bluuebunny/crossref_metadata_2025_split huggingface dataset:
The equivalence of NF-style set theories with “tangled” type theories; the construction of ω-models of predicative NF (and more)
Authors: Holmes M. Randall 
Published: May 2006

AbstractAn ω-model (a model in which all natural numbers are standard) of the predicative fragment of Quine's set theory “New Foundations” (NF) is constructed. Marcel Crabbé has shown that a theory NFI extending predicative NF is consistent, and the model constructed is actually a model of NFI as well. The construction follows the construction of ω-models of NFU (NF with urelements) by R. B. Jensen, and, like the construction of Jensen for NFU, it can be used to construct α-models for any ordinal α. The construction proceeds via a model of a type theory of a peculiar kind; we first discuss such “tangled type theories” in general, exhibiting a “tangled type theory” (and also an extension of Zermelo set theory with Δ0 comprehension) which is equiconsistent with NF (for which the consistency problem seems no easier than the corresponding problem for NF (still open)), and pointing out that “tangled type theory with urelements” has a quite natural interpretation, which seems to provide an explanation for the more natural behaviour of NFU relative to the other set theories of this kind, and can be seen anachronistically as underlying Jensen's consistency proof for NFU.
Some applications of coarse inner model theory
Authors: Hjorth Greg 
Published: May 2006

AbstractThe Martin-Steel coarse inner model theory is employed in obtaining new results in descriptive set theory. determinacy implies that for every thin equivalence relation there is a real, N, over which every equivalence class is generic—and hence there is a good (N#) wellordering of the equivalence classes. Analogous results are obtained for and quasilinear orderings and determinacy is shown to imply that every prewellorder has rank less than .
A logical presentation of the continuous functionals
Authors: Palmgren Erik , Stoltenberg-Hansen Viggo 
Published: May 2006

The Kleene-Kreisel continuous functionals [6, 7] have been given several alternative characterisations: using Kuratowski's limit spaces (Scarpellini [20]), using their generalisation, filter spaces (Hyland [4]), via hyperfinite functionals (Normann [11]) and perhaps most elegantly using Scott-Ershov domains (Ershov [3], later generalised by Berger [1]). We propose to add yet another characterisation to this list, which may be called model-theoretic in contrast to the others, but which is in fact closely related to Ershov's approach.We use the notion of logically presented domains developed in Palmgren and Stoltenberg-Hansen [17] (originating in the work of [15]). Certain logical types, i.e., finitely consistent sets of formulas, over the full type structure built from ℕ correspond to the continuous functionals in the sense of Kreisel, or to Kleene's associates, while the elements realising the types correspond to functionals having associates (cf. [6]). To define these—the total types—we make a nonstandard extension of the full type structure. The set of nonstandard elements is sufficiently rich to single out the total types. The nonstandard extension also makes it possible to relate the logical presentation to Ershov's approach. The logical form of the domain constructions allows us to use a (generalised) Fréchet power as a nonstandard extension. This extension is constructive, in the sense that it avoids the axiom of choice, as distinguished from the one employed in [11].
How to characterize provably total functions by local predicativity
Authors: Weiermann Andreas 
Published: May 2006

AbstractInspired by Pohlers' proof-theoretic analysis of KPω we give a straightforward non-metamathematical proof of the (well-known) classification of the provably total functions of PA, PA + TI(⊰ ↾) (where it is assumed that the well-ordering ⊰ has some reasonable closure properties) and KPω. Our method relies on a new approach to subrecursion due to Buchholz, Cichon and the author.
Countable unions of simple sets in the core model
Authors: Welch P. D. 
Published: May 2006

AbstractWe follow [8] in asking when a set of ordinals X ⊆ α is a countable union of sets in K, the core model. We show that, analogously to L, an X closed under the canonical Σ1 Skolem function for Kα can be so decomposed provided K is such that no ω-closed filters are put on its measure sequence, but not otherwise. This proviso holds if there is no inner model of a weak Erdős-type property.
Prototypes for definable subsets of algebraically closed valued fields
Authors: Holly Jan E. 
Published: May 2006

