \in A \Leftrightarrow_{s,w} \varphi(x)]$ and $\forall x[x \in A \Leftrightarrow_{s,w} \varphi(x, A)]$ is called the defining axioms for the non-classical set $A$.

**Remark 2.1.** Remind that in logic $^2L_{\infty}^{\#}$ with restricted modus ponens rule the statement $\alpha \wedge (\alpha \Rightarrow \beta)$ does not always guarantee that

$$ \alpha, \alpha \Rightarrow \beta \vdash_{\text{RMP}} \beta \qquad (2.1) $$

since for some $\alpha$ and $\beta$ possible

$$ \alpha, \alpha \Rightarrow \beta \nvDash_{\text{RMP}} \beta \qquad (2.2) $$

even if the statement $\alpha \wedge (\alpha \Rightarrow \beta)$ holds.

**Abbreviation 2.1.** We shall write for the sake of brevity instead (2.1) by

$$ \alpha \Rightarrow_s \beta \qquad (2.3) $$

and we shall write instead (2.2) by

$$ \alpha \Rightarrow_w \beta. \qquad (2.4) $$

**Remark 2.2.** Let $A$ be an nonclassical set. Note that in set theory $\mathbf{NC}_{\infty}^{\#}$ the following true formula

$$ \exists A \forall x [x \in A \Leftrightarrow \varphi(x, A)] \qquad (2.5) $$

does not always guarantee that

$$ x \in A, x \in A \Rightarrow \varphi(x, A) \vdash_{\text{RMP}} \varphi(x, A) \qquad (2.6) $$

even if $x \in A$ holds and (or)

$$ \varphi(x, A), \varphi(x, A) \Rightarrow x \in A \vdash_{\text{RMP}} x \in A; \qquad (2.7) $$

even $\varphi(x, A)$ holds, since for nonclassical set $A$ for some $y$ possible

$$ y \in A, y \in A \Rightarrow \varphi(y, A) \nvDash_{\text{RMP}} \varphi(y, A) \qquad (2.8) $$

and (or)

$$ \varphi(y, A), \varphi(y, A) \Rightarrow y \in A \nvDash_{\text{RMP}} y \in A. \qquad (2.9) $$

**Remark 2.3.** Note that in this paper the formulas

$$ \exists a \forall x [x \in a \Leftrightarrow \varphi(x) \wedge x \in u] \qquad (2.10) $$

and more general formulas

$$ \exists a \forall x [x \in a \Leftrightarrow \varphi(x, a) \wedge x \in u] \qquad (2.11) $$

is considered as the defining axioms for the classical set $a$.

**Remark 2.4.** Let $a$ be a classical set. Note that in $\mathbf{NC}_{\infty}^{\#}$: (i) the following true formula
us assume that the proposition is true for all deductions of length less than $n \in \mathbb{N}^#$
If E belongs to $\Gamma$ or is an axiom, then $\Gamma \vdash E$. If E is a direct consequence of
one or two preceding wffs by Gen or MP, then, since E does not depend
upon B, neither do these preceding wfs. By the inductive hypothesis, these
preceding wfs are deducible from $\Gamma$ alone. Consequently, so is E.

**Theorem 1.2.** (Generalized Deduction Theorem 2). Assume that, in some deduction
showing that $\Gamma, B \vdash E$, no application of Gen to a wff that depends upon B has as
its quantified variable a free variable of B. Then $\Gamma \vdash B \Rightarrow_s E$.

**Proof.** Let $D_1, \dots, D_n$ be a deduction of E from $\Gamma$ and B satisfying the assumption of this theorem. In this deduction, $D_n$ is E. Let us show by hyperfinite induction that $\Gamma \vdash B \Rightarrow_s D_i$ for each $i \le n \in \mathbb{N}^#$. If $D_i$ is an axiom or belongs to $\Gamma$, then $\Gamma \vdash B \Rightarrow_s D_i$, since $D_i \Rightarrow_s (B \Rightarrow_s D_i)$ is an axiom. If $D_i$ is B, then $\Gamma \vdash B \Rightarrow_s D_i$, since, by Proposition 1, $\vdash B \Rightarrow_s B$. If there exist $j$ and $k$ less than $i$ such that $D_k$ is $\vdash D_j \Rightarrow_s D_i$, then, by inductive hypothesis, $\Gamma \vdash B \Rightarrow_s D_j$ and $\Gamma \vdash B \Rightarrow_s (D_j \Rightarrow_s D_i)$. Now, by axiom A2,

$\vdash B \Rightarrow_s (D_j \Rightarrow_s D_i) \Rightarrow_s ((B \Rightarrow_s D_j) \Rightarrow_s (B \Rightarrow_s D_i))$. Hence, by MP twice, $\Gamma \vdash B \Rightarrow_s D_i$. Finally, suppose that there is some $j < i$ such that $D_i$ is $\forall x_k D_j$.

By the inductive hypothesis, $\Gamma \vdash B \Rightarrow_s D_j$, and, by the hypothesis of the theorem, either $D_j$ does not depend upon $B$ or $x_k$ is not a free variable of $B$. If $D_j$ does not depend upon $B$, then, by Proposition 2, $\Gamma \vdash D_j$ and, consequently, by Gen, $\Gamma \vdash \forall x_k D_j$. Thus, $\Gamma \vdash D_i$. Now, by axiom A1, $\vdash D_i \Rightarrow_s (B \Rightarrow_s D_i)$.

So, $\Gamma \vdash B \Rightarrow_s D_i$ by MP. If, on the other hand, $x_k$ is not a free variable of $B$, then, by axiom A5, $\vdash \forall x_k (B \Rightarrow_s D_j) \Rightarrow_s (B \Rightarrow_s \forall x_k D_j)$. Since $\Gamma \vdash B \Rightarrow_s D_j$, we have, by Gen, $\Gamma \vdash \forall x_k (B \Rightarrow_s D_j)$, and so, by MP, $\Gamma \vdash B \Rightarrow_s \forall x_k D_j$ that is, $\Gamma \vdash B \Rightarrow_s D_i$. This completes the induction, and our proposition is just the special case $i = n$.

## §2. Set theory $\mathbb{NC}_{\infty}^{\#}$.

Set theory $\mathbb{NC}_{\infty}^{\#}$ is formulated as a system of axioms based on bivalent hyper
infinitary logic $^2L_{\infty}^{\#}$ with restricted modus ponens rule [1],[18]. The language of set
theory $\mathbb{NC}_{\infty}^{\#}$ is a first-order hyper infinitary language $L_{\infty}^{\#}$ with equality $=$, which
includes a binary symbol $\in$. We write $x \neq y$ for $\neg (x = y)$ and $x \not\in y$ for $\neg (x \in y)$.
Individual variables $x, y, z, \dots$, and $x^{\text{cl}}, y^{\text{cl}}, z^{\text{cl}}, \dots$ of $L_{\infty}^{\#}$ will be understood as ranging
over *classical* sets. The unique existential quantifier $\exists!$ is introduced by writing, for any
formula $\varphi(x), \exists!x\varphi(x)$ as an abbreviation of the formula $\exists x[\varphi(x) \ \& \ \forall y(\varphi(y) \Rightarrow_s x = y)]$.
The language $L_{\infty}^{\#}$ will also contains the formation of terms of the form $\{x|\varphi(x)\}^{\text{NCL}}$, for
any formula $\varphi(x)$ containing the free variable $x$.

Such terms are called non-classical sets; we shall use upper case letters $A, B, \dots$, and $A^{\text{NCL}}, B^{\text{NCL}}, \dots$ for such sets. For each non-classical set $A = \{x|\varphi(x)\}^{\text{NCL}}$ the formulas

$\forall x[x \in A \Leftrightarrow_{s,w} \varphi(x)]$ and $\forall x[x \in A \Leftrightarrow_{s,w} \varphi(x,A)]$ is called the defining axioms for the non-classical set $A$.

**Remark 2.1.** Remind that in logic $^2L_{\infty}^{\#}$ with restricted modus ponens rule the statement $\alpha \wedge (\alpha \Rightarrow \beta)$ does not always guarantee that

$$ \alpha, \alpha \Rightarrow \beta \vdash_{\mathrm{RMP}} \beta \qquad (2.1) $$
$$ \exists a \forall x [x \in a \Leftrightarrow \varphi(x, a) \wedge x \in u] \qquad (2.12) $$

always guarantee that

$$ x \in a, x \in a \Rightarrow \varphi(x, a) \vdash_{RMP} \varphi(x) \qquad (2.13) $$

if $x \in a$ holds and

$$ \varphi(x), \varphi(x) \Rightarrow x \in a \vdash_{RMP} x \in a; \qquad (2.14) $$

if $\varphi(x)$ holds;

In order to emphasize this fact mentioned above in Remark 2.1-2.3, we rewrite the defining axioms in general case for the nonclassical sets in the following form

$$ \exists A \forall x \{[x \in A \Leftrightarrow_s \varphi(x, A)] \lor [x \in A \Leftrightarrow_w \varphi(x, A)]\} \qquad (2.15) $$

and similarly we rewrite the defining axioms in general case for the classical sets in the following form

$$ \exists a \forall x [x \in a \Leftrightarrow_s \varphi(x) \wedge (x \in u)]. \qquad (2.16) $$

**Abbreviation 2.2.** We write instead (2.15):

$$ \forall x \{[x \in A \Leftrightarrow_{s,w} \varphi(x, A)]\} \qquad (2.17) $$

**Definition 2.1.** (1) Let A be a nonclassical set defined by formula (2.17). Assum that: (i) for some y statement $\varphi(y)$ and statement $\varphi(y) \Rightarrow y \in A$ holds and (ii) $\varphi(y), \varphi(y) \Rightarrow y \in A \nvDash_{RMP} y \in A, y \in A, y \in A \Rightarrow \varphi(y) \nvDash_{RMP} \varphi(y)$. Then we say that y is a weak member of non-classical set A and abbreviate $y \in_w A$.

**Abbreviation 2.3.** Let A be a nonclassical set defined by formula (2.17) We abbreviate $x \in_{s,w} A$ if the following statement $x \in_s A \lor x \in_w A$ holds, i.e.

$$ x \in_{s,w} A \leftrightarrow_{def} (x \in_s A \lor x \in_w A). \qquad (2.18) $$

**Definition 2.2.** (1) Two nonclassical sets A, B are defined to be equal and we write $A = B$ if $\forall x[x \in_{s,w} A \Leftrightarrow_s x \in_{s,w} B]$. (2) A is a subset of B, and we often write $A \subset_{s,w} B$, if $\forall x[x \in_{s,w} A \Rightarrow_s x \in_{s,w} B]$. (3) We also write **CL. Set**(a) for the formula $\exists u \forall x[x \in a \Leftrightarrow x \in u \wedge \varphi(x)]$. (4) We also write **NCL. Set**(A) for the formulas $\forall x[x \in_{s,w} A \Leftrightarrow_{s,w} \varphi(x)]$ and $\forall x[x \in_{s,w} A \Leftrightarrow_{s,w} \varphi(x, A)]$.

**Remark 2.5.** **CL. Set**(u) asserts that the set u is a classical set. For any classical set u, it follows from the defining axiom for the classical set $u = \{x|x \in_s u \wedge \varphi(x)\}$ that **CL. Set**($\{x|x \in_s u \wedge \varphi(x)\}$).

We shall identify $\{x|x \in_s u\}$ with u, so that sets may be considered as (special sorts of) nonclassical sets and we may introduce assertions such as $u \subset_s A, u \subseteq_s A$, etc.

**Abbreviation 2.4.** Let $\varphi(t)$ be a formula of $NC^\#_\infty$.

(i) $\forall x \varphi(x)$ and $\forall^{CL} x \varphi(x)$ abbreviates $\forall x (\mathbf{CL. Set}(x) \Rightarrow \varphi(x))$

(ii) $\exists x \varphi(x)$ and $\exists^{CL} x \varphi(x)$ abbreviates $\exists x (\mathbf{CL. Set}(x) \Rightarrow \varphi(x))$

(iii) $\forall X \varphi(X)$ and $\forall^{NCL} X \varphi(X)$ abbreviates $\forall X (\mathbf{NCL. Set}(X) \Rightarrow \varphi(X))$

(iv) $\exists X \varphi(X)$ and $\exists^{NCL} X \varphi(X)$ abbreviates $\exists X (\mathbf{NCL. Set}(X) \Rightarrow \varphi(X))$

**Remark 2.6.** If A is a nonclassical set, we write $\exists x \in A \varphi(x, A)$ for $\exists x [x \in A \wedge \varphi(x, A)]$ and $\forall x \in A \varphi(x, A)$ for $\forall x [x \in A \Rightarrow \varphi(x, A)]$.

We define now the following sets:
since for some $\alpha$ and $\beta$ possible

$$ \alpha, \alpha \Rightarrow \beta \Vdash_{\mathrm{RMP}} \beta \qquad (2.2) $$

even if the statement $\alpha \wedge (\alpha \Rightarrow \beta)$ holds.

**Abbreviation 2.1.** We shall write for the sake of brevity instead (2.1) by

$$ \alpha \Rightarrow_s \beta \qquad (2.3) $$

and we shall write instead (2.2) by

$$ \alpha \Rightarrow_w \beta. \qquad (2.4) $$

**Remark 2.2.** Let $A$ be a nonclassical set. Note that in set theory $\mathbf{NC}_{\infty}^{\#}$ the following true formula

$$ \exists A \forall x [x \in A \Leftrightarrow \varphi(x, A)] \qquad (2.5) $$

does not always guarantee that

$$ x \in A, x \in A \Rightarrow \varphi(x, A) \vdash_{\mathrm{RMP}} \varphi(x, A) \qquad (2.6) $$

even if $x \in A$ holds and (or)

$$ \varphi(x, A), \varphi(x, A) \Rightarrow x \in A \vdash_{\mathrm{RMP}} x \in A; \qquad (2.7) $$

even $\varphi(x, A)$ holds, since for nonclassical set $A$ for some $y$ possible

$$ y \in A, y \in A \Rightarrow \varphi(y, A) \Vdash_{\mathrm{RMP}} \varphi(y, A) \qquad (2.8) $$

and (or)

$$ \varphi(y, A), \varphi(y, A) \Rightarrow y \in A \Vdash_{\mathrm{RMP}} y \in A. \qquad (2.9) $$

**Remark 2.3.** Note that in this paper the formulas

$$ \exists a \forall x [x \in a \Leftrightarrow \varphi(x) \wedge x \in u] \qquad (2.10) $$

and more general formulas

$$ \exists a \forall x [x \in a \Leftrightarrow \varphi(x, a) \wedge x \in u] \qquad (2.11) $$

is considered as the defining axioms for the classical set $a$.

**Remark 2.4.** Let $a$ be a classical set. Note that in $\mathbf{NC}_{\infty}^{\#}$: (i) the following true formula

$$ \exists a \forall x [x \in a \Leftrightarrow \varphi(x, a) \wedge x \in u] \qquad (2.12) $$

always guarantee that

$$ x \in a, x \in a \Rightarrow \varphi(x, a) \vdash_{\mathrm{RMP}} \varphi(x) \qquad (2.13) $$

if $x \in a$ holds and

$$ \varphi(x), \varphi(x) \Rightarrow x \in a \vdash_{\mathrm{RMP}} x \in a; \qquad (2.14) $$

if $\varphi(x)$ holds;

In order to emphasize this fact mentioned above in Remark 2.1-2.3, we rewrite the defining axioms in general case for the nonclassical sets in the following

form

$$ \exists A \forall x \{[x \in A \Leftrightarrow_s \varphi(x, A)] \lor [x \in A \Leftrightarrow_w \varphi(x, A)]\} \qquad (2.15) $$

and similarly we rewrite the defining axioms in general case for the classical sets in the
1. $\{u_1, u_2, \dots, u_n\} = \{x|x = u_1 \lor x = u_2 \lor \dots \lor x = u_n\}$. 2. $\{A_1, A_2, \dots, A_n\} =$
$= \{x|x = A_1 \lor x = A_2 \lor \dots \lor x = A_n\}$.

4. $\cap A = \{x | \forall y [y \in A \Rightarrow x \in y]\}$. 5. $A \cup B = \{x | x \in A \lor x \in B\}$.

5. $A \cap B = \{x | x \in A \land x \in B\}$. 6. $A - B = \{x | x \in A \land x \notin B\}$. 7. $u^+ = u \cup \{u\}$.

8. $\mathbf{P}(A) = \{x | x \subseteq A\}$. 9. $\{x \in A | \varphi(x, A)\} = \{x | x \in A \land \varphi(x, A)\}$. 10. $\mathbf{V} = \{x | x = x\}$.

11. $\emptyset = \{x|x \neq x\}$.

The system $\mathbf{NC}_{\infty}^{\#}$ of set theory is based on the following axioms:

**Extensionality 1:** $\forall u \forall v [\forall x (x \in u \Leftrightarrow x \in v) \Rightarrow u = v]$

**Extensionality 2:** $\forall A \forall B [\forall x (x \in A \Leftrightarrow_{s,w} x \in B) \Rightarrow A = B]$

**Universal Set:** NCL. Set($V$)

**Empty Set:** CL. Set($\emptyset$)

**Pairing 1:** $\forall u \forall v \text{ CL. Set}(\{u, v\})$

**Pairing 2:** $\forall A \forall B \text{ NCL. Set}(\{A, B\})$

**Union 1:** $\forall u \text{ CL. Set}(\cup u)$

**Union 2:** $\forall A \text{ NCL. Set}(\cup A)$

**Powerset 1:** $\forall u \text{ CL. Set}(\mathbf{P}(u))$

**Powerset 2:** $\forall A \text{ NCL. Set}(\mathbf{P}(A))$

**Infinity:** $\exists a [\emptyset \in a \land \forall x \in a (x^+ \in a)]$

**Separation 1:** $\forall u_1 \forall u_2, \dots \forall u_n \forall a \exists \mathbf{CL. Set}(\{x \in s \mid a \mid \varphi(x, u_1, u_2, \dots, u_n)\})$

**Separation 2:** $\forall u_1 \forall u_2, \dots \forall u_n \text{ NCL. Set}(\{x \in_{s,w} A \mid \varphi(x, A; u_1, u_2, \dots, u_n)\})$

**Comprehension 1:** $\forall u_1 \forall u_2, \dots \forall u_n \exists A \forall x [x \in_{s,w} A \Leftrightarrow_{s,w} \varphi(x; u_1, u_2, \dots, u_n)]$

**Comprehension 2:** $\forall u_1 \forall u_2, \dots \forall u_n \exists A \forall x [x \in_{s,w} A \Leftrightarrow_{s,w} \varphi(x; u_1, u_2, \dots, u_n)]$

**Comprehension 3:** $\forall u_1 \forall u_2, \dots \forall u_n \exists a \forall x [x \in_s a \Leftrightarrow_s (a \subset u_1) \land \varphi(x, a; u_1, u_2, \dots, u_n)]$

In particular:

**Comprehension 3'**: $\forall u \exists a \forall x [x \in_s a \Leftrightarrow_s (a \subset u) \land \varphi(x, a; u)]$

**Hyperinfinity:** see subsection 2.1.

**Remark 2.7.** Note that the axiom of hyper infinity follows from the schemata
Comprehension 3.

**Definition 2.3.** The ordered pair of two sets $u, v$ is defined as usual by

$$ \langle u, v \rangle = \{ \{u\}, \{u, v\} \}. \qquad (2.19) $$

**Definition 2.4.** We define the Cartesian product of two nonclassical sets $A$ and $B$ as usual by

$$ A \times_{s,w} B = \{ \langle x, y \rangle \mid x \in_{s,w} A \land y \in_{s,w} B \} \qquad (2.20) $$

**Definition 2.5.** A binary relation between two nonclassical sets $A, B$ is a subset $R \subseteq_{s,w} A \times_{s,w} B$. We also write $aR_{s,w}b$ for $<a, b> \in_{s,w} R$. The domain $\textbf{dom}(R)$ and the range $\textbf{ran}(R)$ of $R$ are defined by

$$ \textbf{dom}(R) = \{x \mid \exists y (xR_{s,w}y)\}, \quad \textbf{ran}(R) = \{y : \exists x (xR_{s,w}y)\}. \qquad (2.21) $$

**Definition 2.6.** A relation $F_{s,w}$ is a function, or map, written $\text{Fun}(F_{s,w})$, if for each $a \in_{s,w} \text{dom}(F)$ there is a unique $b$ for which $aF_{s,w}b$. This unique $b$ is written $F(a)$ or $Fa$. We write $F_{s,w}: A \to B$ for the assertion that $F_{s,w}$ is a function with $\text{dom}(F_{s,w}) = A$ and $\text{ran}(F_{s,w}) = B$. In this case we write $a \mapsto F_{s,w}(a)$ for $F_{s,w}a$.

**Definition 2.7.** The identity map $1_A$ on $A$ is the map $A \to A$ given by $a \mapsto a$.
If $X \subseteq_{s,w} A$, the map $x \mapsto x: X \to A$ is called the insertion map of $X$ into $A$.

**Definition 2.8.** If $F_{s,w}: A \to B$ and $X \subseteq_{s,w} A$, the restriction $F_{s,w}|_X$ of $F_{s,w}$ to $X$ is the
following form

$$ \exists a \forall x [x \in a \Leftrightarrow_s \varphi(x) \wedge (x \in u)]. \qquad (2.16) $$

**Abbreviation 2.2.** We write instead (2.15):

$$ \forall x \{[x \in A \Leftrightarrow_{s,w} \varphi(x, A)]\} \qquad (2.17) $$

**Definition 2.1.** (1) Let A be a nonclassical set defined by formula (2.17).

Assum that: (i) for some y statement φ(y) and statement φ(y) ⇒ y ∈ A holds and
(ii) φ(y), φ(y) ⇒ y ∈ A ⊢RMP y ∈ A, y ∈ A, y ∈ A ⇒ φ(y) ⊢RMP φ(y).

Then we say that y is a weak member of non-classical set A and abbreviate y ∈$_w$ A.

**Abbreviation 2.3.** Let A be a nonclassical set defined by formula (2.17) We abbreviate $x \in_{s,w} A$ if the following statement $x \in_s A \lor x \in_w A$ holds, i.e.

$$ x \in_{s,w} A \leftrightarrow_{def} (x \in_s A \lor x \in_w A). \qquad (2.18) $$

**Definition 2.2.** (1) Two nonclassical sets A, B are defined to be equal and we write $A = B$ if $\forall x[x \in_{s,w} A \Leftrightarrow_s x \in_{s,w} B]$. (2) A is a subset of B, and we often write $A \subset_{s,w} B$, if $\forall x[x \in_{s,w} A \Rightarrow_s x \in_{s,w} B]$. (3) We also write **CL. Set**(a) for the formula $\exists u \forall x[x \in a \Leftrightarrow x \in u \wedge \varphi(x)]$. (4) We also write **NCL. Set**(A) for the formulas $\forall x[x \in_{s,w} A \Leftrightarrow_{s,w} \varphi(x)]$ and $\forall x[x \in_{s,w} A \Leftrightarrow_{s,w} \varphi(x, A)]$.

**Remark 2.5.** **CL. Set**(u) asserts that the set u is a classical set. For any classical set u, it follows from the defining axiom for the classical set $u = \{x|x \in_s u \wedge \varphi(x)\}$ that **CL. Set**($\{x|x \in_s u \wedge \varphi(x)\}$).

We shall identify $\{x|x \in_s u\}$ with u, so that sets may be considered as (special sorts of) nonclassical sets and we may introduce assertions such as $u \subset_s A, u \subseteq_s A$, etc.

**Abbreviation 2.4.** Let $\varphi(t)$ be a formula of $NC^#_{\infty}$.

(i) $\forall x\varphi(x)$ and $\forall^{CL}x\varphi(x)$ abbreviates $\forall x(\mathbf{CL}.\mathbf{Set}(x) \Rightarrow \varphi(x))$

(ii) $\exists x\varphi(x)$ and $\exists^{CL}x\varphi(x)$ abbreviates $\exists x(\mathbf{CL}.\mathbf{Set}(x) \Rightarrow \varphi(x))$

(iii) $\forall X\varphi(X)$ and $\forall^{NCL}X\varphi(X)$ abbreviates $\forall X(\mathbf{NCL}.\mathbf{Set}(X) \Rightarrow \varphi(X))$

(iv) $\exists X\varphi(X)$ and $\exists^{NCL}X\varphi(X)$ abbreviates $\exists X(\mathbf{NCL}.\mathbf{Set}(X) \Rightarrow \varphi(X))$

**Remark 2.6.** If A is a nonclassical set, we write $\exists x \in A \varphi(x, A)$ for $\exists x[x \in A \wedge \varphi(x, A)]$ and $\forall x \in A \varphi(x, A)$ for $\forall x[x \in A \Rightarrow \varphi(x, A)]$.

We define now the following sets:

1. $\{u_1, u_2, \dots, u_n\} = \{x|x = u_1 \lor x = u_2 \lor \dots \lor x = u_n\}$. 2. $\{A_1, A_2, \dots, A_n\} =$
$= \{x|x = A_1 \lor x = A_2 \lor \dots \lor x = A_n\}$. 3. $\cup A = \{x|\exists y[y \in A \wedge x \in y]\}$.

4. $\cap A = \{x|\forall y[y \in A \Rightarrow x \in y]\}$. 5. $A \cup B = \{x|x \in A \lor x \in B\}$.

5. $A \cap B = \{x|x \in A \wedge x \in B\}$. 6. $A - B = \{x|x \in A \wedge x \notin B\}$. 7. $u^+ = u \cup \{u\}$.

8. $\mathbf{P}(A) = \{x|x \subseteq A\}$. 9. $\{x \in A|\varphi(x, A)\} = \{x|x \in A \wedge \varphi(x, A)\}$. 10. $\mathbf{V} = \{x|x = x\}$.

11. $\emptyset = \{x|x \neq x\}$.

The system $NC^#_{\infty}$ of set theory is based on the following axioms:

**Extensionality1:** $\forall u \forall v[\forall x(x \in u \Leftrightarrow x \in v) \Rightarrow u = v]$

**Extensionality2:** $\forall A \forall B[\forall x(x \in A \Leftrightarrow_{s,w} x \in B) \Rightarrow A = B]$

**Universal Set:** NCL. Set(V)

**Empty Set:** CL. Set(Ø)

**Pairing1:** $\forall u \forall v \mathbf{CL}.\mathbf{Set}(\{u, v\})$

**Pairing2:** $\forall A \forall B \mathbf{NCL}.\mathbf{Set}(\{A, B\})$

**Union1:** $\forall u \mathbf{CL}.\mathbf{Set}(\cup u)$

**Union2:** $\forall A \mathbf{NCL}.\mathbf{Set}(\cup A)$
map $X \to A$ given by $x \mapsto F_{s,w}(x)$. If $Y \subseteq_{s,w} B$, the inverse image of $Y$ under $F_{s,w}$ is the set

$$F_{s,w}^{-1}[Y] = \{x \in_{s,w} A : F_{s,w}(x) \in_{s,w} Y\}. \qquad (2.22)$$

Given two functions $F_{s,w} : A \to B, G_{s,w} : B \to C$, we define the composite function

$G_{s,w} \circ F_{s,w} : A \to C$ to be the function $a \mapsto G_{s,w}(F_{s,w}(a))$. If $F_{s,w} : A \to A$, we write $F_{s,w}^2$ for $F_{s,w} \circ F_{s,w}, F_{s,w}^3$ for $F_{s,w} \circ F_{s,w} \circ F_{s,w}$ etc.

**Definition 2.9.** A function $F_{s,w} : A \to B$ is said to be monic if for all

$x, y \in_{s,w} A, F_{s,w}(x) = F_{s,w}(y)$ implies $x = y$, epi if for any $b \in_{s,w} B$ there is $a \in_{s,w} A$ for which $b = F_{s,w}(a)$, and bijective, or a bijection, if it is both monic and epi. It is easily shown that

$F_{s,w}$ is bijective if and only if $F_{s,w}$ has an inverse, that is, a map $G_{s,w} : B \to A$ such that $F_{s,w} \circ G_{s,w} = \mathbf{1}_B$ and $G_{s,w} \circ F_{s,w} = \mathbf{1}_A$.

**Definition 2.10.** Two sets $X$ and $Y$ are said to be equipollent, and we write $X \approx_{s,w} Y$, if there is a bijection between them.

**Definition 2.11.** Suppose we are given two sets $I, A$ and an epi map $F_{s,w} : I \to A$.

Then $A = \{F_{s,w}(i)|i \in I\}$ and so, if, for each $i \in_{s,w} I$, we write $a_i$ for $F_{s,w}(i)$, then $A$ can be presented in the form of an indexed set $\{a_i : i \in_{s,w} I\}$. If $A$ is presented as an indexed set of sets $\{X_i|i \in_{s,w} I\}$, then we write $\bigcup_{i \in I} X_i$ and $\bigcap_{i \in I} X_i$ for $\bigcup A$ and $\bigcap A$, respectively.

**Definition 2.12.** The projection maps $\pi_1 : A \times_{s,w} B \to A$ and $\pi_2 : A \times_{s,w} B \to B$ are defined to be the maps $\langle a, b \rangle \mapsto a$ and $\langle a, b \rangle \mapsto b$ respectively.

**Definition 2.13.** For sets $A, B$, the exponential $B^A$ is defined to be the set of all functions from $A$ to $B$.

## Axiom of nonregularity

Remind that a non-empty set $u$ is called regular iff $\forall x[x \neq \emptyset \to (\exists y \in x)(x \cap y = \emptyset)]$.

Let's investigate what it says: suppose there were a non-empty $x$ such that $(\forall y \in x)(x \cap y \neq \emptyset)$. For any $z_1 \in x$ we would be able to get $z_2 \in z_1 \cap x$. Since $z_2 \in x$ we would be able to get $z_3 \in z_2 \cap x$. The process continues forever:

... $\in z_{n+1} \in z_n... \in z_4 \in z_3 \in z_2 \in z_1 \in x$. Thus if we don't wish to rule out such an infinite regress we forced accept the following statement:

$$\exists x[x \neq \emptyset \to (\forall y \in x)(x \cap y \neq \emptyset)]. \qquad (2.23)$$

## Axiom of hyperinfinity.

**Definition 2.14.** (i) A non-empty transitive non regular set $u$ is a well formed non regular set iff:

(i) there is unique countable sequence $\{u_n\}_{n=1}^{\infty}$ such that

$$... \in u_{n+1} \in u_n... \in u_4 \in u_3 \in u_2 \in u_1 \in u, \qquad (2.24)$$

(ii) for any $n \in \mathbb{N}$ and any $u_{n+1} \in u_n$ :

$$u_n = u_{n+1}^+, \qquad (2.25)$$

where $a^+ = a \cup \{a\}$.

(ii) we define a function $a^{+[k]}$ inductively by $a^{+[k]} = (a^{+[k]})^+$.

**Definition 2.15.** Let $u$ and $w$ are well formed non regular sets. We write $w < u$ iff
**Powerset1:** $\forall u \text{ CL. Set}(\mathbf{P}(u))$

**Powerset2:** $\forall A \text{ NCL. Set}(\mathbf{P}(A))$

**Infinity** $\exists a[\emptyset \in a \land \forall x \in a(x^+ \in a)]$

**Separation1** $\forall u_1 \forall u_2, \dots, \forall u_n \forall a \exists \text{CL. Set}(\{x \in s a | \varphi(x, u_1, u_2, \dots, u_n)\})$

**Separation 2** $\forall u_1 \forall u_2, \dots, \forall u_n \text{NCL. Set}(\{x \in s,w A | \varphi(x, A; u_1, u_2, \dots, u_n)\})$

**Comprehension1** $\forall u_1 \forall u_2, \dots, \forall u_n \exists A \forall x [x \in s,w A \Leftrightarrow_s w \varphi(x; u_1, u_2, \dots, u_n)]$

**Comprehension 2** $\forall u_1 \forall u_2, \dots, \forall u_n \exists A \forall x [x \in s,w A \Leftrightarrow_s w \varphi(x, A; u_1, u_2, \dots, u_n)]$

**Comprehension 3** $\forall u_1 \forall u_2, \dots, \forall u_n \exists a \forall x [x \in s a \Leftrightarrow_s (a \subset u_1) \land \varphi(x, a; u_1, u_2, \dots, u_n)]$

In particular:

**Comprehension 3'** $\forall u \exists a \forall x [x \in s a \Leftrightarrow_s (a \subset u) \land \varphi(x, a; u)]$

**Hyperinfinity:** see subsection 2.1.

**Remark 2.7.** Note that the axiom of hyper infinity follows from the schemata Comprehension 3.

**Definition 2.3.** The ordered pair of two sets $u, v$ is defined as usual by

$$ \langle u, v \rangle = \{ \{u\}, \{u, v\} \}. \qquad (2.19) $$

**Definition 2.4.** We define the Cartesian product of two nonclassical sets A and B as usual by

$$ A \times_{s,w} B = \{ \langle x, y \rangle | x \in_{s,w} A \land y \in_{s,w} B \} \qquad (2.20) $$

**Definition 2.5.** A binary relation between two nonclassical sets A, B is a subset $R \subseteq_{s,w} A \times_{s,w} B$. We also write $aR_{s,w} b$ for $< a, b > \in_{s,w} R$. The doman $\text{dom}(R)$ and the range $\text{ran}(R)$ of R are defined by

$$ \text{dom}(R) = \{x | \exists y (xR_{s,w} y)\}, \quad \text{ran}(R) = \{y : \exists x (xR_{s,w} y)\}. \qquad (2.21) $$

**Definition 2.6.** A relation $F_{s,w}$ is a function, or map, written $\text{Fun}(F_{s,w})$, if for each $a \in_{s,w} \text{dom}(F)$ there is a unique $b$ for which $aF_{s,w} b$. This unique $b$ is written $F(a)$ or $Fa$. We write $F_{s,w} : A \to B$ for the assertion that $F_{s,w}$ is a function with $\text{dom}(F_{s,w}) = A$ and $\text{ran}(F_{s,w}) = B$. In this case we write $a \mapsto F_{s,w}(a)$ for $F_{s,w}a$.

**Definition 2.7.** The identity map $1_A$ on A is the map $A \to A$ given by $a \mapsto a$. If $X \subseteq_{s,w} A$, the map $x \mapsto x : X \to A$ is called the insertion map of X into A.

**Definition 2.8.** If $F_{s,w} : A \to B$ and $X \subseteq_{s,w} A$, the restriction $F_{s,w}|_X$ of $F_{s,w}$ to X is the map $X \to A$ given by $x \mapsto F_{s,w}(x)$. If $Y \subseteq_{s,w} B$, the inverse image of Y under $F_{s,w}$ is the set

$$ F_{s,w}^{-1}[Y] = \{x \in_{s,w} A : F_{s,w}(x) \in_{s,w} Y\}. \qquad (2.22) $$

Given two functions $F_{s,w} : A \to B, G_{s,w} : B \to C$, we define the composite function

$G_{s,w} \circ F_{s,w} : A \to C$ to be the function $a \mapsto G_{s,w}(F_{s,w}(a))$. If $F_{s,w} : A \to A$, we write $F_{s,w}^2$ for $F_{s,w} \circ F_{s,w}, F_{s,w}^3$ for $F_{s,w} \circ F_{s,w} \circ F_{s,w}$ etc.

**Definition 2.9.** A function $F_{s,w} : A \to B$ is said to be monic if for all

$x, y \in_{s,w} A, F_{s,w}(x) = F_{s,w}(y)$ implies $x = y$, epi if for any $b \in_{s,w} B$ there is $a \in_{s,w} A$ for which $b = F_{s,w}(a)$, and bijective, or a bijection, if it is both monic and epi. It is easily shown that

$F_{s,w}$ is bijective if and only if $F_{s,w}$ has an inverse, that is, a map $G_{s,w} : B \to A$ such that $F_{s,w} \circ G_{s,w} = 1_B$ and $G_{s,w} \circ F_{s,w} = 1_A$.

**Definition 2.10.** Two sets X and Y are said to be equipollent, and we write $X \approx_{s,w} Y$, if there is a bijection between them.
for any $n \in \mathbb{N}$

$$w \in u_n. \tag{2.26}$$

**Definition 2.16.** We say that a well formed non regular set $u$ is infinite (or hyperfinite) hypernatural number iff:

(I) For any member $w \in u$ one and only one of the following conditions are satisfied:

(i) $w \in \mathbb{N}$ or

(ii) $w = u_n$ for some $n \in \mathbb{N}$ or

(iii) $w < u$.

(II) Let $\prec u$ be a set $\prec u = \{z|z < u\}$, then by relation $(\cdot < \cdot)$ a set $\prec u$ is densely ordered with no first element.

(III) $\mathbb{N} \subset u$.

**Definition 2.17.** Assume $u \in \mathbb{N}^\#$, then $u$ is infinite (hypernatural) number if $u \in \mathbb{N}^\#\mathbb{N}$.

**Axiom of hyperinfinity**

There exists a set $\mathbb{N}^\#$ such that:

(i) $\mathbb{N} \subset \mathbb{N}^\#$,

(ii) if $u \in \mathbb{N}^\#\mathbb{N}$ then there exists infinite (hypernatural) number $v$ such that $v < u$,

(iii) if $u \in \mathbb{N}^\#\mathbb{N}$ then there exists infinite (hypernatural) number $w$ such that for any $n \in \mathbb{N}$: $u^{+[n]} < w$,

(iv) set $\mathbb{N}^\#\mathbb{N}$ is patially ordered by relation $(\cdot < \cdot)$ with no first and no last element.

**Axiom of existence the nonclassical truth predicate**

Let $A, B$ be a closed wff's of $\mathbf{NC}_{\infty}^{\#}$ ($\mathbf{NC}_{\infty}^{\#}$-sentences). There is truth predicate $\mathbf{T}^{\#}[A]$ satisfies the following $\mathbf{T}^{\#}$-schemas:

$$1. \forall x \forall y \{ \mathbf{T}^{\#}[x = y] \Leftrightarrow_{s,w} (x = y) \}$$

$$2. \forall x \forall y \{ \mathbf{T}^{\#}[x \in y] \Leftrightarrow_{s,w} (x \in y) \}$$

$$3. \mathbf{T}^{\#}[\mathbf{T}^{\#}[A]] \Leftrightarrow_s \mathbf{T}^{\#}[A]$$

$$4. \mathbf{T}^{\#}[\neg \mathbf{T}^{\#}[A]] \Leftrightarrow_s \mathbf{T}^{\#}[\neg A] \tag{2.27}$$

$$5. \mathbf{T}^{\#}[\neg A] \Leftrightarrow_s \neg \mathbf{T}^{\#}[A]$$

$$6. \mathbf{T}^{\#}[\neg \neg A] \Leftrightarrow_s \mathbf{T}^{\#}[A]$$

$$7. \mathbf{T}^{\#}[A \wedge B] \Leftrightarrow_s \mathbf{T}^{\#}[A] \wedge \mathbf{T}^{\#}[B]$$

$$8. \mathbf{T}^{\#}[A \vee B] \Leftrightarrow_s \mathbf{T}^{\#}[A] \vee \mathbf{T}^{\#}[B]$$

and

$$9. \mathbf{T}^{\#}[A] \Leftrightarrow_{s,w} A. \tag{2.28}$$

**Definition 2.18.** (i) We say that a $\mathbf{NC}_{\infty}^{\#}$-sentence is a $s$-$\mathbf{NC}_{\infty}^{\#}$-sentence (strong $\mathbf{NC}_{\infty}^{\#}$-sentence relative to $\vdash_{\mathrm{RMP}}$) if

$$\mathbf{T}^{\#}[A] \Leftrightarrow_s A. \tag{2.29}$$

(ii) We say that a $\mathbf{NC}_{\infty}^{\#}$-sentence is a $w$-$\mathbf{NC}_{\infty}^{\#}$-sentence (weak $\mathbf{NC}_{\infty}^{\#}$-sentence relative to $\vdash_{\mathrm{RMP}}$) if

$$\mathbf{T}^{\#}[A] \Leftrightarrow_w A. \tag{2.30}$$

**Notations 2.1.** (i) We write $x =_s y$ if $\mathbf{T}^{\#}[x = y] \Leftrightarrow_s (x = y)$.
**Definition 2.11.** Suppose we are given two sets $I, A$ and an epi map $F_{s,w} : I \to A$. Then $A = \{F_{s,w}(i) | i \in I\}$ and so, if, for each $i \in_{s,w} I$, we write $a_i$ for $F_{s,w}(i)$, then $A$ can be presented in the form of an indexed set $\{a_i : i \in_{s,w} I\}$. If $A$ is presented as an indexed set of sets $\{X_i | i \in_{s,w} I\}$, then we write $\bigcup_{i \in I} X_i$ and $\bigcap_{i \in I} X_i$ for $\bigcup A$ and $\bigcap A$, respectively.

**Definition 2.12.** The projection maps $\pi_1 : A \times_{s,w} B \to A$ and $\pi_2 : A \times_{s,w} B \to B$ are defined to be the maps $\langle a, b \rangle \mapsto a$ and $\langle a, b \rangle \mapsto b$ respectively.

**Definition 2.13.** For sets $A, B$, the exponential $B^A$ is defined to be the set of all functions from $A$ to $B$.

## Axiom of nonregularity

Remind that a non-empty set $u$ is called regular iff $\forall x[x \neq \emptyset \to (\exists y \in x)(x \cap y = \emptyset)]$. Let's investigate what it says: suppose there were a non-empty $x$ such that $(\forall y \in x)(x \cap y \neq \emptyset)$. For any $z_1 \in x$ we would be able to get $z_2 \in z_1 \cap x$. Since $z_2 \in x$ we would be able to get $z_3 \in z_2 \cap x$. The process continues forever: $\dots \in z_{n+1} \in z_n \dots \in z_4 \in z_3 \in z_2 \in z_1 \in x$. Thus if we don't wish to rule out such an infinite regress we forced accept the following statement:

$$ \exists x[x \neq \emptyset \to (\forall y \in x)(x \cap y \neq \emptyset)]. \qquad (2.23) $$

## Axiom of hyperinfinity.

**Definition 2.14.** (i) A non-empty transitive non regular set $u$ is a well formed non regular set iff:

(i) there is unique countable sequence $\{u_n\}_{n=1}^{\infty}$ such that

$$ \dots \in u_{n+1} \in u_n \dots \in u_4 \in u_3 \in u_2 \in u_1 \in u, \qquad (2.24) $$

(ii) for any $n \in \mathbb{N}$ and any $u_{n+1} \in u_n$ :

$$ u_n = u_{n+1}^+, \qquad (2.25) $$

where $a^+ = a \cup \{a\}$.

(ii) we define a function $a^{+[k]}$ inductively by $a^{+[k]} = (a^{+[k]})^+$.

**Definition 2.15.** Let $u$ and $w$ are well formed non regular sets. We write $w < u$ iff for any $n \in \mathbb{N}$

$$ w \in u_n. \qquad (2.26) $$

**Definition 2.16.** We say that a well formed non regular set $u$ is infinite (or hyperfinite) hypernatural number iff:

(I) For any member $w \in u$ one and only one of the following conditions are satisfied:

(i) $w \in \mathbb{N}$ or

(ii) $w = u_n$ for some $n \in \mathbb{N}$ or

(iii) $w < u$.

(II) Let $\prec u$ be a set $\prec u = \{z | z < u\}$, then by relation $(\cdot < \cdot)$ a set $\prec u$ is densely ordered with no first element.

(III) $\mathbb{N} \subset u$.

**Definition 2.17.** Assume $u \in \mathbb{N}^\#$, then $u$ is infinite (hypernatural) number if $u \in \mathbb{N}^\#\mathbb{N}$.

## Axiom of hyperinfinity

There exists a set $\mathbb{N}^\#$ such that:
(ii) We write $x =_w y$ and if $T^{\#}[x = y] \Leftrightarrow_w (x = y)$.

**Notations 2.2.** (i) We write $x \in_s y$ and will be say that a set if $T^{\#}[x \in y] \Leftrightarrow_s (x \in y)$.

(ii) We write $x \in_w y$ and will be say that if $T^{\#}[x \in y] \Leftrightarrow_w (x \in y)$.

**Definition 2.19.** (i) We will be say that a set $y$ is a $s$-set if

$$ \forall x[x \in y \Leftrightarrow_s x \in_s y] \quad (2.31) $$

(ii) We will be say that a set $y$ is a $w$-set if

$$ \forall x[x \in y \Leftrightarrow_s x \in_s y] \quad (2.32) $$

(iii) We will be say that a set $y$ is a $s, w$-set if

$$ \forall x[x \in y \Leftrightarrow_s x \in_{s,w} y] \quad (2.33) $$

**Remark 2.8.** For any model $M$ in a first-order language, the definition of the truth predicate of $M$ is the same - we define the elementary diagram of $M$ as the set of all sentences with parameters from $M$ that are true in $M$, using Tarski's recursive definition of truth, using the $T$ schema. This is the same for a model of ZFC as for any other model in first-order logic. Symbolically

$$ T[A] \Leftrightarrow M \vdash A, \quad (2.34) $$

where $M \vdash A$ stands to $A$ true in model $M$.

**Remark 2.9.** Remind that classical truth predicate $T[A]$ unrestrictedly satisfies the following $T$-schema [25-27]:

$$ T[A] \Leftrightarrow A, \quad (2.35) $$

i.e., the sentence $A \Leftrightarrow T[A]$ is true for every sentence $A$ of language $L$, where $T[A]$ stands for "the sentence (denoted by) $A$ is true". Unfortunately $T$-schema incorrect by well known Curry's paradox.

Assume, too, that we have the principle called Assertion (also known as pseudo modus ponens): $(A \land (A \Rightarrow B)) \Rightarrow B$. By diagonalization, self-reference we can get a sentence $C$ such that $C \Leftrightarrow (T[C] \Rightarrow F)$ where $F$ is anything you like. (For effect, though, make $F$ something obviously false, e.g. $F \equiv \bot = 0 = 1$) By an instance of the $T$-schema: $T[C] \Leftrightarrow C$ we immediately get: $T[C] \Leftrightarrow (T[C] \Rightarrow F)$. Again, using the

same

instance of the $T$-schema, we can substitute $C[T, F]$ for $T[C]$ in the above to get (1).

(1) $\vdash C[T, F] \Leftrightarrow (C[T, F] \Rightarrow F)$ [by $T$-schema and substitution]

(2) $\vdash (C[T, F] \land (C[T, F] \Rightarrow F)) \Rightarrow F$ [by assertion]

(3) $\vdash (C[T, F] \land C[T, F]) \Rightarrow F$ [by substitution, from (2)]

(4) $\vdash C[T, F] \Rightarrow F$ [by equivalence of $C$ and $C \land C$, from (3)]

(5) $\vdash C[T, F]$ [by unrestricted Modus Ponens, from (1) and (4)]

(6) $\vdash F$ [by unrestricted Modus Ponens, from (4) and (5)]

Letting $F$ be anything entailing triviality Curry's paradox quickly 'shows' that the world is trivial.

**Remark 2.10.** Curry's paradox easily avoided by restricted MP such that:

1. $C[T, F] \Rightarrow F$, $(C[T, F] \Rightarrow F) \Rightarrow C[T, F] \vdash_{RMP} C[T, F]$ and

2. $C[T, F], C[T, F] \Rightarrow F \vdash_{RMP} F$,

**Remark 2.11.** The set of all $T$-sentences $T[\phi] \Leftrightarrow \phi$, where $\phi$ is any sentence of the language $L_T$, that is, where $\phi$ may contain $T$, is inconsistent with PA (or any theory that proves the diagonal lemma) because of the Liar paradox [28].

In formal languages, self-reference is also very easy to come by. Any language
(i) $N \subset N^#,$

(ii) if $u \in N^# \setminus N$ then there exists infinite (hypernatural) number $v$ such that $v < u$,

(iii) if $u \in N^# \setminus N$ then there exists infinite (hypernatural) number $w$ such that for any $n \in N: u^{+[n]} < w$,

(iv) set $N^# \setminus N$ is patially ordered by relation $(\cdot < \cdot)$ with no first and no last element.

Axiom of existence the nonclassical truth predicate

Let A, B be a closed wff's of $NC_{\infty}^{#}(NC_{\infty}^{#}-sentences)$. There is truth predicate $T^{#}[A]$ satisfies the following $T^{#}$-schemas:

$$1. \forall x \forall y \{ \mathbf{T}^{#}[x = y] \Leftrightarrow_{s,w} (x = y) \}$$

$$2. \forall x \forall y \{ \mathbf{T}^{#}[x \in y] \Leftrightarrow_{s,w} (x \in y) \}$$

$$3. \mathbf{T}^{#}[\mathbf{T}^{#}[A]] \Leftrightarrow_s \mathbf{T}^{#}[A]$$

$$4. \mathbf{T}^{#}[\neg \mathbf{T}^{#}[A]] \Leftrightarrow_s \mathbf{T}^{#}[\neg A] \qquad (2.27)$$

$$5. \mathbf{T}^{#}[\neg A] \Leftrightarrow_s \neg \mathbf{T}^{#}[A]$$

$$6. \mathbf{T}^{#}[\neg \neg A] \Leftrightarrow_s \mathbf{T}^{#}[A]$$

$$7. \mathbf{T}^{#}[A \wedge B] \Leftrightarrow_s \mathbf{T}^{#}[A] \wedge \mathbf{T}^{#}[B]$$

$$8. \mathbf{T}^{#}[A \vee B] \Leftrightarrow_s \mathbf{T}^{#}[A] \vee \mathbf{T}^{#}[B]$$

and

$$9. \mathbf{T}^{#}[A] \Leftrightarrow_{s,w} A. \qquad (2.28)$$

**Definition 2.18.** (i) We say that a $NC_{\infty}^{#}$-sentence is a $s$-$NC_{\infty}^{#}$-sentence (strong $NC_{\infty}^{#}$-sentence relative to $\vdash_{RMP}$) if

$$\mathbf{T}^{#}[A] \Leftrightarrow_s A. \qquad (2.29)$$

(ii) We say that a $NC_{\infty}^{#}$-sentence is a $w$-$NC_{\infty}^{#}$-sentence (weak $NC_{\infty}^{#}$-sentence relative to $\vdash_{RMP}$) if

$$\mathbf{T}^{#}[A] \Leftrightarrow_w A. \qquad (2.30)$$

**Notations 2.1.** (i) We write $x =_s y$ if $\mathbf{T}^{#}[x = y] \Leftrightarrow_s (x = y)$.

(ii) We write $x =_w y$ and if $\mathbf{T}^{#}[x = y] \Leftrightarrow_w (x = y)$.

**Notations 2.2.** (i) We write $x \in_s y$ and will be say that a set if $\mathbf{T}^{#}[x \in y] \Leftrightarrow_s (x \in y)$.

(ii) We write $x \in_w y$ and will be say that if $\mathbf{T}^{#}[x \in y] \Leftrightarrow_w (x \in y)$.

**Definition 2.19.** (i) We will be say that a set $y$ is a $s$-set if

$$\forall x[x \in y \Leftrightarrow_s x \in_s y] \qquad (2.31)$$

(ii) We will be say that a set $y$ is a $w$-set if

$$\forall x[x \in y \Leftrightarrow_s x \in_s y] \qquad (2.32)$$

(iii) We will be say that a set $y$ is a $s, w$-set if

$$\forall x[x \in y \Leftrightarrow_s x \in_{s,w} y] \qquad (2.33)$$

**Remark 2.8.** For any model $M$ in a first-order language, the definition of the truth predicate of $M$ is the same - we define the elementary diagram of $M$ as the set of all sentences with parameters from $M$ that are true in $M$, using Tarski's recursive definition of truth, using the $T$ schema. This is the same for a model of $ZFC$ as for any other model in first-order logic. Symbolically
capable of expressing some basic syntax can generate self-referential sentences via so-called diagonalization (or more properly, any language together with an appropriate theory of syntax or arithmetic). A language containing a truth predicate and this basic syntax will thus have a sentence *L* such that

$$L \Leftrightarrow \neg \mathbf{Tr}[L] \qquad (2.36)$$

This is a 'fixed point' of (the compound predicate) $\neg Tr$, and is, in effect, our simple-untruth Liar.

Other conspicuous ingredients in common Liar paradoxes concern logical behavior of basic connectives or features of implication. A few of the relevant principles are:

Modus ponens (MP): $A, A \Rightarrow B \vdash B$

Excluded middle (LEM): $\vdash A \lor \neg A$

Explosion (EFQ): $A, \neg A \vdash B$

Disjunction principle (DP): If $A \vdash C$ and $B \vdash C$ then $A \lor B \vdash C$

Adjunction: If $A \vdash B$ and $A \vdash C$ then $A \vdash B \land C$.

An argument that Liar sentence *L* implies a contradiction runs as follows.

1. $\mathbf{Tr}[L] \lor \neg \mathbf{Tr}[L]$ [LEM]

2. Case One:

a. $\mathbf{Tr}[L]$

b. *L* [2a: release by MP from *T* schema (2.35)]

c. $\neg \mathbf{Tr}[L]$ [2b: definition of *L*]

d. $\neg \mathbf{Tr}[L] \land \mathbf{Tr}[L]$ [2a, 2c: adjunction]

Case Two:

a. $\neg \mathbf{Tr}[L]$

b. *L* [3a: definition of *L* by MP]

c. $\mathbf{Tr}[L]$ [3b: by MP from *T* schema (2.35)]

d. $\neg \mathbf{Tr}[L] \land \mathbf{Tr}[L]$ [3a, 3c: adjunction]

4. $\neg \mathbf{Tr}[L] \land \mathbf{Tr}[L]$ [1–3: DP]

**Remark 2.12.** Liar easily avoided by restricted MP such that:

1. $\mathbf{Tr}[L] \dashv \mathbf{RMP} L$

2. $L \dashv \mathbf{RMP} \mathbf{Tr}[L]$

3. $\neg \mathbf{Tr}[L] \dashv \mathbf{RMP} L$

4. $L \dashv \mathbf{RMP} \neg \mathbf{Tr}[L]$

## §3. Nonconservative extension of the model theoretical NSA based on bivalent hyper Infinitary first-order logic $^{2}L_{\infty}^{\#}$ with restricted canonical rules of conclusion.

Extending the classical real numbers $\mathbb{R}$ to include infinite and infinitesimal quantities originally enabled D. Laugwitz [1] to view the delta distribution $\delta(x)$ as a nonstandard point function. Independently A. Robinson [2] demonstrated that distributions could be viewed as generalized polynomials. Luxemburg [3] and Sloan [4] presented an alternate representative of distributions as internal functions within the context of canonical Robinson's theory of nonstandard analysis. For further information on classical nonstandard real analysis, we refer to [8]–[11].

**Abbreviation 3.1.** In this paper we adopt the following notations. For a standard set *E*
$$ T[A] \Leftrightarrow M \vdash A, \qquad (2.34) $$

where $M \vdash A$ stands to A true in model M.

**Remark 2.9.** Remind that classical truth predicate $T[A]$ unrestrictedly satisfies the following T-schema [25-27]:

$$ T[A] \Leftrightarrow A, \qquad (2.35) $$

i.e., the sentence $A \Leftrightarrow T[A]$ is true for every sentence A of language L, where $T[A]$ stands for "the sentence (denoted by) A is true". Unfortunately T-schema incorrect by well known Curry's paradox.

Assume, too, that we have the principle called Assertion (also known as pseudo modus ponens): $(A \land (A \Rightarrow B)) \Rightarrow B$. By diagonalization, self-reference we can get a sentence C such that $C \Leftrightarrow (T[C] \Rightarrow F)$ where F is anything you like. (For effect, though, make F something obviously false, e.g. $F \equiv \bot = 0 = 1$) By an instance of the T-schema: $T[C] \Leftrightarrow C$ we immediately get: $T[C] \Leftrightarrow (T[C] \Rightarrow F)$. Again, using the same

instance of the T-schema, we can substitute $C[T,F]$ for $\neg T[C]$ in the above to get (1).

(1) $\vdash C[T,F] \Leftrightarrow (C[T,F] \Rightarrow F)$ [by T-schema and substitution]

(2) $\vdash (C[T,F] \land (C[T,F] \Rightarrow F)) \Rightarrow F$ [by assertion]

(3) $\vdash (C[T,F] \land C[T,F]) \Rightarrow F$ [by substitution, from (2)]

(4) $\vdash C[T,F] \Rightarrow F$ [by equivalence of C and $C \land C$, from (3)]

(5) $\vdash C[T,F]$ [by unrestricted Modus Ponens, from (1) and (4)]

(6) $\vdash F$ [by unrestricted Modus Ponens, from (4) and (5)]

Letting F be anything entailing triviality Curry's paradox quickly 'shows' that the world is trivial.

**Remark 2.10.** Curry's paradox easily avoided by restricted MP such that:

1. $C[T,F] \Rightarrow F$, $(C[T,F] \Rightarrow F) \Rightarrow C[T,F] \vdash_{RMP} C[T,F]$ and

2. $C[T,F], C[T,F] \Rightarrow F \vdash_{RMP} F$,

**Remark 2.11.** The set of all T-sentences $T[\phi] \Leftrightarrow \phi$, where $\phi$ is any sentence of the language $L_T$, that is, where $\phi$ may contain T, is inconsistent with PA (or any theory that proves the diagonal lemma) because of the Liar paradox [28].

In formal languages, self-reference is also very easy to come by. Any language capable of expressing some basic syntax can generate self-referential sentences via so-called diagonalization (or more properly, any language together with an appropriate theory of syntax or arithmetic). A language containing a truth predicate and this basic syntax will thus have a sentence L such that

$$ L \Leftrightarrow \neg \mathbf{Tr}[L] \qquad (2.36) $$

This is a 'fixed point' of (the compound predicate) $\neg Tr$, and is, in effect, our simple-untruth Liar.

Other conspicuous ingredients in common Liar paradoxes concern logical behavior of basic connectives or features of implication. A few of the relevant principles are:

Modus ponens (MP): $A, A \Rightarrow B \vdash B$

Excluded middle (LEM): $\vdash A \lor \neg A$

Explosion (EFQ): $A, \neg A \vdash B$

Disjunction principle (DP): If $A \vdash C$ and $B \vdash C$ then $A \lor B \vdash C$

Adjunction: If $A \vdash B$ and $A \vdash C$ then $A \vdash B \land C$.

An argument that Liar sentence L implies a contradiction runs as follows.
we often write $E_{\text{st}}$. For a set $E_{\text{st}}$ let $^{\sigma}E_{\text{st}}$ be a set $^{\sigma}E_{\text{st}} = \{^*x|x \in E_{\text{st}}\}$. We identify $z$ with $^{\sigma}z$ i.e., $z \equiv^{\sigma} z$ for all $z \in \mathbb{C}$. Hence, $^{\sigma}E_{\text{st}} = E_{\text{st}}$ if $E \subseteq \mathbb{C}$, e.g., $^{\sigma}\mathbb{C} = \mathbb{C}$, $^{\sigma}\mathbb{R} = \mathbb{R}$, $^{\sigma}P = P$, $^{\sigma}L_{\uparrow}^{+} = L_{\uparrow}^{+}$, etc. Let $^{*}\mathbb{R}_{\approx}$, $^{*}\mathbb{R}_{\approx,+}$, $^{*}\mathbb{R}_{\text{fin}}$, $^{*}\mathbb{R}_{\infty}$, and $^{*}\mathbb{N}_{\infty}$ denote the sets of infinitesimal hyper-real numbers, positive infinitesimal hyper-real numbers, finite hyper-real numbers, infinite hyper-real numbers and infinite hyper natural numbers, respectively.
Note that $^{*}\mathbb{R}_{\text{fin}} = ^{*}\mathbb{R}/^{*}\mathbb{R}_{\infty}$, $^{*}\mathbb{C} = ^{*}\mathbb{R} + i^{*}\mathbb{R}$, $^{*}\mathbb{C}_{\text{fin}} = ^{*}\mathbb{R}_{\text{fin}} + i^{*}\mathbb{R}_{\text{fin}}$.

Remind that Robinson nonstandard analysis (RNA) many developed using set-theoretical objects called superstructures [8]-[11]. A superstructure $V(S)$ over a set $S$ is defined in the following way

$$V_0(S) = S, V_{n+1}(S) = V_n(S) \cup (P(V_n(S)), V(S) = \bigcup_{n \in \mathbb{N}} V_n(S)). \quad (3.1)$$

Superstructures of the empty set consist of sets of infinite rank in the cumulative hierarchy and therefore do not satisfy the in...nity axiom. Making $S = \mathbb{R}$ will suffice for virtually any construction necessary in analysis.

Bounded formulas are formulas where all quantifiers occur in the form

$$\forall x(x \in y \Rightarrow \dots), \exists x(x \in y \Rightarrow \dots). \quad (3.2)$$

A nonstandard embedding is a mapping

$$*: V(X) \to V(Y)$$

from a superstructure $V(X)$ called the standard universum, into another superstructure

$V(Y)$, called nonstandard universum, satisfying the following postulates:

**1.** $Y = ^{*}X$

**2. Transfer Principle.** For every bounded formula $\Phi(x_1, \dots, x_n)$ and elements $a_1, \dots, a_n \in V(X)$, the property $\Phi$ is true for $a_1, \dots, a_n$ in the standard universum if and only if it is true for $^{*}a_1, \dots, ^{*}a_n$ in the nonstandard universum:

$$\langle V(X), \in \rangle \models \Phi(a_1, \dots, a_n) \Leftrightarrow \langle V(Y), \in \rangle \models \Phi(^{*}a_1, \dots, ^{*}a_n).$$

**3. Non-triviality.** For every infinite set $A$ in the standard universum, the set $\{^*a|a \in A\}$ is a proper subset of $^*A$.

**Definition 3.1.** [10]. A set $x$ is **internal** if and only if $x$ is an element of $^*A$ for some element $A$ of $V(\mathbb{R})$. Let $X$ be a set with $A = \{A_i\}_{i \in I}$ a family of subsets of $X$. Then the collection $A$ has the infinite intersection property, if any infinite subcollection $J \subset I$ has non-empty intersection. Nonstandard universum is $\kappa$-saturated if whenever $\{A_i\}_{i \in I}$ is a collection of internal sets with the infinite intersection property and the cardinality of $I$ is less than or equal to $\kappa$, $\bigcap_{i \in I} A_i \neq \emptyset$.

**Remark 3.1.** Remind that: (i) for each standard universum $U = V(X)$ there exists canonical language $\mathcal{L} = \mathcal{L}_U$, (ii) for each nonstandard universum $W = V(Y)$ there exists corresponding canonical nonstandard language $^{*}\mathcal{L} = \mathcal{L}_W$ [10].

**3\*. The restricted rules of conclusion.**

If $W \models A$ then $\neg A \not\models B$, where $B \in \mathcal{L} \wedge B \in ^{*}\mathcal{L}$.

Thus if $A$ holds in $W$ we cannot obtain from $\neg A$ any formula $B$ whatsoever.

**Remark 3.2.** We write $^{*}A \models A$ instead $W \models A$.

In this paper we apply the following hyper inductive definitions of a sets [18]

$$\exists S \forall \beta (\beta \in ^{*}\mathbb{N}) \left[ \beta \in S \Leftrightarrow_s \bigwedge_{0 \le \alpha < \beta} (\alpha \in S \Rightarrow_s \alpha^+ \in S) \right].$$
1. $\mathbf{Tr}[L] \lor \neg\mathbf{Tr}[L]$ [LEM]

2. Case One:

a $\mathbf{Tr}[L]$

b $L$ [2a: release by MP from T schema (2.35)]

c $\neg\mathbf{Tr}[L]$ [2b: definition of $L$]

d $\neg\mathbf{Tr}[L] \land \mathbf{Tr}[L]$ [2a, 2c: adjunction]

Case Two:

a $\neg\mathbf{Tr}[L]$

b $L$ [3a: definition of $L$ by MP]

c $\mathbf{Tr}[L]$ [3b: by MP from T schema (2.35)]

d $\neg\mathbf{Tr}[L] \land \mathbf{Tr}[L]$ [3a, 3c: adjunction]

4. $\neg\mathbf{Tr}[L] \land \mathbf{Tr}[L]$ [1-3: DP]

**Remark 2.12.** Liar easily avoided by restricted MP such that:

1. $\mathbf{Tr}[L] \not\vdash_{RMP} L$

2. $L \not\vdash_{RMP} \mathbf{Tr}[L]$

3. $\neg\mathbf{Tr}[L] \not\vdash_{RMP} L$

4. $L \not\vdash_{RMP} \neg\mathbf{Tr}[L]$

## §3. Nonconservative extension of the model theoretical NSA based on bivalent hyper Infinitary first-order logic $^{2}L_{\infty}^{\#}$ with restricted canonical rules of conclusion.

Extending the classical real numbers $\mathbb{R}$ to include infinite and infinitesimal quantities originally enabled D. Laugwitz [1] to view the delta distribution $\delta(x)$ as a nonstandard point function. Independently A. Robinson [2] demonstrated that distributions could be viewed as generalized polynomials. Luxemburg [3] and Sloan [4] presented an alternate representative of distributions as internal functions within the context of canonical Robinson's theory of nonstandard analysis. For further information on classical nonstandard real analysis, we refer to [8]-[11].

**Abbreviation 3.1.** In this paper we adopt the following notations. For a standard set $E$ we often write $E_{st}$. For a set $E_{st}$ let $^{\sigma}E_{st}$ be a set $^{\sigma}E_{st} = \{^{\sigma}x|x \in E_{st}\}$. We identify $z$ with $^{\sigma}z$ i.e., $z \equiv^{\sigma} z$ for all $z \in \mathbb{C}$. Hence, $^{\sigma}E_{st} = E_{st}$ if $E \subseteq \mathbb{C}$, e.g., $^{\sigma}\mathbb{C} = \mathbb{C}$, $^{\sigma}\mathbb{R} = \mathbb{R}$, $^{\sigma}P = P$, $^{\sigma}L_{\uparrow}^{+} = L_{\uparrow}^{+}$, etc. Let $^{\sigma}\mathbb{R}_{\approx}$, $^{\sigma}\mathbb{R}_{\approx_{+}}$, $^{\sigma}\mathbb{R}_{\text{fin}}$, $^{\sigma}\mathbb{R}_{\infty}$, and $^{\sigma}\mathbb{N}_{\infty}$ denote the sets of infinitesimal hyper-real numbers, positive infinitesimal hyper-real numbers, finite hyper-real numbers, infinite hyper-real numbers and infinite hyper natural numbers, respectively.

Note that $^{\sigma}\mathbb{R}_{\text{fin}} = ^{\sigma}\mathbb{R}/^{\sigma}\mathbb{R}_{\infty}$, $^{\sigma}\mathbb{C} = ^{\sigma}\mathbb{R} + i^{\sigma}\mathbb{R}$, $^{\sigma}\mathbb{C}_{\text{fin}} = ^{\sigma}\mathbb{R}_{\text{fin}} + i^{\sigma}\mathbb{R}_{\text{fin}}$.

Remind that Robinson nonstandard analysis (RNA) many developed using set-theoretical objects called superstructures [8]-[11]. A superstructure $V(S)$ over a set $S$ is defined in the following way

$$V_0(S) = S, V_{n+1}(S) = V_n(S) \cup (P(V_n(S)), V(S) = \bigcup_{n \in \mathbb{N}} V_n(S)). \quad (3.1)$$

Superstructures of the empty set consist of sets of infinite rank in the cumulative hierarchy and therefore do not satisfy the in...nity axiom. Making $S = \mathbb{R}$ will suffice for virtually any construction necessary in analysis.

Bounded formulas are formulas where all quantifiers occur in the form
**Definition 3.2.** [18]. A set $S \subset *\mathbb{N}$ is a hyper inductive if the following statement holds

$$ \bigwedge_{\alpha \in *\mathbb{N}} (\alpha \in S \Rightarrow_s \alpha^+ \in S), \qquad (3.3) $$

where $\alpha^+ \triangleq \alpha + 1$. Obviously a set $*\mathbb{N}$ is a hyper inductive. As we see later there is just one hyper inductive subset of $*\mathbb{N}$, namely $*\mathbb{N}$ itself.

We extend up Robinson nonstandard analysis (RNA) by adding the following postulate:

**4. Any hyper inductive set $S$ is internal.**

**Remark 3.3.** The statement 4 is not provable in ZFC but provable in set theory $NC_{\infty}^{\#}$, see [2]-[3]. Thus postulates 1-4 gives an nonconservative extension of RNA and we denote such extension by **NERNA**.

**Remark 3.4.** Note that NERNA of course based on the same hyper infinitary logic with Restricted Modus Ponens Rule as set theory $NC_{\infty}^{\#}$ [1]-[3].

Remind that in RNA the following induction principle holds.

**Theorem 3.1.** [6]. Assume that $S \subset *\mathbb{N}$ is internal set, then

$$ (1 \in S) \wedge \forall x[x \in S \Rightarrow x + 1] \Rightarrow S = *\mathbb{N}. \qquad (3.4) $$

In NERNA Theorem 1.1 also holds.

**Remark 3.5.** It follows from postulate 4 and Theorem 1.1 that any hyper inductive set $S$ is equivalent to $*\mathbb{N} : S = *\mathbb{N}$.

**Remark 3.6.** Note that the following statements are provable in $NC_{\infty}^{\#}$ [2]-[3]:

**5 Axiom of $\omega$-induction**

$$ \forall S(S \subset_s \mathbb{N}) \left\{ \forall \beta(\beta \in_s \mathbb{N}) \left[ \bigwedge_{0 \le \alpha < \beta} (\alpha \in_s S \Rightarrow_s \alpha^+ \in_s S) \right] \Rightarrow_s S = \mathbb{N} \right\}. \qquad (3.5) $$

**6 Axiom of hyper infinite induction**

$$ \forall S(S \subset *\mathbb{N}) \left\{ \forall \beta(\beta \in *\mathbb{N}) \left[ \bigwedge_{0 \le \alpha < \beta} (\alpha \in S \Rightarrow_s \alpha^+ \in S) \right] \Rightarrow_s S = *\mathbb{N} \right\}. \qquad (3.6) $$

Thus postulate 5 of the theory NERNA is provable in $NC_{\infty}^{\#}$.

**Rules of conclusion**

(1) Restricted Modus Ponens Rule (denoted by $\vdash_{RMP}$) the same as in set theory $NC_{\infty}^{\#}$.

(2) Restricted Modus Tollens Rule (denoted by $\vdash_{RMT}$) the same as in set theory $NC_{\infty}^{\#}$.

(3) MRR1 (1. Main Restricted rule of conclusion)

Let $\varphi(x)$ be a wff with one free variable $x$ and such that $\exists \bar{n}(\bar{n} \in *\mathbb{N}\mathbb{N}) \wedge \mathbf{V}(Y) \models \varphi(\bar{n})$, then for all $n \ge \bar{n} : \neg\varphi(n) \vdash_{RMP} B$, i.e., if statement $\varphi(\bar{n})$ holds in $\mathbf{V}(Y)$ we cannot obtain from $\neg\varphi(n)$, with $n \ge \bar{n}$ any formula $B$ whatsoever.

(4) MRR2 (2. Main Restricted rule of conclusion)

Let $\varphi(x)$ be a wff with one free variable $x$ and such that $\exists \bar{n}(\bar{n} \in *\mathbb{N}) \wedge \mathbf{V}(Y) \models \varphi(\bar{n})$, then for all $n \ge \bar{n} : \neg\varphi(n) \vdash_{RMP} B$, i.e., if statement $\varphi(\bar{n})$ holds in $\mathbf{V}(Y)$ we cannot obtain from $\neg\varphi(n)$, with $n \ge \bar{n}$ any formula $B$ whatsoever.

**Remark 3.5.** The MRR1,2 is necessarily in natural way, since by assumption $\neg\varphi(n)$ one obtains directly the apparent contradiction $\varphi(n) \wedge \neg\varphi(n)$ from which by unrestricted modus ponens rule (UMPR) one obtains $\varphi(n) \wedge \neg\varphi(n) \vdash_{UMPR} B$.
$$ \forall x(x \in y \Rightarrow \dots), \exists x(x \in y \Rightarrow \dots). \qquad (3.2) $$

A nonstandard embedding is a mapping

$$*: \mathbf{V}(X) \to \mathbf{V}(Y)$$

from a superstructure $\mathbf{V}(X)$ called the standard universum, into another superstructure

$\mathbf{V}(Y)$, called nonstandard universum, satisfying the following postulates:

1. $Y = *X$

2. Transfer Principle. For every bounded formula $\Phi(x_1, \dots, x_n)$ and elements $a_1, \dots, a_n \in \mathbf{V}(X)$, the property $\Phi$ is true for $a_1, \dots, a_n$ in the standard universum if and only if it is true for $*a_1, \dots, *a_n$ in the nonstandard universum:

$$ \langle \mathbf{V}(X), \in \rangle \models \Phi(a_1, \dots, a_n) \Leftrightarrow \langle \mathbf{V}(Y), \in \rangle \models \Phi(*a_1, \dots, *a_n). $$

3. Non-triviality. For every infinite set $A$ in the standard universum, the set $\{*a|a \in A\}$ is a proper subset of $*A$.

**Definition 3.1.** [10]. A set $x$ is *internal* if and only if $x$ is an element of $*A$ for some element $A$ of $\mathbf{V}(\mathbb{R})$. Let $X$ be a set with $A = \{A_i\}_{i \in I}$ a family of subsets of $X$. Then the collection $A$ has the infinite intersection property, if any infinite subcollection $J \subset I$ has non-empty intersection. Nonstandard universum is $\kappa$-saturated if whenever $\{A_i\}_{i \in I}$ is a collection of internal sets with the infinite intersection property and the cardinality of $I$ is less than or equal to $\kappa$, $\bigcap_{i \in I} A_i \neq \emptyset$.

**Remark 3.1.** Remind that: (i) for each standard universum $U = \mathbf{V}(X)$ there exists canonical language $\mathcal{L} = \mathcal{L}_U$, (ii) for each nonstandard universum $W = \mathbf{V}(Y)$ there exists corresponding canonical nonstandard language $*\mathcal{L} = \mathcal{L}_W$ [10].

**3\*. The restricted rules of conclusion.**

If $W \models A$ then $\neg A \not\vdash B$, where $B \in \mathcal{L} \wedge B \in *\mathcal{L}$.

Thus if $A$ holds in $W$ we cannot obtain from $\neg A$ any formula $B$ whatsoever.

**Remark 3.2.** We write $* \models A$ instead $W \models A$.

In this paper we apply the following hyper inductive definitions of a sets [18]

$$ \exists S \forall \beta (\beta \in *\mathbb{N}) \left[ \beta \in S \Leftrightarrow_s \bigwedge_{0 \le \alpha < \beta} (\alpha \in S \Rightarrow_s \alpha^+ \in S) \right]. $$

**Definition 3.2.** [18]. A set $S \subset *\mathbb{N}$ is a *hyper inductive* if the following statement holds

$$ \bigwedge_{\alpha \in *\mathbb{N}} (\alpha \in S \Rightarrow_s \alpha^+ \in S), \qquad (3.3) $$

where $\alpha^+ \triangleq \alpha + 1$. Obviously a set $*\mathbb{N}$ is a hyper inductive. As we see later there is just one hyper inductive subset of $*\mathbb{N}$, namely $*\text{Nitself}$.

We extend up Robinson nonstandard analysis (RNA) by adding the following postulate:

**4. Any hyper inductive set $S$ is internal.**

**Remark 3.3.** The statement 4 is not provable in ZFC but provable in set theory $\mathbb{NC}^#_{\infty}$, see [2]-[3]. Thus postulates 1-4 gives an nonconservative extension of RNA and we denote such extension by **NERNA**.

**Remark 3.4.** Note that NERNA of course based on the same hyper infinitary logic with Restricted Modus Ponens Rule as set theory $\mathbb{NC}^#_{\infty}$ [1]-[3]. Remind that in RNA the following induction principle holds.
**Example 3.1.** Remind the proof of the following statement:

**Theorem 3.2.** The structure $(\mathbb{N}, <)$ is a well-ordered set.

**Proof.** Let $X$ be a nonempty subset of $\mathbb{N}$. Suppose $X$ does not have a $<$-least element. Then consider the set $\mathbb{N}\setminus X$.

Case (1) $\mathbb{N}\setminus X = \emptyset$. Then $X = \mathbb{N}$ and so $0$ is a $<$-least element. Contradiction.

Case (2) $\mathbb{N}\setminus X \neq \emptyset$. Then $1 \in \mathbb{N}\setminus X$ otherwise $1$ is a $<$-least element. Contradiction.

Case (3) $\mathbb{N}\setminus X \neq \emptyset$. Assume now that there exists an $n \in \mathbb{N}\setminus X$ such that $n \neq 1$.

Since we have supposed that $X$ does not have a least element, thus $n+1 \notin X$.

Thus we see that for all $n: n \in \mathbb{N}\setminus X$ implies that $n+1 \in \mathbb{N}\setminus X$. We can conclude by induction that $n \in \mathbb{N}\setminus X$ for all $n \in \mathbb{N}$. Thus $\mathbb{N}\setminus X = \mathbb{N}$ implies $X = \emptyset$.

This is a contradiction to $X$ being a nonempty subset of $\mathbb{N}$.

**Remark 3.6.** (i) The proof of the Theorem 3.2 is an example proof by a contradiction.

Remind that a mathematical proof employing proof by contradiction usually proceeds as follows:

1. The proposition to be proved is $P$.

2. We assume $P$ to be false, i.e., we assume $\models \neg P$.

3. It is then shown that $\neg P$ implies falsehood. This is typically accomplished by deriving two mutually contradictory assertions, $Q$ and $\neg Q$, and appealing to the law of noncontradiction.

4. Since assuming $P$ to be false leads to a contradiction, it is concluded that $P$ is in fact true.

(ii) The statement of the Theorem 3.2 obviously is unprovable by a contradiction under MRR2. Note that in the Case (3) there is an $\bar{n} \neq 1, \bar{n} \in X$ and $\bar{n} \notin \mathbb{N}\setminus X$. Thus induction hypothesis $\models \bar{n} \in \mathbb{N}\setminus X$ is not holds since $\bar{n} \notin \mathbb{N}\setminus X \land \bar{n} \in \mathbb{N}\setminus X$ is a contradiction and by MRR2

$$ \bar{n} \in \mathbb{N}\setminus X \vdash_{\mathrm{RMP}} \bar{n} + 1 \in \mathbb{N}\setminus X. $$

(iii) Note that proof of the Theorem 3.2 mentioned above completely abnormal in fact even in point view of classical proof theory, since basic assumption $\bar{n} \notin \mathbb{N}\setminus X$ which is employed in proof by contradiction, contradicts with induction hypothesis $\models \bar{n} \in \mathbb{N}\setminus X$.

**Example 3.2.** (i) We set now $X_1 = *\mathbb{N}\setminus \emptyset$, thus $*\mathbb{N}\setminus X_1 = \mathbb{N}$. In contrast with a set $X$ mentioned in Example 3.1, the assumption $n \in *\mathbb{N}\setminus X_1$ implies that $n+1 \in *\mathbb{N}\setminus X_1$ if and only if $n$ is finite, since for any infinite $n \in *\mathbb{N}\setminus \emptyset$ the assumption $n \in *\mathbb{N}\setminus X_1$ contradicts with a true statement $V(Y) \models n \notin *\mathbb{N}\setminus X_1 = \mathbb{N}$ and therefore in accordance with MRR we cannot obtain for any infinite $n$ from formula $n \in *\mathbb{N}\setminus X_1$ any formula $B$ whatsoever.

**Remark 3.7.** Notice in order to prove an statement $G = \forall n(n \in *\mathbb{N})P(n)$ by induction one needs to proof that: $P(n) \vdash_{\mathrm{RMP}} P(n+1)$, i.e. by assuming that $P(n)$ is true and then by RMP proving $P(n+1)$. Thus:

(i) any proof by hyperinfinite induction bused on additional assumption that

$$ \vdash_{\mathrm{RMP}} \exists \bar{n} (\bar{n} \in_s *\mathbb{N}) [\neg P(\bar{n})]. \qquad (3.7) $$

(ii) any proof by $\omega$-induction bused on additional assumption that

$$ \vdash_{\mathrm{RMP}} \exists \bar{n} (\bar{n} \in_s \mathbb{N}) [\neg P(\bar{n})]. \qquad (3.8) $$

**Definition 3.3.** $\kappa$ is a natural number if $\kappa \in_s X$ for every set $X$ such that $0 \in_s X$ and, for any $\lambda$, if $\lambda \in_s X$ then $\lambda + 1 \in_s X$, i.e. $\lambda \in_s X \vdash_{\mathrm{RMP}} \lambda + 1 \in_s X$.
**Theorem 3.1.[6].** Assume that $S \subset *\mathbb{N}$ is internal set, then

$$ (1 \in S) \wedge \forall x[x \in S \Rightarrow x + 1] \Rightarrow S = *\mathbb{N}. \qquad (3.4) $$

In NERNA Theorem 1.1 also holds.

**Remark 3.5.** It follows from postulate 4 and Theorem 1.1 that any hyper inductive set $S$ is equivalent to $*\mathbb{N} : S = *\mathbb{N}$.

**Remark 3.6.** Note that the following statements are provable in $NC_{\infty}^{\#}$ [2]-[3]:

**5 Axiom of ω-induction**

$$ \forall S(S \subset_s \mathbb{N}) \left\{ \forall \beta(\beta \in_s \mathbb{N}) \left[ \bigwedge_{0 \le \alpha < \beta} (\alpha \in_s S \Rightarrow_s \alpha^+ \in_s S) \right] \Rightarrow_s S = \mathbb{N} \right\}. \quad (3.5) $$

**6 Axiom of hyper infinite induction**

$$ \forall S(S \subset *\mathbb{N}) \left\{ \forall \beta(\beta \in *\mathbb{N}) \left[ \bigwedge_{0 \le \alpha < \beta} (\alpha \in S \Rightarrow_s \alpha^+ \in S) \right] \Rightarrow_s S = *\mathbb{N} \right\}. \quad (3.6) $$

Thus postulate 5 of the theory NERNA is provable in $NC_{\infty}^{\#}$.

**Rules of conclusion**

(1) Restricted Modus Ponens Rule (denoted by ⊢RMP) the same as in set theory $NC_{\infty}^{\#}$.

(2) Restricted Modus Tollens Rule (denoted by ⊢RMT) the same as in set theory $NC_{\infty}^{\#}$.

(3) MRR1 (1. Main Restricted rule of conclusion)

Let $\varphi(x)$ be a wff with one free variable $x$ and such that $\exists \bar{n} (\bar{n} \in *\mathbb{N} \wedge \mathbb{N}) \wedge \mathbf{V}(Y) \vdash \varphi(\bar{n})$, then for all $n \ge \bar{n} : \neg \varphi(n) \vdash_{RMP} B$, i.e., if statement $\varphi(\bar{n})$ holds in $\mathbf{V}(Y)$ we cannot obtain from $\neg \varphi(n)$, with $n \ge \bar{n}$ any formula $B$ whatsoever.

(4) MRR2 (2. Main Restricted rule of conclusion)

Let $\varphi(x)$ be a wff with one free variable $x$ and such that $\exists \bar{n} (\bar{n} \in *\mathbb{N}) \wedge \mathbf{V}(Y) \vdash \varphi(\bar{n})$, then for all $n \ge \bar{n} : \neg \varphi(n) \vdash_{RMP} B$, i.e., if statement $\varphi(\bar{n})$ holds in $\mathbf{V}(Y)$ we cannot obtain from $\neg \varphi(n)$, with $n \ge \bar{n}$ any formula $B$ whatsoever.

**Remark 3.5.** The MRR1,2 is necessarily in natural way, since by assumption $\neg \varphi(n)$ one obtains directly the apparent contradiction $\varphi(n) \wedge \neg \varphi(n)$ from which by unrestricted modus ponens rule (UMPR) one obtains $\varphi(n) \wedge \neg \varphi(n) \vdash_{UMPR} B$.

**Example 3.1.** Remind the proof of the following statement:

**Theorem 3.2.** The structure $(\mathbb{N}, <)$ is a well-ordered set.

**Proof.** Let $X$ be a nonempty subset of $\mathbb{N}$. Suppose $X$ does not have a $<$-least element. Then consider the set $\mathbb{N} \setminus X$.

Case (1) $\mathbb{N} \setminus X = \emptyset$. Then $X = \mathbb{N}$ and so 0 is a $<$-least element. Contradiction.

Case (2) $\mathbb{N} \setminus X \neq \emptyset$. Then $1 \in \mathbb{N} \setminus X$ otherwise 1 is a $<$-least element. Contradiction.

Case (3) $\mathbb{N} \setminus X \neq \emptyset$. Assume now that there exists an $n \in \mathbb{N} \setminus X$ such that $n \neq 1$. Since we have supposed that $X$ does not have a least element, thus $n + 1 \notin X$.

Thus we see that for all $n : n \in \mathbb{N} \setminus X$ implies that $n + 1 \in \mathbb{N} \setminus X$. We can conclude by induction that $n \in \mathbb{N} \setminus X$ for all $n \in \mathbb{N}$. Thus $\mathbb{N} \setminus X = \mathbb{N}$ implies $X = \emptyset$. This is a contradiction to $X$ being a nonempty subset of $\mathbb{N}$.

**Remark 3.6.** (i) The proof of the Theorem 3.2 is an example proof by a contradiction. Remind that a mathematical proof employing proof by contradiction usually proceeds as follows:

1. The proposition to be proved is $P$.
**We remind now some basic theorem and definitions related to classical naturals.**

**Definition 3.4.** [20]. $κ$ is a natural number if $κ$ belongs to every set $X$ such that $0 \in X$ and, for any $λ$, if $λ \in X$ then $λ + 1 \in X$.

(As usual, $j, k, \dots, n$ will denote natural numbers.)

**Remark 3.8.** [20]. If the set of all natural numbers exists, we call it $\mathbb{N}$. But it is not necessary for us to assume now that $\mathbb{N}$ exists. The assumption that $\mathbb{N}$ exists is a form of what is called the Axiom of Infinity.

**Proposition 3.1.** [20] For any $κ$, $\{λ|λ ≤ κ\}$ exists.

**Proof.** Let $\overline{A} = κ$. The desired set is $\{\overline{B}|B \in_s P(A)\}$, which exists by the Axiom of Replacement.

**Theorem 3.3.** [20].

(a) 0 is a natural number, $0 ≤ k$

(b) If $k$ is a natural number so is $k + 1$, if $k < λ$ then $k + 1 ≤ λ$.

(c) (Induction) Suppose that $P(0)$ ('P holds for 0'); and that, for any natural number $n$, $P(n) \Rightarrow P(n + 1)$ holds. Then for every $n$, $P(n)$ holds.

**Proof** [20] (a) and (b) are very easy. For (c), suppose that the whole hypothesis of (c) holds, but that, for some particular $\overline{n}, P(\overline{n})$ fails, i.e. $\neg P(\overline{n})$ holds. Put

$$X = \{m < \overline{n}|P(m)\} \qquad (3.9)$$

$X$ exists since $X = \{λ|λ < η$ and $λ$ is a natural number and $P(λ)\}$, which exists by Proposition 3.1 and the Separation Axiom. Obviously, $0 \in X$ by Theorem 3.3 (a). It will be enough to show that: $λ + 1 \in X$ whenever $λ \in X$ - as then $X$ is 'an $X$' as in Definition 3.3, so, by Definition 3.3, the natural number $\overline{n} \in X$, and so $P(\overline{n})$ holds, a contradiction. Suppose then that $λ \in X$, so that $λ ≤ \overline{n}, λ$ is a natural number, and $P(λ)$. By our hypothesis (in (c)), $P(λ + 1)$. By (b), $λ + 1$ is a natural number. Also, $λ < \overline{n}$, as $P(\overline{n})$ fails. Hence $λ + 1 ≤ n$, by Theorem 3.3 (b). So $λ + 1 \in X$, as desired.

**Remark 3.9.** Note that proof of the Theorem 3.3 mentioned above completely abnormal since definition (3.9) incorrect. Correct definition reads

$$X = \{m < \overline{n}|T[P(m)] \wedge P(m)\} \qquad (3.10)$$

where $T[A]$ is a truth predicate such that for any well formed closed formula $A$ of ZFC [24]

$$T[A] \Leftrightarrow A. \qquad (3.11)$$

However as well known such truth predicate is not exists by Curry's paradox. Thus a set $X$ is not exists in general case.

**Definition 3.5.** An element $x$ is said to be a first element of the linearly ordered set $A$ (with respect to the relation $R$) if $xRy$ for all $y \in A$. On the other hand, if $yRx$ for all $y$, then $x$ is said to be a last element of $A$ (with respect to $R$). Generally speaking, not every set has a first or last element; but if such an element exists, then it is uniquely determined.

**Theorem 3.4.** [20]-[22]. In a finite non-empty subset $X$ of a linearly ordered set $A$ there is a first element and a last element of $X$.

**Proof.** The proof is by induction on the number of elements of $X$. If $X$ has only one
2. We assume $P$ to be false, i.e., we assume $\neg P$.

3. It is then shown that $\neg P$ implies falsehood. This is typically accomplished by deriving two mutually contradictory assertions, $Q$ and $\neg Q$, and appealing to the law of noncontradiction.

4. Since assuming $P$ to be false leads to a contradiction, it is concluded that $P$ is in fact true.

(ii) The statement of the Theorem 3.2 obviously is unprovable by a contradiction under MRR2. Note that in the Case (3) there is an $\bar{n} \neq 1, \bar{n} \in X$ and $\bar{n} \notin \mathbb{N} \setminus X$. Thus induction hypothesis $\models \bar{n} \in \mathbb{N} \setminus X$ is not holds since $\bar{n} \notin \mathbb{N} \setminus X \land \bar{n} \in \mathbb{N} \setminus X$ is a contradiction and by MRR2

$$ \bar{n} \in \mathbb{N} \setminus X \vdash_{\mathrm{RMP}} \bar{n} + 1 \in \mathbb{N} \setminus X. $$

(iii) Note that proof of the Theorem 3.2 mentioned above completely abnormal in fact even in point view of classical proof theory, since basic assumption $\bar{n} \notin \mathbb{N} \setminus X$ which is employed in proof by contradiction, contradicts with induction hypothesis $\models \bar{n} \in \mathbb{N} \setminus X$.

**Example 3.2.** (i) We set now $X_1 = *\mathbb{N}\mathbb{N}$, thus $*\mathbb{N}\mathbb{N}X_1 = \mathbb{N}$. In contrast with a set $X$ mentioned in Example 3.1, the assumption $n \in *\mathbb{N}\mathbb{N}X_1$ implies that $n + 1 \in *\mathbb{N}\mathbb{N}X_1$ if and only if $n$ is finite, since for any infinite $n \in *\mathbb{N}\mathbb{N}$ the assumption $n \in *\mathbb{N}\mathbb{N}X_1$ contradicts with a true statement $V(Y) \models n \notin *\mathbb{N}\mathbb{N}X_1 = \mathbb{N}$ and therefore in accordance with MRR we cannot obtain for any infinite $n$ from formula $n \in *\mathbb{N}\mathbb{N}X_1$ any formula $B$ whatsoever.

**Remark 3.7.** Notice in order to prove an statement $G = \forall n(n \in *\mathbb{N})P(n)$ by induction one needs to proof that: $P(n) \vdash_{\mathrm{RMP}} P(n + 1)$, i.e. by assuming that $P(n)$ is true and then by RMP proving $P(n + 1)$. Thus:

(i) any proof by hyperinfinite induction bused on additional assumption that

$$ \vdash_{\mathrm{RMP}} \exists \bar{n} (\bar{n} \in_s *\mathbb{N}) [\neg P(\bar{n})]. \qquad (3.7) $$

(ii) any proof by $\omega$-induction bused on additional assumption that

$$ \vdash_{\mathrm{RMP}} \exists \bar{n} (\bar{n} \in_s \mathbb{N}) [\neg P(\bar{n})]. \qquad (3.8) $$

**Definition 3.3.** $\kappa$ is a natural number if $\kappa \in_s X$ for every set $X$ such that $0 \in_s X$ and, for any $\lambda$, if $\lambda \in_s X$ then $\lambda + 1 \in_s X$, i.e. $\lambda \in_s X \vdash_{\mathrm{RMP}} \lambda + 1 \in_s X$.

**We remind now some basic theorem and definitions related to classical naturals.**

**Definition 3.4.** [20]. $\kappa$ is a natural number if $\kappa$ belongs to every set $X$ such that $0 \in_s X$ and, for any $\lambda$, if $\lambda \in_s X$ then $\lambda + 1 \in_s X$.

(As usual, $j, k, \dots, n$ will denote natural numbers.)

**Remark 3.8.** [20]. If the set of all natural numbers exists, we call it $\mathbb{N}$. But it is not necessary for us to assume now that $\mathbb{N}$ exists. The assumption that $\mathbb{N}$ exists is a form of what is called the Axiom of Infinity.

**Proposition 3.1.** [20] For any $\kappa$, $\{\lambda|\lambda \leq \kappa\}$ exists.

**Proof.** Let $\overline{\mathbb{A}} = \kappa$. The desired set is $\{\overline{\mathbb{B}}|B \in_s P(A)\}$, which exists by the Axiom of Replacement.

**Theorem 3.3.** [20].

(a) 0 is a natural number, $0 \leq k$

(b) If $k$ is a natural number so is $k + 1$, if $k < \lambda$ then $k + 1 \leq \lambda$.
element, then the theorem is obvious. Suppose that the theorem holds for subsets with $n$ elements. Let $X = Y \cup \{a\}$ where $a \notin Y$ and $Y$ has $n$ elements. Let $b_1$ be the first and $b_2$ the last element of $Y$. Since $A$ is linearly ordered, either a precedes $b_1$ or $b_1$ precedes $a$. That element which precedes the other is clearly the first element of $Y$. Similarly we show that one of the elements $a$ and $b_2$ is the last element of $X$.

**Corollary 3.1.** Every finite subset $A$ of $\mathbb{N}$ has a first element and also a last element.

**Proof.** From Theorem 3.4 by definitions.

**Theorem 3.5.** [20]. (a) (**The least element principle**). If for some $\bar{n}, P(\bar{n})$, then there is a minimal (which is here the same as minimum) $n$ such that $P(n)$.

(b) (**Course-of-values induction**). If, for any $n$, if $Q(m)$ holds for all $m < n$, then $Q(n)$; then, for all $n, Q(n)$.

**Proof** of (a). Suppose $P(\bar{n})$. If ($P(m)$ for no $m < \bar{n}$, then $\bar{n}$ is minimal as desired. Otherwise $\{k \in W(\bar{n}) | P(k)\} (W(\bar{n}) = \{m | m < n\})$ is non-empty, and so, being finite, has a least element $m$, by Corollary 3.1. It is easy to see that $m$ is the least number with the property $P$, as desired.

**Proof** of (b). Assume the hypothesis of (b) holds and that, for some $n$, $Q(n)$. fails. By (a) let $k$ be the least such $n$. Thus $Q(m)$ holds for all $m < k$, so by our hypothesis, $Q(k)$ holds, a contradiction.

**Theorem 3.6.** (s-Induction) Let $P(x)$ be wff of $\mathbf{NC}_{\infty}^{\#}$ with a free variable $x$. Suppose that

$$ \mathbf{T}^{\#}[P(0)] \wedge \mathbf{T}^{\#}[P(0)] \qquad (3.12) $$

($P$ holds for $0$); $\mathbf{T}^{\#}[P(0)] \iff_s P(0)$, and that, for any natural number $n$,

$$ P(n) \Rightarrow_s P(n + 1) \qquad (3.13) $$

and for every $n$,

$$ \mathbf{T}^{\#}[P(n)] \iff_s P(n), \qquad (3.14) $$

i.e. or every given $n, P(n)$ is s-sentence. Then for every $n \in \mathbb{N}$, $P(n)$ holds, i.e. $\forall n \{\mathbf{T}^{\#}[P(n)] \iff_s P(n)\}$.

**Proof.** Suppose that the whole hypothesis mentioned above holds, but that, for some particular $\bar{n}, P(\bar{n})$ fails, i.e. $\neg P(\bar{n})$ holds. Put

$$ X = \{m < \bar{n} | \mathbf{T}^{\#}[P(m)] \wedge P(m)\} \qquad (3.15) $$

$X$ exists since $X = \{\lambda|\lambda < \eta$ and $\lambda$ is a natural number and $\mathbf{T}^{\#}[P(m)] \wedge P(\lambda)\}$, which exists by Proposition 3.1 and the Separation Axiom. Obviously, $0 \in X$ by Theorem 3.3 (a). It will be enough to show that: $\lambda + 1 \in_s X$ whenever $\lambda \in_s X$ - as then $X$ is 'an $X$' as in Definition 3.3, so, by Definition 3.3, the natural number $\bar{n} \in_s X$,

and so $P(\bar{n})$ holds, a contradiction. Suppose then that $\lambda \in_s X$, so that $\lambda \le \bar{n}, \lambda$ is a natural number, and $P(\lambda)$. By our hypothesis (in (3.13)), $P(\lambda + 1)$. By (b), $\lambda + 1$ is a natural number. Also, $\lambda < \bar{n}$, as $P(\bar{n})$ fails. Hence $\lambda + 1 \le n$, by Theorem 3.3 (b). So $\lambda + 1 \in_s X$, as desired.

**Theorem 3.7.** [23] Any finite nonempty subset $X$ of $\mathbb{N}$ has minimal and maximal members.

**Proof** [23]. Let $X_n$ consist of $x_1, \dots, x_n$. Define $m_1 = x_1$ and $m_k$ as $x_k$ if $x_k < m_{k-1}$ and $m_{k-1}$ otherwise. Then $m_n$ will be minimal. Similarly, $X$ has a maximal element.

**Remark 3.7.** This proof in fact based on assumption (the induction hypothesis) that the theorem holds for $X_{k-1}$ consist of $x_1, \dots, x_{k-1}$, i.e. $m_{k-1} = \min\{x_1, \dots, x_{k-1}\}$, then it
(c) (Induction) Suppose that $P(0)$ ('P holds for 0'); and that, for any natural number $n$, $P(n) \Rightarrow P(n+1)$ holds. Then for every $n$, $P(n)$ holds.

**Proof** [20] (a) and (b) are very easy. For (c), suppose that the whole hypothesis of (c) holds, but that, for some particular $\bar{n}, P(\bar{n})$ fails, i.e. $\neg P(\bar{n})$ holds. Put

$$X = \{m < \bar{n} | P(m)\} \qquad (3.9)$$

$X$ exists since $X = \{\lambda|\lambda < \eta$ and $\lambda$ is a natural number and $P(\lambda)\}$, which exists by Proposition 3.1 and the Separation Axiom. Obviously, $0 \in X$ by Theorem 3.3 (a). It will be enough to show that: $\lambda + 1 \in X$ whenever $\lambda \in X$ - as then $X$ is 'an $X'$ as in Definition 3.3, so, by Definition 3.3, the natural number $\bar{n} \in X$, and so $P(\bar{n})$ holds, a contradiction. Suppose then that $\lambda \in X$, so that $\lambda \le \bar{n}, \lambda$ is a natural number, and $P(\lambda)$. By our hypothesis (in (c)), $P(\lambda + 1)$. By (b), $\lambda + 1$ is a natural number. Also, $\lambda < \bar{n}$, as $P(\bar{n})$ fails. Hence $\lambda + 1 \le n$, by Theorem 3.3 (b). So $\lambda + 1 \in X$, as desired.

**Remark 3.9.** Note that proof of the Theorem 3.3 mentioned above completely abnormal since definition (3.9) incorrect. Correct definition reads

$$X = \{m < \bar{n} | \mathbf{T}[P(m)] \wedge P(m)\} \qquad (3.10)$$

where $\mathbf{T}[A]$ is a truth predicate such that for any well formed closed formula $A$ of ZFC [24]

$$\mathbf{T}[A] \Leftrightarrow A. \qquad (3.11)$$

However as well known such truth predicate is not exists by Curry's paradox. Thus a set $X$ is not exists in general case.

**Definition 3.5.** An element $x$ is said to be a first element of the linearly ordered set $A$ (with respect to the relation $R$) if $xRy$ for all $y \in A$. On the other hand, if $yRx$ for all $y$, then $x$ is said to be a last element of $A$ (with respect to $R$). Generally speaking, not every set has a first or last element; but if such an element exists, then it is uniquely determined.

**Theorem 3.4.** [20]-[22]. In a finite non-empty subset $X$ of a linearly ordered set $A$ there is a first element and a last element of $X$.

**Proof.** The proof is by induction on the number of elements of $X$. If $X$ has only one element, then the theorem is obvious. Suppose that the theorem holds for subsets with $n$ elements. Let $X = Y \cup \{a\}$ where $a \notin Y$ and $Y$ has $n$ elements. Let $b_1$ be the first and $b_2$ the last element of $Y$. Since $A$ is linearly ordered, either a precedes $b_1$ or $b_1$ precedes $a$. That element which precedes the other is clearly the first element of $Y$. Similarly we show that one of the elements $a$ and $b_2$ is the last element of $X$.

**Corollary 3.1.** Every finite subset $A$ of $\mathbb{N}$ has a first element and also a last element. **Proof.** From Theorem 3.4 by definitions.

**Theorem 3.5.** [20]. (a) **(The least element principle)**. If for some $\bar{n}, P(\bar{n})$, then there is a minimal (which is here the same as minimum) $n$ such that $P(n)$. (b) **(Course-of-values induction)**. If, for any $n$, if $Q(m)$ holds for all $m < n$, then $Q(n)$; then, for all $n, Q(n)$.

**Proof** of (a). Suppose $P(\bar{n})$. If $(P(m)$ for no $m < \bar{n}$, then $\bar{n}$ is minimal as desired. Otherwise $\{k \in W(\bar{n}) | P(k)\} (W(\bar{n}) = \{m|m < n\})$ is non-empty, and so, being finite, has a least element $m$, by Corollary 3.1. It is easy to see that $m$ is the least number with the property $P$, as desired.
follows $m_{k-1} = \min\{x_1, \dots, x_{k-1}\} \Rightarrow m_k = \min\{x_1, \dots, x_k\}$ and by induction we conclude
that for all $n \in \mathbb{N}$, $m_n = \min\{x_1, \dots, x_n\}$.

**Definition 3.6.** An element $x$ is said to be a first element of the linearly s-ordered set $A$ (with respect to the s-relation $R$) if $xRy$ for all $y \in_s A$. On the other hand, if $yRx$ for all $y \in_s A$, then $x$ is said to be a last element of $A$ (with respect to $R$). Generally speaking, not every set has a first or last element; but if such an element exists, then it is uniquely determined.

**Abbreviation 3.2** Let $X_n(A), \bar{X}_n(A) = n$ be s-finite non-empty subset of a linearly s-ordered set $A$ such that there is a first element and a last element of $X_n$. We shall abbreviate: $[X_n(A), (\bar{X}_n(A) = n)$ is a s-finite non-empty subset of a linearly s-ordered set $A$ such that there is a first element and a last element of $X_n(A)] \Leftrightarrow \hat{X}_n(A)$. Under assumption

$$ \Vdash_{\mathrm{RMP}} \exists m(m \in_s \mathbb{N}) \exists X_m(A) [\neg \hat{X}_m(A)]. \qquad (3.16) $$

by axiom of $\omega$-induction we obtain

$$ \forall X_n(A) \left[ \bigwedge_{n \in \mathbb{N}} (\hat{X}_n(A) \Rightarrow_s \hat{X}_{n+1}(A)) \right] \Rightarrow_s \forall n \forall X_n(A) [\hat{X}_n(A)]. \qquad (3.17) $$

In particular for $A = \mathbb{N}$ under assumption

$$ \Vdash_{\mathrm{RMP}} \exists m(m \in_s \mathbb{N}) \exists X_m(\mathbb{N}) [\neg \hat{X}_m(\mathbb{N})]. \qquad (3.18) $$

by axiom of $\omega$-induction we obtain

$$ \forall X_n(\mathbb{N}) \left[ \bigwedge_{n \in \mathbb{N}} (\hat{X}_n(\mathbb{N}) \Rightarrow_s \hat{X}_{n+1}(\mathbb{N})) \right] \Rightarrow_s \forall n \forall X_n(A) [\hat{X}_n(\mathbb{N})]. \qquad (3.19) $$

**7 Axiom of existence non well-ordered s-finite subset of $\mathbb{N}$**

$$ \exists m(m \in_s \mathbb{N}) \exists X_m(\mathbb{N}) [\neg \hat{X}_m(\mathbb{N})]. \qquad (3.20) $$

## §4. Internal Set Theory IST.

The axiomatics IST (Internal Set Theory) was presented in 1977 [19] and in a sense formulates within first-order language the behaviour of standard and internal sets of a nonstandard model of ZFC. This were done by adding the unary standardness predicate "st" to the language of ZFC as well as adding to the axioms of ZFC three new axiom schemes involving the predicate "st": **Idealization**, **Standardization** and **Transfer**.

**Remark 4.1.** Formulas which do not use the predicate st are called **internal formulas** (or $\epsilon$-formulas) and formulas that use this new predicate are called **external formulas** (or st-$\epsilon$-formulas). A formula $\varphi$ is standard if only standard constants occur in $\varphi$.

**Abbreviation 4.1.** We denote a set of the all naturals by $\mathbb{N}^\# $ and a set of the all finite naturals by $\mathbb{N}$.

**Abbreviation 4.2.** We write fin($x$) meaning 'x is finite'. Let $\varphi(x)$ be a st-$\epsilon$-formula:

1. $\forall^{\mathrm{st}} x \varphi(x)$ abbreviates $\forall x (\mathrm{st}(x) \Rightarrow \varphi(x))$. 2. $\exists^{\mathrm{st}} x \varphi(x)$ abbreviates $\exists x (\mathrm{st}(x) \wedge \varphi(x))$.

3. $\forall^{\mathrm{fin}} x \varphi(x)$ abbreviates $\forall x (\mathrm{fin}(x)) \Rightarrow \varphi(x)$. 4. $\exists^{\mathrm{fin}} x \varphi(x)$ abbreviates $\exists x (\mathrm{fin}(x) \wedge \varphi(x))$.

5. $\forall^{\mathrm{stfin}} x \varphi(x)$ abbreviates $\forall x (\mathrm{st}(x) \wedge \mathrm{fin}(x)) \Rightarrow \varphi(x)$.

6. $\exists^{\mathrm{stfin}} x \varphi(x)$ abbreviates $\exists x (\mathrm{st}(x) \wedge \mathrm{fin}(x) \wedge \varphi(x))$.

The fundamental axioms of IST :
**Proof** of (b). Assume the hypothesis of (b) holds and that, for some $n$, $Q(n)$. fails. By (a) let $k$ be the least such $n$. Thus $Q(m)$ holds for all $m < k$, so by our hypothesis, $Q(k)$ holds, a contradiction.

**Theorem 3.6.** (s-Induction) Let $P(x)$ be wff of $\mathbf{NC}_{\infty}^{\#}$ with a free variable $x$. Suppose that

$$ \mathbf{T}^{\#}[P(0)] \wedge \mathbf{T}^{\#}[P(0)] \qquad (3.12) $$

($P$ holds for $0$); $\mathbf{T}^{\#}[P(0)] \Leftrightarrow_P P(0)$, and that, for any natural number $n$,

$$ P(n) \Rightarrow_P P(n+1) \qquad (3.13) $$

and for every $n$,

$$ \mathbf{T}^{\#}[P(n)] \Leftrightarrow_P P(n), \qquad (3.14) $$

i.e. or every given $n$, $P(n)$ is s-sentence. Then for every $n \in \mathbb{N}$, $P(n)$ holds, i.e. $\forall n \{\mathbf{T}^{\#}[P(n)] \Leftrightarrow_P P(n)\}$.

**Proof.** Suppose that the whole hypothesis mentioned above holds, but that, for some particular $\bar{n}, P(\bar{n})$ fails, i.e. $\neg P(\bar{n})$ holds. Put

$$ X = \{m < \bar{n} | \mathbf{T}^{\#}[P(m)] \wedge P(m)\} \qquad (3.15) $$

$X$ exists since $X = \{\lambda | \lambda < \eta$ and $\lambda$ is a natural number and $\mathbf{T}^{\#}[P(m)] \wedge P(\lambda)\}$, which exists by Proposition 3.1 and the Separation Axiom. Obviously, $0 \in X$ by Theorem 3.3 (a). It will be enough to show that: $\lambda + 1 \in_S X$ whenever $\lambda \in_S X$ - as then $X$ is 'an $X'$ as in Definition 3.3, so, by Definition 3.3, the natural number $\bar{n} \in_S X$,

and so $P(\bar{n})$ holds, a contradiction. Suppose then that $\lambda \in_S X$, so that $\lambda \le \bar{n}$, $\lambda$ is a natural number, and $P(\lambda)$. By our hypothesis (in (3.13)), $P(\lambda + 1)$. By (b), $\lambda + 1$ is a natural number. Also, $\lambda < \bar{n}$, as $P(\bar{n})$ fails. Hence $\lambda + 1 \le n$, by Theorem 3.3 (b). So $\lambda + 1 \in_S X$, as desired.

**Theorem 3.7.** [23] Any finite nonempty subset $X$ of $\mathbb{N}$ has minimal and maximal members.

**Proof** [23]. Let $X_n$ consist of $x_1, \dots, x_n$. Define $m_1 = x_1$ and $m_k$ as $x_k$ if $x_k < m_{k-1}$ and $m_{k-1}$ otherwise. Then $m_n$ will be minimal. Similarly, $X$ has a maximal element.

**Remark 3.7.** This proof in fact based on assumption (the induction hypothesis) that the theorem holds for $X_{k-1}$ consist of $x_1, \dots, x_{k-1}$, i.e. $m_{k-1} = \min\{x_1, \dots, x_{k-1}\}$, then it follows $m_{k-1} = \min\{x_1, \dots, x_{k-1}\} \Rightarrow m_k = \min\{x_1, \dots, x_k\}$ and by induction we conclude that for all $n \in \mathbb{N}$, $m_n = \min\{x_1, \dots, x_n\}$.

**Definition 3.6.** An element $x$ is said to be a first element of the linearly s-ordered set $A$ (with respect to the s-relation $R$) if $xRy$ for all $y \in_S A$. On the other hand, if $yRx$ for all $y \in_S A$, then $x$ is said to be a last element of $A$ (with respect to $R$). Generally speaking, not every set has a first or last element; but if such an element exists, then it is uniquely determined.

**Abbreviation 3.2** Let $X_n(A), \bar{X}_n(A) = n$ be s-finite non-empty subset of a linearly s-ordered set $A$ such that there is a first element and a last element of $X_n$. We shall abbreviated: $[X_n(A), (\bar{X}_n(A) = n)$ is a s-finite non-empty subset of a linearly s-ordered set $A$ such that there is a first element and a last element of $X_n(A)] \Leftrightarrow \hat{X}_n(A)$. Under assumption

$$ \mathbf{RMP} \exists m(m \in_S \mathbb{N}) \exists X_m(A) [-\hat{X}_m(A)]. \qquad (3.16) $$
(I) **Idealization**

$$ \forall^{\text{stfin}} F \exists y \forall x \in F[R(x, y) \Leftrightarrow \exists b \forall^{\text{st}} x R(x, b)] \quad (4.1) $$

for any internal relation $R$.

**Remark 4.2.** The idealization axiom obviously states that saying that for any fixed finite set $F$ there is a $y$ such that $R(x,y)$ holds for all $x \in F$ is the same as saying that there is a $b$ such that for all fixed $x$ the relation $R(x,b)$ holds.

(II) **Standardization**

$$ \forall^{\text{st}} A \exists^{\text{st}} B \forall^{\text{st}} x (x \in B \Leftrightarrow x \in A \wedge \varphi(x)) \quad (4.2) $$

for every st-epsilon formula $\varphi$ with arbitrary (internal) parameters.

(III) **Transfer**

$$ \forall^{\text{st}} y_1, \dots, y_n \forall^{\text{st}} x [\varphi(x, y_1, \dots, y_n)] \Rightarrow \forall x \varphi(x, y_1, \dots, y_n) \quad (4.3) $$

for all internal $\varphi(x, y_1, \dots, y_n)$.

**Remark 5.3.** An important consequence of (I) is the principle of **External Induction**, which states that for any (external or internal) formula $\varphi$, one has

$$ \varphi(0) \wedge [\forall^{\text{st}} n (\varphi(n) \Rightarrow \varphi(n+1))] \Rightarrow \forall^{\text{st}} n \varphi(n). \quad (4.4) $$

**Boundedness**

$$ \forall x \exists^{\text{st}} y (x \in y) \quad (4.5) $$

and since (2.5) contradicts idealization the following (bounded) form is taken instead:

(IV) **Bounded Idealization**

For every $\epsilon$-formula $R$:

$$ \forall^{\text{st}} Y [\forall^{\text{stfin}} F \exists y \in Y (\forall x \in FR(x, y) \Leftrightarrow \exists b (b \in Y) \forall^{\text{st}} x R(x, b))] \quad (4.6) $$

This gives a subsystem BST, which corresponds to the bounded sets of IST.

## §5. Internal Set Theory IST#

The axiomatics IST# formulates within infinitary first-order language the behaviour of standard and internal sets of a nonstandard model of $NC_{\infty}^{\#}$. This done by adding the unary standardness predicate "st" to the language of $NC_{\infty}^{\#}$ as well as adding to the axioms of $NC_{\infty}^{\#}$ three new axiom schemes involving the predicate "st":

**Idealization, Standardization, Transfer** and **Axiom of internal hyper infinite induction**.

**Remark 5.1.** Formulas which do not use the predicate st are called **internal formulas** (or $\in_{sw}$-formulas) and formulas that use this new predicate are called **external formulas** (or st-$\in_{sw}$-formulas). A formula $\varphi$ is standard if only standard constants occur in $\varphi$.

**Abbreviation 5.1.** We write fin($x$) meaning 'x is finite'. Let $\varphi(x)$ be a st-$\in_{sw}$-formula:

1. $\forall_s^{\text{st}} x \varphi(x)$ abbreviates $\forall x (\text{st}(x) \Rightarrow_s \varphi(x))$.

2. $\forall_{s,w}^{\text{st}} x \varphi(x)$ abbreviates $\forall x (\text{st}(x) \Rightarrow_{s,w} \varphi(x))$.

3. $\exists^{\text{st}} x \varphi(x)$ abbreviates $\exists x (\text{st}(x) \wedge \varphi(x))$.

4. $\forall_s^{\text{fin}} x \varphi(x)$ abbreviates $\forall x (\text{fin}(x)) \Rightarrow_s \varphi(x)$.

5. $\forall_{s,w}^{\text{fin}} x \varphi(x)$ abbreviates $\forall x (\text{fin}(x)) \Rightarrow_{s,w} \varphi(x)$.

6. $\exists^{\text{fin}} x \varphi(x)$ abbreviates $\exists x (\text{fin}(x) \wedge \varphi(x))$.

7. $\forall_s^{\text{stfin}} x \varphi(x)$ abbreviates $\forall x (\text{st}(x) \wedge \text{fin}(x)) \Rightarrow_s \varphi(x)$.
by axiom of ω-induction we obtain

$$ \forall X_n(A) \left[ \bigwedge_{n \in \mathbb{N}} (\hat{X}_n(A) \Rightarrow_s \hat{X}_{n+1}(A)) \right] \Rightarrow_s \forall n \forall X_n(A) [\hat{X}_n(A)]. \quad (3.17) $$

In particular for $A = \mathbb{N}$ under assumption

$$ \forall_{\text{RMP}} \exists m (m \in_s \mathbb{N}) \exists X_m(\mathbb{N}) [\neg \hat{X}_m(\mathbb{N})]. \quad (3.18) $$

by axiom of ω-induction we obtain

$$ \forall X_n(\mathbb{N}) \left[ \bigwedge_{n \in \mathbb{N}} (\hat{X}_n(\mathbb{N}) \Rightarrow_s \hat{X}_{n+1}(\mathbb{N})) \right] \Rightarrow_s \forall n \forall X_n(A) [\hat{X}_n(\mathbb{N})]. \quad (3.19) $$

**7 Axiom of existence non well-ordered s-finite subset of $\mathbb{N}$**

$$ \exists m (m \in_s \mathbb{N}) \exists X_m(\mathbb{N}) [\neg \hat{X}_m(\mathbb{N})]. \quad (3.20) $$

## §4. Internal Set Theory IST.

The axiomatics IST (Internal Set Theory) was presented in 1977 [19] and in a sense formulates within first-order language the behaviour of standard and internal sets of a nonstandard model of ZFC. This were done by adding the unary standardness predicate "st" to the language of ZFC as well as adding to the axioms of ZFC three new axiom schemes involving the predicate "st": **Idealization**, **Standardization** and **Transfer**.

**Remark 4.1.** Formulas which do not use the predicate st are called **internal formulas** (or $\epsilon$-formulas) and formulas that use this new predicate are called **external formulas** (or st-$\epsilon$-formulas). A formula $\varphi$ is standard if only standard constants occur in $\varphi$.

**Abbreviation 4.1.** We denote a set of the all naturals by $\mathbb{N}^\# $ and a set of the all finite naturals by $\mathbb{N}$.

**Abbreviation 4.2.** We write **fin(x)** meaning 'x is finite'. Let $\varphi(x)$ be a st-$\epsilon$-formula:

1. $\forall^{\text{st}} x \varphi(x)$ abbreviates $\forall x (\text{st}(x) \Rightarrow \varphi(x))$. 2. $\exists^{\text{st}} x \varphi(x)$ abbreviates $\exists x (\text{st}(x) \wedge \varphi(x))$.

3. $\forall^{\text{fin}} x \varphi(x)$ abbreviates $\forall x (\text{fin}(x)) \Rightarrow \varphi(x)$. 4. $\exists^{\text{fin}} x \varphi(x)$ abbreviates $\exists x (\text{fin}(x) \wedge \varphi(x))$.

5. $\forall^{\text{stfin}} x \varphi(x)$ abbreviates $\forall x (\text{st}(x) \wedge \text{fin}(x)) \Rightarrow \varphi(x)$.

6. $\exists^{\text{stfin}} x \varphi(x)$ abbreviates $\exists x (\text{st}(x) \wedge \text{fin}(x) \wedge \varphi(x))$.

The fundamental axioms of IST :

### (I) Idealization

$$ \forall^{\text{stfin}} F \exists y \forall x \in F [R(x, y) \iff \exists b \forall^{\text{st}} x R(x, b)] \quad (4.1) $$

for any internal relation $R$.

**Remark 4.2.** The idealization axiom obviously states that saying that for any fixed finite set $F$ there is a $y$ such that $R(x, y)$ holds for all $x \in F$ is the same as saying that there is a $b$ such that for all fixed $x$ the relation $R(x, b)$ holds.

### (II) Standardization

$$ \forall^{\text{st}} A \exists^{\text{st}} B \forall^{\text{st}} x (x \in B \Leftrightarrow x \in A \wedge \varphi(x)) \quad (4.2) $$

for every st-$\epsilon$-formula $\varphi$ with arbitrary (internal) parameters.

### (III) Transfer

$$ \forall^{\text{st}} y_1, \dots, y_n \forall^{\text{st}} x [\varphi(x, y_1, \dots, y_n)] \Rightarrow \forall x \varphi(x, y_1, \dots, y_n) \quad (4.3) $$

for all internal $\varphi(x, y_1, \dots, y_n)$.

**Remark 5.3.** An important consequence of (I) is the principle of **External Induction**,
8. $\forall_{s,w}^{\text{stfin}} x\varphi(x)$ abbreviates $\forall x(\text{st}(x) \land \text{fin}(x)) \Rightarrow_{s,w} \varphi(x)$.

9. $\exists^{\text{stfin}} x\varphi(x)$ abbreviates $\exists x(\text{st}(x) \land \text{fin}(x) \land \varphi(x))$.

The fundamental axioms of IST# :

(I) **Idealization for classical sets**

$$ \forall_s^{\text{stfin}} F^{\text{CL}} \exists y^{\text{CL}} \forall x^{\text{CL}} \in_s F [R^{\text{CL}}(x, y) \iff_s \exists b^{\text{CL}} \forall_s^{\text{st}} x R^{\text{CL}}(x, b)] \quad (5.1) $$

for any internal classical relation $R^{\text{CL}}(x, y)$.

**Remark 5.2.** The idealization axiom obviously states that saying that for any fixed classical finite set $F$ there is a classical $y$ such that $R^{\text{CL}}(x, y)$ holds for all classical $x \in_s F$ is the same as saying that there is a classical $b$ such that for all fixed classical $x$ the classical relation $R^{\text{CL}}(x, b)$ holds.

(II) **Standardization for classical sets**

$$ \forall^{\text{st}} A^{\text{CL}} \exists^{\text{st}} B^{\text{CL}} \forall^{\text{st}} x^{\text{CL}} (x \in B \iff_s x \in A \land \varphi(x)) \quad (5.2) $$

for every st-$\in$-formula $\varphi$ with arbitrary (internal) parameters.

(III) **Transfer for classical sets**

$$ \forall^{\text{st}} y_1^{\text{CL}}, \dots, y_n^{\text{CL}} \forall^{\text{st}} x^{\text{CL}} [\varphi(x, y_1, \dots, y_n)] \Rightarrow_s \forall x^{\text{CL}} \varphi(x, y_1, \dots, y_n) \quad (5.3) $$

for all internal $\varphi(x, y_1, \dots, y_n)$.

**Boundedness**

$$ \forall x^{\text{CL}} \exists^{\text{st}} y^{\text{CL}} (x \in_s y) \quad (5.4) $$

and since (5.4) contradicts idealization the following (bounded) form is taken instead:

(IV) **Bounded Idealization for classical sets**

For every $\in$-formula $R$ :

$$ \forall^{\text{st}} Y^{\text{CL}} [\forall^{\text{stfin}} F^{\text{CL}} \exists y^{\text{CL}} \in Y (\forall x^{\text{CL}} (x \in F) R(x, y) \iff_s \exists b^{\text{CL}} (b \in Y) \forall^{\text{st}} x R(x, b))] \quad (5.5) $$

(V) **Idealization for nonclassical sets**

$$ \forall_{s,w}^{\text{stfin}} F^{\text{NCL}} \exists y^{\text{NCL}} \forall x^{\text{NCL}} \in_{s,w} F [R^{\text{NCL}}(x, y) \iff_{s,w} \exists b^{\text{NCL}} \forall_{s,w}^{\text{st}} x R^{\text{NCL}}(x, b)] \quad (5.6) $$

for any internal nonclassical relation $R^{\text{NCL}}(x, y)$.

**Remark 5.3.** The idealization axiom obviously states that saying that for any fixed nonclassical finite set $F$ there is a classical $y$ such that $R^{\text{NCL}}(x, y)$ holds for all classical $x \in_s F$ is the same as saying that there is a classical $b$ such that for all fixed classical $x$ the nonclassical relation $R^{\text{NCL}}(x, b)$ holds.

(VI) **Standardization for nonclassical sets**

$$ \forall_{s,w}^{\text{st}} A^{\text{NCL}} \exists^{\text{st}} B^{\text{NCL}} \forall_{s,w}^{\text{st}} x^{\text{NCL}} (x \in_{s,w} B \iff_{s,w} x \in_{s,w} A \land \varphi(x)) \quad (5.7) $$

for every st-$\in_{s,w}$-formula $\varphi$ with arbitrary (internal) parameters.

(VII) **Transfer for nonclassical sets**

$$ \forall_{s,w}^{\text{st}} y_1^{\text{NCL}}, \dots, y_n^{\text{NCL}} \forall^{\text{st}} x^{\text{NCL}} [\varphi(x, y_1, \dots, y_n)] \Rightarrow_{s,w} \forall_{s,w}^{\text{st}} x^{\text{NCL}} \varphi(x, y_1, \dots, y_n) \quad (5.8) $$

for all internal $\varphi(x, y_1, \dots, y_n)$.

**Boundedness for nonclassical sets**

$$ \forall_{s,w}^{\text{NCL}} \exists^{\text{st}} y^{\text{NCL}} (x \in_{s,w} y) \quad (5.9) $$

and since (5.9) contradicts idealization the following (bounded) form is taken instead:

(VIII) **Bounded Idealization for nonclassical sets**

For every $\in_{s,w}$-formula $R$ :
which states that for any (external or internal) formula $\varphi$, one has

$$ \varphi(0) \wedge [\forall^{\text{st}} n(\varphi(n) \Rightarrow \varphi(n+1))] \Rightarrow \forall^{\text{st}} n\varphi(n). \quad (4.4) $$

## Boundedness

$$ \forall x \exists^{\text{st}} y (x \in y) \qquad (4.5) $$

and since (2.5) contradicts idealization the following (bounded) form is taken instead:

### (IV) Bounded Idealization

For every $\epsilon$-formula $R$:

$$ \forall^{\text{st}} Y[\forall^{\text{stfin}} F \exists y \in Y(\forall x \in FR(x, y) \Leftrightarrow \exists b (b \in Y) \forall^{\text{st}} x R(x, b))]. \quad (4.6) $$

This gives a subsystem BST, which corresponds to the bounded sets of IST.

# §5. Internal Set Theory IST#

The axiomatics IST# formulates within infinitary first-order language the behaviour of standard and internal sets of a nonstandard model of $NC_{\infty}^{\#}$. This done by adding the unary standardness predicate "st" to the language of $NC_{\infty}^{\#}$ as well as adding to the axioms of $NC_{\infty}^{\#}$ three new axiom schemes involving the predicate "st":

**Idealization, Standardization, Transfer and Axiom of internal hyper infinite induction.**

**Remark 5.1.** Formulas which do not use the predicate st are called *internal formulas* (or $\epsilon \in_{sw}$-formulas) and formulas that use this new predicate are called *external formulas* (or st-$\epsilon \in_{sw}$-formulas). A formula $\varphi$ is standard if only standard constants occur in $\varphi$.

**Abbreviation 5.1.** We write fin($x$) meaning 'x is finite'. Let $\varphi(x)$ be a st-$\epsilon \in_{sw}$-formula:

1. $\forall_s^{\text{st}} x \varphi(x)$ abbreviates $\forall x (\text{st}(x) \Rightarrow_s \varphi(x))$.

2. $\forall_{s,w}^{\text{st}} x \varphi(x)$ abbreviates $\forall x (\text{st}(x) \Rightarrow_{s,w} \varphi(x))$.

3. $\exists^{\text{st}} x \varphi(x)$ abbreviates $\exists x (\text{st}(x) \wedge \varphi(x))$.

4. $\forall_s^{\text{fin}} x \varphi(x)$ abbreviates $\forall x (\text{fin}(x) \Rightarrow_s \varphi(x))$.

5. $\forall_{s,w}^{\text{fin}} x \varphi(x)$ abbreviates $\forall x (\text{fin}(x) \Rightarrow_{s,w} \varphi(x))$.

6. $\exists^{\text{fin}} x \varphi(x)$ abbreviates $\exists x (\text{fin}(x) \wedge \varphi(x))$.

7. $\forall_s^{\text{stfin}} x \varphi(x)$ abbreviates $\forall x (\text{st}(x) \wedge \text{fin}(x) \Rightarrow_s \varphi(x))$.

8. $\forall_{s,w}^{\text{stfin}} x \varphi(x)$ abbreviates $\forall x (\text{st}(x) \wedge \text{fin}(x) \Rightarrow_{s,w} \varphi(x))$.

9. $\exists^{\text{stfin}} x \varphi(x)$ abbreviates $\exists x (\text{st}(x) \wedge \text{fin}(x) \wedge \varphi(x))$.

The fundamental axioms of IST# :

### (I) Idealization for classical sets

$$ \forall_s^{\text{stfin}} F^{\text{CL}} \exists y^{\text{CL}} \forall x^{\text{CL}} \in_s F[R^{\text{CL}}(x, y) \Leftrightarrow_s \exists b^{\text{CL}} \forall_s^{\text{st}} x R^{\text{CL}}(x, b)] \quad (5.1) $$

for any internal classical relation $R^{\text{CL}}(x, y)$.

**Remark 5.2.** The idealization axiom obviously states that saying that for any fixed classical finite set $F$ there is a classical $y$ such that $R^{\text{CL}}(x, y)$ holds for all classical $x \in_s F$ is the same as saying that there is a classical $b$ such that for all fixed classical $x$ the classical relation $R^{\text{CL}}(x, b)$ holds.

### (II) Standardization for classical sets

$$ \forall^{\text{st}} A^{\text{CL}} \exists^{\text{st}} B^{\text{CL}} \forall^{\text{st}} x^{\text{CL}} (x \in B \Leftrightarrow_s x \in A \wedge \varphi(x)) \quad (5.2) $$

for every st-$\epsilon \in$-formula $\varphi$ with arbitrary (internal) parameters.
$$ \forall_{s,w}^{\mathbf{st}} Y^{\mathbf{NCL}} \left[ \begin{array}{l} \forall_{s,w}^{\mathbf{stfin}} F^{\mathbf{NCL}} \exists y^{\mathbf{NCL}} \in_{s,w} Y(\forall_{s,w} x^{\mathbf{NCL}}(x \in F) R(x, y) \iff_{s,w} \\ \phantom{\forall_{s,w}^{\mathbf{stfin}} F^{\mathbf{NCL}} \exists} \exists b^{\mathbf{NCL}} (b \in Y) \forall_{s,w}^{\mathbf{st}} x R(x, b)) \end{array} \right]. \qquad (5.10) $$

### (IX) Internal Hyper Infinite Induction

$$ \forall S(S \subset_s \mathbb{N}^{\#}) \left\{ \forall \beta(\beta \in \mathbb{N}^{\#}) \left[ \bigwedge_{0 \le \alpha < \beta} (\alpha \in_s S \Rightarrow \alpha^+ \in_s S) \right] \Rightarrow_s S =_s \mathbb{N}^{\#} \right\}. \qquad (5.11) $$

**The main restricted rules of conclusion.**

If $IST^\# \vdash A$ then $\neg A \vdash B$, where $B \in \mathcal{L}^\#$.

Thus if statement A holds in $IST^\#$ we cannot obtain from $\neg A$ any formula B whatsoever.

**Abbreviation 5.2** Let $X_n(A), \bar{X}_n(A) = n$ be a s-finite non-empty subset of a linearly s-ordered set A such that there is a first element and a last element of $X_n$. We shall abbreviated: $[X_n(A), (\bar{X}_n(A) = n)$ is a s-finite non-empty subset of a linearly s-ordered set A such that there is a first element and a last element of $X_n(A)] \Leftrightarrow \hat{X}_n(A)$.

(X) **Axiom of existence non well-ordered s-finite subset of $\mathbb{N}$**.

$$ \exists m(m \in_s \mathbb{N}) \exists X_m(\mathbb{N}) [\neg \hat{X}_m(\mathbb{N})]. \qquad (5.12) $$

## §6. Hypernaturals $\mathbb{N}^\#$. Axiom of hyperinfinity

**Definition 6.1.** (i) A non-empty transitive non regular set $u$ is a well formed non regular set iff:

(i) there is unique countable sequence $\{u_n\}_{n=1}^\infty$ such that

$$ \dots \in u_{n+1} \in u_n \dots \in u_4 \in u_3 \in u_2 \in u_1 \in u, \qquad (6.1) $$

(ii) for any $n \in \mathbb{N}$ and any $u_{n+1} \in u_n$ :

$$ u_n = u_{n+1}^+, \qquad (6.2) $$

where $a^+ = a \cup \{a\}$.

(ii) we define a function $a^{+[k]}$ inductively by $a^{+[k]} = (a^{+[k]})^+$.

**Definition 6.2.** Let $u$ and $w$ are well formed non regular sets. We write $w < u$ iff for any $n \in \mathbb{N}$

$$ w \in u_n. \qquad (6.3) $$

**Definition 6.3.** We say that a well formed non regular set $u$ is infinite (or hyperfinite) hypernatural number iff:

(I) For any member $w \in u$ one and only one of the following conditions are satisfied:

(i) $w \in \mathbb{N}$ or

(ii) $w = u_n$ for some $n \in \mathbb{N}$ or

(iii) $w < u$.

(II) Let $\prec u$ be a set $\prec u = \{z | z < u\}$, then by relation $(\cdot < \cdot)$ a set $\prec u$ is densely ordered with no first element.

(III) $\mathbb{N} \subset u$.

**Definition 6.4.** Assume $u \in \mathbb{N}^\#$, then $u$ is infinite (hypernatural) number if $u \in \mathbb{N}^\# \setminus \mathbb{N}$.

### Axiom of hyperinfinity
(III) Transfer for classical sets

$$
\forall^{\mathrm{st}} y_1^{\mathrm{CL}}, \dots, y_n^{\mathrm{CL}} \forall^{\mathrm{st}} x^{\mathrm{CL}} [\varphi(x, y_1, \dots, y_n)] \Rightarrow_s \forall x^{\mathrm{CL}} \varphi(x, y_1, \dots, y_n) \quad (5.3)
$$

for all internal $\varphi(x, y_1, \dots, y_n)$.

Boundedness

$$
\forall x^{\mathrm{CL}} \exists^{\mathrm{st}} y^{\mathrm{CL}} (x \in_s y) \qquad (5.4)
$$

and since (5.4) contradicts idealization the following (bounded) form is taken instead:

(IV) Bounded Idealization for classical sets

For every $\epsilon$-formula $R$ :

$$
\forall^{\mathrm{st}} Y^{\mathrm{CL}}[\forall^{\mathrm{stfin}} F^{\mathrm{CL}} \exists y^{\mathrm{CL}} \in Y (\forall x^{\mathrm{CL}} (x \in F) R(x, y) \Leftrightarrow_s \exists b^{\mathrm{CL}} (b \in Y) \forall^{\mathrm{st}} x R(x, b))]. \quad (5.5)
$$

(V) Idealization for nonclassical sets

$$
\forall_{s,w}^{\text{stfin}} F^{\text{NCL}} \exists y^{\text{NCL}} \forall x^{\text{NCL}} \in_{s,w} F [R^{\text{NCL}}(x,y) \Leftrightarrow_{s,w} \exists b^{\text{NCL}} \forall_{s,w}^{\text{st}} x R^{\text{NCL}}(x,b)] \quad (5.6)
$$

for any internal nonclassical relation $R^{\text{NCL}}(x, y)$.

**Remark 5.3.** The idealization axiom obviously states that saying that for any fixed nonclassical finite set *F* there is a classical *y* such that *R*<sup>NCL</sup>(*x*,*y*) holds for all classical *x* ∈ *S* *F* is the same as saying that there is a classical *b* such that for all fixed classical *x* the nonclassical relation *R*<sup>NCL</sup>(*x*,*b*) holds.

(VI) Standardization for nonclassical sets

$$
\forall_{s,w}^{\mathrm{st}} A^{\mathrm{NCL}} \exists^{\mathrm{st}} B^{\mathrm{NCL}} \forall_{s,w}^{\mathrm{st}} x^{\mathrm{NCL}} (x \in_{s,w} B \Leftrightarrow_{s,w} x \in_{s,w} A \wedge \varphi(x)) \quad (5.7)
$$

for every st-`ε`<sub>s,w</sub> -formula *φ* with arbitrary (internal) parameters.

(VII) Transfer for nonclassical sets

$$
\forall_{s,w}^{\mathrm{st}} y_1^{\mathrm{NCL}}, \dots, y_n^{\mathrm{NCL}} \forall^{\mathrm{st}} x^{\mathrm{NCL}} [\varphi(x, y_1, \dots, y_n)] \Rightarrow_{s,w} \forall_{s,w} x^{\mathrm{NCL}} \varphi(x, y_1, \dots, y_n) \quad (5.8)
$$

for all internal $\varphi(x, y_1, \dots, y_n)$.

Boundedness for nonclassical sets

$$
\forall_{s,w} x^{\mathrm{NCL}} \exists^{\mathrm{st}} y^{\mathrm{NCL}} (x \in_{s,w} y) \qquad (5.9)
$$

and since (5.9) contradicts idealization the following (bounded) form is taken instead:

(VIII) Bounded Idealization for nonclassical sets

For every $\epsilon \in_{s,w}$ -formula $R$ :

$$
\forall_{s,w}^{\mathrm{st}} Y^{\mathrm{NCL}} \left[ \begin{array}{l} \forall_{s,w}^{\mathrm{stfin}} F^{\mathrm{NCL}} \exists y^{\mathrm{NCL}} \in_{s,w} Y(\forall_{s,w} x^{\mathrm{NCL}}(x \in F) R(x, y) \Leftrightarrow_{s,w} \\ \phantom{\forall_{s,w}^{\mathrm{stfin}} F^{\mathrm{NCL}} \exists y^{\mathrm{NCL}} \in_{s,w} Y(\forall_{s,w} x^{\mathrm{NCL}}(}} \exists b^{\mathrm{NCL}} (b \in Y) \forall_{s,w}^{\mathrm{st}} x R(x, b)) \end{array} \right]. \quad (5.10)
$$

(IX) Internal Hyper Infinite Induction

$$
\forall S (S \subset_s \mathbb{N}^#) \left\{ \forall \beta (\beta \in \mathbb{N}^#) \left[ \bigwedge_{0 \le \alpha < \beta} (\alpha \in_S S \Rightarrow_S \alpha^+ \in_S S) \right] \Rightarrow_S S =_S \mathbb{N}^# \right\}. \quad (5.11)
$$

The main restricted rules of conclusion.

If IST<sup>#</sup> ⊢ A then ¬A ⊢ B, where B ∈ ℒ<sup>#</sup>.

Thus if statement A holds in IST<sup>#</sup> we cannot obtain from ¬A any formula B
whatsoever.

**Abbreviation 5.2** Let $X_n(A), \bar{X}_n(A) = n$ be a s-finite non-empty subset of a linearly s-ordered set $A$ such that there is a first element and a last element of $X_n$. We shall abbreviated: $[X_n(A), (\bar{X}_n(A) = n)$ is a s-finite non-empty subset of a linearly s-ordered
There exists a set $N^#$ such that:

(i) $N \subset N^#,$

(ii) if $u \in N^# \setminus N$ then there exists infinite (hypernatural) number $v$ such that $v < u,$

(iii) if $u \in N^# \setminus N$ then there exists infinite (hypernatural) number $w$ such that for any $n \in N: u^{[n]} < w,$

(iv) set $N^# \setminus N$ is patially ordered by relation $(\cdot < \cdot)$ with no first and no last element.

## §7. Axioms of the nonstandard arithmetic $A^#.$

Axioms of the nonstandard arithmetic $A^#:$ are:

### Axiom of hyperinfinity

There exists a set $N^#$ such that:

(i) $N \subset N^#$

(ii) if $u$ is infinite (hypernatural) number then there exists infinite (hypernatural) number $v$ such that $v < u$

(iii) if $u$ is infinite hypernatural number then there exists infinite (hypernatural) number $w$ such that $u < w$

(iv) set $N^# \setminus N$ is patially ordered by relation $(\cdot < \cdot)$ with no first and no last element.

### Axioms of infinite $\omega$-induction

(i)

$$ \forall S(S \subset N) \{ \left[ \bigwedge_{n \in \omega} (n \in S \Rightarrow_s n^+ \in S) \right] \Rightarrow_s S = N \}. \quad (7.1) $$

(ii) Let $F(x)$ be a wff of the set theory $NC^#_{\infty}$, then

$$ \left[ \bigwedge_{n \in \omega} (F(n) \Rightarrow_s F(n^+)) \right] \Rightarrow_s \forall n(n \in \omega) F(n). \quad (7.2) $$

**Definition 7.1.** (i) Let $\beta$ be a hypernatural such that $\beta \in N^# \setminus N$. Let $[0, \beta] \subset N^#$ be a set such that $\forall x[x \in [0, \beta] \Leftrightarrow 0 \le x \le \beta]$ and let $[0, \beta)$ be a set $[0, \beta) = [0, \beta] \setminus \{\beta\}$.

(ii) Let $\beta \in N^# \setminus N$ and let $\beta_\infty \subset N^#$ be a set such that

$$ \forall x\{x \in \beta_{\infty} \Leftrightarrow \exists k(k \ge 0)[0 \le x \le \beta^{[k]}]\}. \quad (7.3) $$

**Definition 7.2.** Let $F(x)$ be a wff of $NC^#_{\infty}$ with unique free variable $x$. We will say that a wff $F(x)$ is restricted on a classical set $S$ such that $S \subsetneq_s N^#$ iff the following condition is satisfied

$$ \forall \alpha[\alpha \in N^# \setminus S \Rightarrow_s \neg F(\alpha)]. \quad (7.4) $$

**Definition 7.3.** Let $F(x)$ be a wff of $NC^#_{\infty}$ with unique free variable $x$. We will say that a wff $F(x)$ is *strictly* restricted on a set $S$ such that $S \subsetneq_s N^#$ iff there is no proper subset $S' \subset S$ such that a wff $F(x)$ is restricted on a set $S'$.

**Example 7.1.** (i) Let $\text{fin}(\alpha), \alpha \in N^#$ be a wff formula such that $\text{fin}(\alpha) \Leftrightarrow_s \alpha \in N$.

Obviously wff $\text{fin}(\alpha)$ is strictly restricted on a set $N$ since $\forall \alpha[\alpha \in N^# \setminus N \Rightarrow_s \neg \text{fin}(\alpha)]$.

Let $\text{hfin}(\alpha), \alpha \in N^#$ be a wff formula such that $\text{hfin}(\alpha) \Leftrightarrow_s \alpha \in N^# \setminus N$ since $\forall \alpha[\alpha \in N \Rightarrow_s \neg \text{hfin}(\alpha)]$.

**Definition 7.4.** Let $F(x)$ be a wff of $NC^#_{\infty}$ with unique free variable $x$. We will say that a wff $F(x)$ is unrestricted if wff $F(x)$ is not restricted on any set $S$ such that $S \subsetneq N^#$.
set A such that there is a first element and a last element of $X_n(A)] \Leftrightarrow \hat{X}_n(A)$.

(X) **Axiom of existence non well-ordered s-finite subset of $\mathbb{N}$**.

$$ \exists m(m \in_s \mathbb{N}) \exists X_m(\mathbb{N}) [\neg \hat{X}_m(\mathbb{N})]. \qquad (5.12) $$

## §6. Hypernaturals $\mathbb{N}^#$. Axiom of hyperinfinity

**Definition 6.1.** (i) A non-empty transitive non regular set $u$ is a well formed non regular set iff:

(i) there is unique countable sequence $\{u_n\}_{n=1}^\infty$ such that

$$ \dots \in u_{n+1} \in u_n \dots \in u_4 \in u_3 \in u_2 \in u_1 \in u, \qquad (6.1) $$

(ii) for any $n \in \mathbb{N}$ and any $u_{n+1} \in u_n$ :

$$ u_n = u_{n+1}^+, \qquad (6.2) $$

where $a^+ = a \cup \{a\}$.

(ii) we define a function $a^{+[k]}$ inductively by $a^{+[k]} = (a^{+[k]})^+$.

**Definition 6.2.** Let $u$ and $w$ are well formed non regular sets. We write $w < u$ iff for any $n \in \mathbb{N}$

$$ w \in u_n. \qquad (6.3) $$

**Definition 6.3.** We say that a well formed non regular set $u$ is infinite (or hyperfinite) hypernatural number iff:

(I) For any member $w \in u$ one and only one of the following conditions are satisfied:

(i) $w \in \mathbb{N}$ or

(ii) $w = u_n$ for some $n \in \mathbb{N}$ or

(iii) $w < u$.

(II) Let $\prec u$ be a set $\prec u = \{z|z < u\}$, then by relation $(\prec \cdot \prec \cdot)$ a set $\prec u$ is densely ordered with no first element.

(III) $\mathbb{N} \subset u$.

**Definition 6.4.** Assume $u \in \mathbb{N}^#$, then $u$ is infinite (hypernatural) number if $u \in \mathbb{N}^#\mathbb{N}$.

### Axiom of hyperinfinity

There exists a set $\mathbb{N}^#$ such that:

(i) $\mathbb{N} \subset \mathbb{N}^#$,

(ii) if $u \in \mathbb{N}^#\mathbb{N}$ then there exists infinite (hypernatural) number $v$ such that $v < u$,

(iii) if $u \in \mathbb{N}^#\mathbb{N}$ then there exists infinite (hypernatural) number $w$ such that for any $n \in \mathbb{N}$ : $u^{+[n]} < w$,

(iv) set $\mathbb{N}^#\mathbb{N}$ is patially ordered by relation $(\prec \cdot \prec \cdot)$ with no first and no last element.

## §7. Axioms of the nonstandard arithmetic $\mathbb{A}^#$.

Axioms of the nonstandard arithmetic $\mathbb{A}^#$ are:

### Axiom of hyperinfinity

There exists a set $\mathbb{N}^#$ such that:

(i) $\mathbb{N} \subset \mathbb{N}^#$

(ii) if $u$ is infinite (hypernatural) number then there exists infinite (hypernatural) number $v$ such that $v < u$
**Axiom of hyperfinite induction 1**

$$ \forall S (S \subseteq_s [0, \beta]) \forall \beta (\beta \in_s \mathbb{N}^#) \searrow \\ \left\{ \forall \alpha (\alpha \in_s [0, \beta]) \left[ \bigwedge_{0 \le \alpha < \beta} (\alpha \in_s S \Rightarrow \alpha^+ \in_s S) \right] \Rightarrow_s S = [0, \beta] \right\}. \qquad (7.5) $$

**Axiom of hyperfinite induction 1'**

$$ \forall S (S \subseteq_s [0, \beta_\infty]) \forall \beta (\beta \in \mathbb{N}^#) \searrow \\ \left\{ \forall \alpha (\alpha \in [0, \beta_\infty]) \left[ \bigwedge_{0 \le \alpha < \beta_\infty} (\alpha \in S \Rightarrow \alpha^+ \in S) \right] \Rightarrow_S S = [0, \beta_\infty] \right\}. \qquad (7.6) $$

**Axiom of hyper infinite induction 1**

$$ \forall S (S \subseteq_s \mathbb{N}^#) \left\{ \forall \beta (\beta \in \mathbb{N}^#) \left[ \bigwedge_{0 \le \alpha < \beta} (\alpha \in_s S \Rightarrow \alpha^+ \in_s S) \right] \Rightarrow_s S =_s \mathbb{N}^# \right\}. \qquad (7.7) $$

**Definition 7.5.** A set $S \subseteq_s \mathbb{N}^#$ is a hyper inductive if the following statement holds

$$ \bigwedge_{\alpha \in \mathbb{N}^#} (\alpha \in_s S \Rightarrow_s \alpha^+ \in_s S). \qquad (7.8) $$

Obviously a set $\mathbb{N}^#$ is a hyper inductive. Thus axiom of hyper infinite induction 1 asserts that a set $\mathbb{N}^#$ this is the smallest hyper inductive set.

**Axioms of hyperfinite induction 2**

Let $F(x)$ be a wff of the set theory $\mathbb{N}\mathcal{C}_{\infty}^{\#}$ strictly restricted on a set $[0, \beta]$ then

$$ \left[ \forall \beta (\beta \in [0, \beta]) \left[ \bigwedge_{0 \le \alpha < \beta} (F(\alpha) \Rightarrow_s F(\alpha^+)) \right] \right] \Rightarrow_s \forall \alpha (\alpha \in [0, \beta]) F(\alpha). \qquad (7.9) $$

Let $F(x)$ be a wff of the set theory $\mathbb{N}\mathcal{C}_{\infty}^{\#}$ strictly restricted on a set $[0, \beta_{\infty}]$ then

$$ \left[ \forall \beta (\beta \in [0, \beta_{\infty}]) \left[ \bigwedge_{0 \le \alpha < \beta_{\infty}} (F(\alpha) \Rightarrow_s F(\alpha^+)) \right] \right] \Rightarrow_s \forall \alpha (\alpha \in [0, \beta_{\infty}]) F(\alpha). \qquad (7.10) $$

**Axiom of hyper infinite induction 2**

Let $F(x)$ be anrestricted wff of the set theory $\mathbb{N}\mathcal{C}_{\infty}^{\#}$ then

$$ \left[ \forall \beta (\beta \in \mathbb{N}^#) \left[ \bigwedge_{0 \le \alpha < \beta} (F(\alpha) \Rightarrow_s F(\alpha^+)) \right] \right] \Rightarrow_s \forall \beta (\beta \in \mathbb{N}^#) F(\beta). \qquad (7.11) $$

**The main restricted rules of conclusion.**

If $A^# \vdash A$ then $\neg A \vdash B$, where $B \in \mathcal{L}^#$.

Thus if statement A holds in $A^#$ we cannot obtain from $\neg A$ any formula B whatsoever.

**§8. The Generalized Recursion Theorem.**

**Theorem 1.** Let $S$ be a set, $c \in S$ and $G: S \to S$ is any function with $\text{dom}(G) = S$ and $\text{range}(G) \subseteq S$. Let $W[G] \in \mathbb{N}^# \times S$ be a binary relation such that:

(a) $(1, c) \in W[G]$ and

(b) if $(x, y) \in W[G]$ then $(\mathbf{Sc}(x), G(y)) \in W[G]$.

Then there exists a function $\mathcal{F}: \mathbb{N}^# \to S$ such that:

(i) $\text{dom}(\mathcal{F}) = \mathbb{N}^# \text{ and } \text{range}(\mathcal{F}) \subseteq S$;

(ii) $\mathcal{F}(1) = c$;
(iii) if $u$ is infinite hypernatural number then there exists infinite (hypernatural) number $w$ such that $u < w$

(iv) set $\mathbb{N}^{\#}\mathbb{N}$ is partially ordered by relation $(\cdot, <)$ with no first and no last element.

**Axioms of infinite $\omega$-induction**

(i)

$$ \forall S(S \subset \mathbb{N}) \left\{ \left[ \bigwedge_{n \in \omega} (n \in S \Rightarrow_s n^+ \in S) \right] \Rightarrow_s S = \mathbb{N} \right\}. \quad (7.1) $$

(ii) Let $F(x)$ be a wff of the set theory $\mathbb{N}\mathbb{C}_{\infty}^{\#}$, then

$$ \left[ \bigwedge_{n \in \omega} (F(n) \Rightarrow_s F(n^+)) \right] \Rightarrow_s \forall n(n \in \omega) F(n). \quad (7.2) $$

**Definition 7.1.** (i) Let $\beta$ be a hypernatural such that $\beta \in \mathbb{N}^{\#}\mathbb{N}$. Let $[0, \beta] \subset \mathbb{N}^{\#}$ be a set such that $\forall x[x \in [0, \beta] \Leftrightarrow 0 \le x \le \beta]$ and let $[0, \beta)$ be a set $[0, \beta) = [0, \beta]\setminus\{\beta\}$.

(ii) Let $\beta \in \mathbb{N}^{\#}\mathbb{N}$ and let $\beta_{\infty} \subset \mathbb{N}^{\#}$ be a set such that

$$ \forall x\{x \in \beta_{\infty} \Leftrightarrow \exists k(k \ge 0)[0 \le x \le \beta^{+[k]}]\}. \quad (7.3) $$

**Definition 7.2.** Let $F(x)$ be a wff of $\mathbb{N}\mathbb{C}_{\infty}^{\#}$ with unique free variable $x$. We will say that a wff $F(x)$ is restricted on a classical set $S$ such that $S \subsetneq_s \mathbb{N}^{\#}$ iff the following condition is satisfied

$$ \forall \alpha[\alpha \in \mathbb{N}^{\#}\setminus S \Rightarrow_s \neg F(\alpha)]. \quad (7.4) $$

**Definition 7.3.** Let $F(x)$ be a wff of $\mathbb{N}\mathbb{C}_{\infty}^{\#}$ with unique free variable $x$. We will say that a wff $F(x)$ is _strictly_ restricted on a set $S$ such that $S \subsetneq_s \mathbb{N}^{\#}$ iff there is no proper subset $S' \subset S$ such that a wff $F(x)$ is restricted on a set $S'$.

**Example 7.1.** (i) Let $\mathit{fin}(\alpha)$, $\alpha \in \mathbb{N}^{\#}$ be a wff formula such that $\mathit{fin}(\alpha) \Leftrightarrow_s \alpha \in \mathbb{N}$. Obviously wff $\mathit{fin}(\alpha)$ is strictly restricted on a set $\mathbb{N}$ since $\forall \alpha[\alpha \in \mathbb{N}^{\#}\mathbb{N} \Rightarrow_s \neg \mathit{fin}(\alpha)]$. Let $\mathit{hfin}(\alpha)$, $\alpha \in \mathbb{N}^{\#}$ be a wff formula such that $\mathit{hfin}(\alpha) \Leftrightarrow_s \alpha \in \mathbb{N}^{\#}\mathbb{N}$ since $\forall \alpha[\alpha \in \mathbb{N} \Rightarrow_s \neg \mathit{hfin}(\alpha)]$.

**Definition 7.4.** Let $F(x)$ be a wff of $\mathbb{N}\mathbb{C}_{\infty}^{\#}$ with unique free variable $x$. We will say that a wff $F(x)$ is unrestricted if wff $F(x)$ is not restricted on any set $S$ such that $S \subsetneq \mathbb{N}^{\#}$.

**Axiom of hyperfinite induction 1**

$$ \forall S(S \subseteq_s [0, \beta])\forall \beta(\beta \in_s \mathbb{N}^\#) \searrow \\ \left\{ \forall \alpha(\alpha \in_s [0, \beta)) \left[ \bigwedge_{0 \le \alpha < \beta} (\alpha \in_s S \Rightarrow \alpha^+ \in_s S) \right] \Rightarrow_s S = [0, \beta] \right\}. \quad (7.5) $$

**Axiom of hyperfinite induction 1'**

$$ \forall S(S \subseteq_s [0, \beta_{\infty}])\forall \beta(\beta \in_s \mathbb{N}^\#) \searrow \\ \left\{ \forall \alpha(\alpha \in_s [0, \beta_{\infty}]) \left[ \bigwedge_{0 \le \alpha < \beta_{\infty}} (\alpha \in_s S \Rightarrow \alpha^+ \in_s S) \right] \Rightarrow_s S = [0, \beta_{\infty}] \right\}. \quad (7.6) $$

**Axiom of hyper infinite induction 1**

$$ \forall S(S \subset_s \mathbb{N}^{\#}) \left\{ \forall \beta(\beta \in_s \mathbb{N}^{\#}) \left[ \bigwedge_{0 \le \alpha < \beta} (\alpha \in_s S \Rightarrow \alpha^+ \in_s S) \right] \Rightarrow_s S =_s \mathbb{N}^{\#} \right\}. \quad (7.7) $$
(iii) for all $x \in \mathbb{N}^\#$, $\mathcal{F}(\mathbf{Sc}(x)) = G(\mathcal{F}(x))$.

1. The desired function $\mathcal{F}$ is a certain hyper inductive relation $\mathbf{W} \subseteq \mathbb{N}^\# \times S$. It is to have the properties:

(ii') $(1, c) \in \mathbf{W}$;

(iii') if $(x, y) \in \mathbf{W}$ then $(\mathbf{Sc}(x), G(y)) \in \mathbf{W}$.

**Remark 1.** The latter is just another way of expressing (iii), that if

$$ \mathcal{F}(x) = y \qquad (1) $$

then

$$ \mathcal{F}(\mathbf{Sc}(x)) = G(y). \qquad (2) $$

**Remark 2.** Note that any relation $\mathbf{W}$ mentioned above is hyper inductive relation since the hyper inductivity conditions (ii')-(iii') are satisfied.

However there are many hyper inductive relations which satisfy the conditions (ii')-(iii'); on such is $\mathbb{N}^\# \times S$. What distinguishes the desired function from all these other relations is that we want $(a, b)$ to be on it only as required by (ii') and (iii'). In other words, it is to be the smallest relation satisfying (ii')-(iii'). This can be expressed precisely as follows:

(1) Let $\mathbf{M}$ be a set of the relations $\mathbf{W}$ satisfying the conditions (ii') and (iii'); then we define

$$ \mathcal{F} = \bigcap_{W \in \mathbf{M}} \mathbf{W}. $$

Hence

(2) whenever $\mathbf{W} \in \mathbf{M}$ then $\mathcal{F} \subseteq \mathbf{W}$.

We shall now show that we can derived from (1) that $\mathcal{F}$ is also one relation in $\mathbf{M}$.

(3) $(1, c) \in \mathcal{F}$.

This follows immediately from the definition of $\bigcap_{W \in \mathbf{M}}$ and the fact that $(1, c) \in \mathbf{W}$ for all $\mathbf{W} \in \mathbf{M}$.

(4) If $(x, y) \in \mathcal{F}$ then $(\mathbf{Sc}(x), G(y)) \in \mathcal{F}$.

For if $(x, y) \in \mathcal{F}$ then $(x, y) \in \mathbf{W}$ for all $\mathbf{W} \in \mathbf{M}$; hence by (iii')

$(\mathbf{Sc}(x), G(y)) \in \mathbf{W}$ for all $\mathbf{W} \in \mathbf{M}$ so that $(\mathbf{Sc}(x), G(y)) \in \mathcal{F}$ by (1).

We must now verify that $\mathcal{F}$ is actually a function, i.e., we wish to show that for any $x, z_1, z_2 \in \mathbb{N}^\#$, if $(x, z_1) \in \mathcal{F}$ and $(x, z_2) \in \mathcal{F}$, then $z_1 = z_2$.

We shall prove this by **hyper infinite induction** on $x$. Let

(5) $A = \{x | x \in \mathbb{N}^\# \text{ and for all } z_1, z_2 \in \mathbb{N}^\#, \text{ if } (x, z_1) \in \mathcal{F} \text{ and } (x, z_2) \in \mathcal{F} \text{ then } z_1 = z_2\}$.

We shall show $A = \mathbb{N}^\#$ by applying hyper infinite induction. First we have

(6) $1 \in A$.

To prove (6), it suffices to show that for any $z$, if $(1, z) \in \mathcal{F}$ then $z = c$.

We prove this by contradiction; in other words, suppose to be the contrary that there is some $z$ with $(1, z) \in \mathcal{F}$ but $z \neq c$. Consider the relation $W = \mathcal{F} \setminus \{(1, z)\}$. Since $(1, c) \in \mathcal{F}$ and $(1, c) \neq (1, z)$, it follows that $(1, c) \in W$. Moreover, whenever $(u, y) \in W$ then $(u, y) \in \mathcal{F}$ and hence $(\mathbf{Sc}(u), G(y)) \in \mathcal{F}$ but $\mathbf{Sc}(u) \neq 1$, so $(\mathbf{Sc}(u), G(y)) \neq (1, z)$, and hence $(\mathbf{Sc}(u), G(y)) \in W$. Thus $W$ satisfies both conditions (ii') and (iii'); in other words, $\mathbf{W} \in \mathbf{M}$. But then it follows from (2) that $\mathcal{F} \subseteq \mathbf{W}$ however this is clearly false since $(1, z) \in \mathcal{F}$ and $(1, z) \notin W$. Thus our hypothesis has led us to a
**Definition 7.5.** A set $S \subset_s \mathbb{N}^\#$ is a hyper inductive if the following statement holds

$$ \bigwedge_{\alpha \in \mathbb{N}^\#} (\alpha \in_s S \Rightarrow_s \alpha^+ \in_s S). \qquad (7.8) $$

Obviously a set $\mathbb{N}^\#$ is a hyper inductive. Thus axiom of hyper infinite induction 1 asserts that a set $\mathbb{N}^\#$ this is the smallest hyper inductive set.

**Axioms of hyperfinite induction 2**

Let $F(x)$ be a wff of the set theory $\mathbf{NC}^\#_{\infty\#}$ strictly restricted on a set $[0, \beta]$ then

$$ \left[ \forall \beta (\beta \in [0, \beta]) \left[ \bigwedge_{0 \le \alpha < \beta} (F(\alpha) \Rightarrow_s F(\alpha^+)) \right] \right] \Rightarrow_s \forall \alpha (\alpha \in [0, \beta]) F(\alpha). \qquad (7.9) $$

Let $F(x)$ be a wff of the set theory $\mathbf{NC}^\#_{\infty\#}$ strictly restricted on a set $[0, \beta_\infty]$ then

$$ \left[ \forall \beta (\beta \in [0, \beta_\infty]) \left[ \bigwedge_{0 \le \alpha < \beta_\infty} (F(\alpha) \Rightarrow_s F(\alpha^+)) \right] \right] \Rightarrow_s \forall \alpha (\alpha \in [0, \beta_\infty]) F(\alpha). \qquad (7.10) $$

**Axiom of hyper infinite induction 2**

Let $F(x)$ be anrestricted wff of the set theory $\mathbf{NC}^\#_{\infty\#}$ then

$$ \left[ \forall \beta (\beta \in \mathbb{N}^\#) \left[ \bigwedge_{0 \le \alpha < \beta} (F(\alpha) \Rightarrow_s F(\alpha^+)) \right] \right] \Rightarrow_s \forall \beta (\beta \in \mathbb{N}^\#) F(\beta). \qquad (7.11) $$

**The main restricted rules of conclusion.**

If $\mathbf{A}^\# \vdash A$ then $\neg A \vdash B$, where $B \in \mathcal{L}^\#$.

Thus if statement A holds in $\mathbf{A}^\#$ we cannot obtain from $\neg A$ any formula B whatsoever.

## §8. The Generalized Recursion Theorem.

**Theorem 1.** Let $S$ be a set, $c \in S$ and $G: S \to S$ is any function with $\text{dom}(G) = S$ and $\text{range}(G) \subseteq S$. Let $W[G] \in \mathbb{N}^\# \times S$ be a binary relation such that:

(a) $(1, c) \in W[G]$ and

(b) if $(x, y) \in W[G]$ then $(\mathbf{Sc}(x), G(y)) \in W[G]$.

Then there exists a function $\mathcal{F}: \mathbb{N}^\# \to S$ such that:

(i) $\text{dom}(\mathcal{F}) = \mathbb{N}^\#$ and $\text{range}(\mathcal{F}) \subseteq S$;

(ii) $\mathcal{F}(1) = c$;

(iii) for all $x \in \mathbb{N}^\#$, $\mathcal{F}(\mathbf{Sc}(x)) = G(\mathcal{F}(x))$.

1. The desired function $\mathcal{F}$ is a certain hyper inductive relation $W \subseteq \mathbb{N}^\# \times S$. It is to have the properties:

(ii') $(1, c) \in W$;

(iii') if $(x, y) \in W$ then $(\mathbf{Sc}(x), G(y)) \in W$.

**Remark 1.** The latter is just another way of expressing (iii), that if

$$ \mathcal{F}(x) = y \qquad (1) $$

then

$$ \mathcal{F}(\mathbf{Sc}(x)) = G(y). \qquad (2) $$

**Remark 2.** Note that any relation $W$ mentioned above is hyper inductive relation since the hyper inductivity conditions (ii')-(iii') are satisfied.

However there are many hyper inductive relations which satisfy the conditions (ii')-(iii'); on such is $\mathbb{N}^\# \times S$. What distinguishes the desired function from all
contradiction, and hence (6) is proved. Next we show that
(7) for any $x \in \mathbb{N}^#$ if $x \in A$ then $\mathbf{Sc}(x) \in A$.

Suppose that $x \in A$, so that whenever $(x, z_1) \in \mathcal{F}$ and $(x, z_2) \in \mathcal{F}$ then
$z_1 = z_2$. We must show that whenever $(\mathbf{Sc}(x), w_1) \in \mathcal{F}$ and $(\mathbf{Sc}(x), w_2) \in \mathcal{F}$
then $w_1 = w_2$. To prove this, it suffices to show that
(8) whenever $(\mathbf{Sc}(x), w) \in \mathcal{F}$ then there exists a $z$ with $w = G(z)$ and $(x, z) \in \mathcal{F}$.
For if (8) is true, we would have for the given $w_1, w_2$ some $z_1 = z_2$ with
$w_1 = G(z_1)$, $w_2 = G(z_2)$, $(x, z_1) \in \mathcal{F}$ and $(x, z_2) \in \mathcal{F}$. Then, since $x \in A$, $z_1 = z_2$
and hence $G(z_1) = G(z_2)$, that is, $w_1 = w_2$.

Now to prove (8) suppose, to the contrary, that it is not true; in other words,
suppose that we have some $w$ with $(\mathbf{Sc}(x), w) \in \mathcal{F}$ but such that for all
$z$ which $(x, z) \in \mathcal{F}$ we have $w \neq G(z)$. Consider the relation $\mathbf{W} = \mathcal{F}\{(\mathbf{Sc}(x), w)\}$.
We shall show that $\mathbf{W} \in \mathbf{M}$. First of all $(1, c) \in \mathcal{F}$ and $(1, c) \neq (\mathbf{Sc}(x), w)$; hence
$(1, c) \in \mathbf{W}$. Suppose tbat $(u, y) \in \mathbf{W}$; then $(u, y) \in \mathcal{F}$ and $(\mathbf{Sc}(u), G(y)) \in \mathcal{F}$.
Clearly if $u \neq x$ then $(\mathbf{Sc}(u), G(y)) \neq (\mathbf{Sc}(x), w)$, so that in this case $(\mathbf{Sc}(u), G(y)) \in \mathbf{W}$.
On the other hand, if $u = x$ and $(\mathbf{Sc}(u), G(y)) = (\mathbf{Sc}(x), w)$, then $w = G(y)$, where
$(x, y) \in \mathcal{F}$, contrary to the choice of $w$ henee $(\mathbf{Sc}(u), G(y)) \neq (\mathbf{Sc}(x), w)$, so again
$(\mathbf{Sc}(u), G(y)) \in \mathbf{W}$. Thus whenever $(u, y) \in \mathbf{W}$, also $(\mathbf{Sc}(u), G(y)) \in \mathbf{W}$. Now that we
have shown $\mathbf{W} \in \mathbf{M}$ we see by (2) that $\mathcal{F} \subseteq \mathbf{W}$ but this is false since $(\mathbf{Sc}(x), w) \in \mathcal{F}$
and $(\mathbf{Sc}(x), w) \notin \mathbf{W}$. Thus our hypothesis that (8) is incorrect has led to a
contradiction, and now (8) is proved. Since (7) follows from (8), we have
by *hyper infinite induction from (6)* that $A = \mathbb{N}^#$. Hence
(9) $\mathcal{F}$ is a function.

We have still to prove that $\mathcal{F}$ satisfies condition (i); we must show that
for each $x \in \mathbb{N}^#$ there is a $y$ with $(x, y) \in \mathcal{F}$. Since $\mathcal{F} \subseteq \mathbb{N}^# \times S$, it will
then follow that $\text{dom}(\mathcal{F}) = \mathbb{N}^#$ and $\text{range}(\mathcal{F}) \subseteq S$. Let $B = \text{dom}(\mathcal{F})$, that is,
(10) $B = \{x | x \in \mathbb{N}^# \text{ and for some } y, (x, y) \in \mathcal{F}\}$.

We prove now by **hyper infinite induction** that $B = \mathbb{N}^#$. First, $1 \in B$, *sínce* $(1, c) \in \mathcal{F}$
by (3). Next, if $x \in B$, pick some $y$ with $(x, y) \in \mathcal{F}$; then by (4), $(\mathbf{Sc}(x), G(y)) \in \mathcal{F}$,
and henee $\mathbf{Sc}(x) \in B$.

Thus concludes the first part of the proof, that there is at least one function $\mathcal{F}$
satisfying conditions (i)-(iii).

**Part 2.** We prove that there cannot be more than one such function.
Suppose that $\mathcal{F}_1$ and $\mathcal{F}_2$ both satisfy the conditions (i)-(iii) we wish to show
$\mathcal{F}_1 = \mathcal{F}_2$, i.e., that for all $x \in \mathbb{N}^#$, $\mathcal{F}_1(x) = \mathcal{F}_2(x)$. Thus
is proved by **hyper infinite induction** on $X$. By (ii), $\mathcal{F}_1(1) = c$ and $\mathcal{F}_2(1) = c$, so
$\mathcal{F}_1(1) = \mathcal{F}_2(1)$. Suppose that $\mathcal{F}_1(x) = \mathcal{F}_2(x)$; then $\mathcal{F}_1(\mathbf{Sc}(x)) = G(\mathcal{F}_1(x))$
and $\mathcal{F}_2(\mathbf{Sc}(x)) = G(\mathcal{F}_2(x))$, so $\mathcal{F}_1(\mathbf{Sc}(x)) = \mathcal{F}_2(\mathbf{Sc}(x))$.

**Theorem 2.** Let $S$ be a set, $c \in S$ and $G : S \times \mathbb{N}^# \to S$ is a binary function with
$\text{dom}(G) = S \times \mathbb{N}^# \text{ and } \text{range}(G) \subseteq S$.

Then there exists a function $\mathcal{F} : \mathbb{N}^# \to S$ such that:

(i) $\text{dom}(\mathcal{F}) = \mathbb{N}^# \text{ and } \text{range}(\mathcal{F}) \subseteq S$;

(ii) $\mathcal{F}(1) = c$;

(iii) for all $x \in \mathbb{N}^#$, $\mathcal{F}(\mathbf{Sc}(x)) = G(\mathcal{F}(x), x)$.

We omit the proof of the Theorem 3.4.2 since it can be given by simple modification
of the proof to Theorem 3.4.1.
these other relations is that we want $(a, b)$ to be on it only as required by (ii') and
(iii'). In other words, it is to be the smallest relation satisfying
(ii')-(iii'). This can be expressed precisely as follows:

(1) Let $\mathbf{M}$ be a set of the relations $\mathbf{W}$ satisfying the conditions (ii') and (iii');
then we define

$$
\mathcal{F} = \bigcap_{W \in \mathcal{M}} \mathbf{W}.
$$

Hence

(2) whenever $\mathbf{W} \in \mathbf{M}$ then $\mathcal{F} \subseteq \mathbf{W}$.

We shall now show that we can derived from (1) that $\mathcal{F}$ is also one relation in $\mathbf{M}$.

(3) $(1, c) \in \mathcal{F}$.

This follows immediately from the definition of $\bigcap_{W \in \mathcal{M}}$ and the fact that $(1, c) \in \mathbf{W}$ for all $\mathbf{W} \in \mathcal{M}$.

(4) If $(x, y) \in \mathcal{F}$ then $(\mathbf{Sc}(x), G(y)) \in \mathcal{F}$.

For if $(x, y) \in \mathcal{F}$ then $(x, y) \in \mathbf{W}$ for all $\mathbf{W} \in \mathcal{M}$; hence by (iii')

$(\mathbf{Sc}(x), G(y)) \in \mathbf{W}$ for all $\mathbf{W} \in \mathcal{M}$ so that $(\mathbf{Sc}(x), G(y)) \in \mathcal{F}$ by (1).

We must now verify that $\mathcal{F}$ is actually a function, i,e., we wish to show
that for any $x, z_1, z_2 \in \mathbb{N}^\#$, if $(x, z_1) \in \mathcal{F}$ and $(x, z_2) \in \mathcal{F}$, then $z_1 = z_2$.

We shall prove this by **hyper infinite induction** on $x$. Let

(5) $A = \{x | x \in \mathbb{N}^{\#} \text{ and for all } z_1, z_2 \in \mathbb{N}^{\#}, \text{ if } (x, z_1) \in \mathcal{F} \text{ and } (x, z_2) \in \mathcal{F} \\ \text{then } z_1 = z_2\}$.

We shall show $A = \mathbb{N}^\#$ by applying hyper infinite induction. First we have

(6) $1 \in A$.

To prove (6), it suffices to show that for any $z$, if $(1, z) \in \mathcal{F}$ then $z = c$.

We prove this by contradiction; in other words, suppose to be contrary that there
is some $z$ with $(1, z) \in \mathcal{F}$ but $z \neq c$. Consider the relation $W = \mathcal{F} \setminus \{(1, z)\}$. Since
$(1, c) \in \mathcal{F}$ and $(1, c) \neq (1, z)$, it follows that $(1, c) \in W$. Moreover, whenever $(u, y) \in W$
then $(u, y) \in \mathcal{F}$ and hence $(\mathbf{Sc}(u), G(y)) \in \mathcal{F}$ but $\mathbf{Sc}(u) \neq 1$, so $(\mathbf{Sc}(u), G(y)) \neq (1, z)$,
and hence $(\mathbf{Sc}(u), G(y)) \in W$. Thus $W$ satisfies both conditions (ii') and (iii'); in other
words, $W \in \mathcal{M}$. But then it follows from (2) that $\mathcal{F} \subseteq W$ however this
is clearly false since $(1, z) \in \mathcal{F}$ and $(1, z) \notin W$. Thus our hypothesis has led us to a
contradiction, and hence (6) is proved. Next we show that

(7) for any $x \in \mathbb{N}^\#$ if $x \in A$ then $\mathbf{Sc}(x) \in A$.

Suppose that $x \in A$, so that whenever $(x, z_1) \in \mathcal{F}$ and $(x, z_2) \in \mathcal{F}$ then
$z_1 = z_2$. We must show that whenever $(\mathbf{Sc}(x), w_1) \in \mathcal{F}$ and $(\mathbf{Sc}(x), w_2) \in \mathcal{F}$
then $w_1 = w_2$. To prove this, it suffices to show that

(8) whenever $(\mathbf{Sc}(x), w) \in \mathcal{F}$ then there exists a $z$ with $w = G(z)$ and $(x, z) \in \mathcal{F}$.
For if (8) is true, we would have for the given $w_1, w_2$ some $z_1 = z_2$ with
$w_1 = G(z_1)$, $w_2 = G(z_2)$, $(x, z_1) \in \mathcal{F}$ and $(x, z_2) \in \mathcal{F}$. Then, since $x \in A, z_1 = z_2$
and hence $G(z_1) = G(z_2)$, that is, $w_1 = w_2$.

Now to prove (8) suppose, to the contrary, that it is not true; in other words,
suppose that we have some $w$ with $(\mathbf{Sc}(x), w) \in \mathcal{F}$ but such that for all
$z$ which $(x, z) \in \mathcal{F}$ we have $w \neq G(z)$. Consider the relation $W = \mathcal{F} \setminus \{(\mathbf{Sc}(x), w)\}$.

We shall show that $\mathbf{W} \in \mathcal{M}$. First of all $(1, c) \in \mathcal{F}$ and $(1, c) \neq (\mathbf{Sc}(x), w)$; hence
$(1, c) \in \mathbf{W}$. Suppose that $(u, y) \in \mathbf{W}$; then $(u, y) \in \mathcal{F}$ and $(\mathbf{Sc}(u), G(y)) \in \mathcal{F}$.
§9. General associative and commutative laws.

**Definition C.1.** Suppose that $\mathcal{S}$ is a set on which a binary operation $+$ is defined and under which $\mathcal{S}$ is closed. Let $\{x_k\}_{k \in \mathbb{N}^#}$ be any hyper infinite sequence of terms of $\mathcal{S}$. For every $n \in \mathbb{N}^#$ we denote by $\text{Ext}-\sum_{k=1}^{n} x_k$ the element of $\mathcal{S}$ uniquely determined by the following conditions:

(i) $\text{Ext}-\sum_{k=1}^{1} x_k = x_1$; (ii) $\text{Ext}-\sum_{k=1}^{n+1} x_k = \text{Ext}-\sum_{k=1}^{n} x_k + x_{n+1}$ for all $n \in \mathbb{N}^#$.

**Remark 9.1.** This definition is justified on the following grounds. The sequence $\{x_k\}_{k \in \mathbb{N}^#}$ is a given external function $H$ with domain $\mathbb{N}^#$, $H(x_k) = x_k$ for every $k$. We seek a function $F$ with domain $\mathbb{N}^#$ whose value $F(n)$ is to be $\text{Ext}-\sum_{k=1}^{n} x_k$. Then the conditions

(i), (ii) above correspond to the following conditions on $F$:

(i') $F(1) = H(1)$; (ii') $F(n+1) = F(n) + H(n+1)$, for all $n \in \mathbb{N}^#$.

Let (1) $c = H(1)$; (2) $G(n, z) = z + H(n+1)$.

Thus the conditions (i') and (ii') are equivalent to

(i'') $F(1) = c$;

(ii'') $F(n+1) = G(n, F(n))$ for all $n \in \mathbb{N}^#$.

Given the function $H$, the element $c$ of $\mathcal{S}$ and the function $G$ are well-defined by (1)-(2).

Then by Theorem B.1 we see that there is a unique function $F$ satisfying (1)-(2) with $\text{dom}(F) = \mathbb{N}^#$ and $\text{range}(F) \subseteq \mathcal{S}$. Thus (i')-(ii') is just another form of recursive definition.

(Hence it should be expected that various properties of $\text{Ext}-\sum_{k=1}^{n} x_k$ will have to be verified by hyper infinite induction on $n \in \mathbb{N}^#$.)

**Definition 9.2.** Suppose that $\mathcal{S}$ is a set on which a binary operation $\times$ is defined and under which $\mathcal{S}$ is dosed. Let $\{x_k\}_{k \in \mathbb{N}^#}$ be an hyper infinite sequence of terms of $\mathcal{S}$. For every $n \in \mathbb{N}^#$ we denote by $\text{Ext}-\prod_{k=1}^{n} x_k$ the element of $\mathcal{S}$ uniquely determined by the following conditions:

(i) $\text{Ext}-\prod_{k=1}^{n} x_k = x_1$; (ii) $\text{Ext}-\prod_{k=1}^{n+1} x_k = \left(\text{Ext}-\prod_{k=1}^{n} x_k\right) \times x_{n+1}$ for all $n \in \mathbb{N}^#$.

**Theorem 1.** (1) Suppose that $\mathcal{S}$ is a set closed under a binary operation $+$ and that $+$ is associative on $\mathcal{S}$, i.e., for all $x, y, z \in \mathcal{S}, x + (y + z) = (x + y) + z$. Let $\{x_k\}_{k \in \mathbb{N}^#}$ be any hyper infinite sequence of terms in $\mathcal{S}$. Then for any $n, m \in \mathbb{N}^#$. we have

$$ \text{Ext}-\sum_{k=1}^{n+m} x_k = \left( \text{Ext}-\sum_{k=1}^{n} x_k \right) + \left( \text{Ext}-\sum_{k=1}^{m} x_{n+k} \right). \qquad (9.1) $$

(2) Suppose that $\mathcal{S}$ is a set closed under a binary operation $\times$ and that $\times$ is associative on $\mathcal{S}$, i.e., for all $x, y, z \in \mathcal{S}, x \times (y \times z) = (x \times y) \times z$. Let $\{x_k\}_{k \in \mathbb{N}^#}$ be any hyper infinite sequence of terms in $\mathcal{S}$. Then for any $n, m \in \mathbb{N}^#$. we have