AbstractElimination of imaginaries for 1-variable definable equivalence relations is proved for a theory of algebraically closed valued fields with new sorts for the disc spaces. The proof is constructive, and is based upon a new framework for proving elimination of imaginaries, in terms of prototypes which form a canonical family of formulas for defining each set that is definable with parameters. The proof also depends upon the formal development of the tree-like structure of valued fields, in terms of valued trees, and a decomposition of valued trees which is used in the coding of certain sets of discs.
Proofs of strong normalisation for second order classical natural deduction
Authors: Parigot Michel 
Published: May 2006

AbstractWe give two proofs of strong normalisation for second order classical natural deduction. The first one is an adaptation of the method of reducibility candidates introduced in [9] for second order intuitionistic natural deduction; the extension to the classical case requires in particular a simplification of the notion of reducibility candidate. The second one is a reduction to the intuitionistic case, using a Kolmogorov translation.
Definability of types, and pairs of O-minimal structures
Authors: Pillay Anand 
Published: May 2006

AbstractLet T be a complete O-minimal theory in a language L. We first give an elementary proof of the result (due to Marker and Steinhorn) that all types over Dedekind complete models of T are definable. Let L* be L together with a unary predicate P. Let T* be the L*-theory of all pairs (N, M), where M is a Dedekind complete model of T and N is an ⅼMⅼ+-saturated elementary extension of N (and M is the interpretation of P). Using the definability of types result, we show that T* is complete and we give a simple set of axioms for T*. We also show that for every L*-formula ϕ(x) there is an L-formula ψ(x) such that T* ⊢ (∀x)(P(x) → (ϕ(x) ↔ ψ(x)). This yields the following result:Let M be a Dedekind complete model of T. Let ϕ(x, y) be an L-formula where l(y) – k. Let X = {X ⊂ Mk: for some a in an elementary extension N of M, X = ϕ(a, y)N ∩ Mk}. Then there is a formula ψ(y, z) of L such that X = {ψ(y, b)M: b in M}.
On countable fractions from an elementary class
Authors: Ash C. J. 
Published: May 2006

The following fairly elementary result seems to raise possibilities for the study of countable models of a theory in a countable language. For the terminology of model theory we refer to [CK].Let L be a countable first-order language. Let L′ be the relational language having, for each formula φ of L and each sequence υ1,…,υn of variables including the free variables of φ, an n-ary relation symbol Pφ. For any L-structure and any formula Ψ(υ) of L, we define the Ψ-fraction of to be the L′-structure Ψ whose universe consists of those elements of satisfying Ψ(υ) and whose relations {Rφ}φϵL are defined by letting a1,…,an satisfy Rφ in Ψ if, and only if, a1,…, an satisfy φ in .An L-elementary class means the class of all L-structures satisfying each of some set of sentences of L. The countable part of an L-elementary class K means the class of all countable L-structures from K.Theorem. Let K be an L-elementary class and let Ψ(υ) be a formula of L. Then the class of countable Ψ-fractions of structures in K is the countable part of some L′-elementary class.Comment. By the downward Löwenheim-Skolem theorem, the countable Ψ-fractions of structures in K are the same as the Ψ-fractions of countable structures in K.Proof. We give a set Σ′ of L′-sentences whose countable models are exactly the countable Ψ-fractions of structures in K.
More on imaginaries in p-adic fields
Authors: Scowcroft Philip 
Published: May 2006

According to [4, p. 1154], a complete L-theory T eliminates imaginaries just in case for every L-formula φ(x1,… , xm, y1, …, yn), every model M of T, and every ā Є Mn, there is a subset A of M's domain with the following property: if N ≽ M and f is an automorphism of N, thenif and only ifAmong the several equivalent conditions discussed in [4, p. 1155], one may single out the following: if T is a complete theory in which two distinct objects are definable, T eliminates imaginaries just in case every T-definable n-ary equivalence relation may be defined by a formulawhere g is a T-definable n-ary function taking k-tuples as values (for some natural number k).Say that an L-structure M eliminates imaginaries just in case Th(M) does. If L is the language of rings with unit, [4, p. 1158] shows that any algebraically closed field eliminates imaginaries, and [2, p. 629] points out that any real-closed field eliminates imaginaries.
Une correspondance entre anneaux partiels et groupes
Authors: Simonetta Patrick 
Published: May 2006

AbstractThis work is inspired by the correspondence of Malcev between rings and groups. Let A be a domain with unit, and S a multiplicative group of invertible elements. We define AS as the structure obtained from A by restraining the multiplication to A × S, and σ(AS) as the group of functions from A to A of the form x → xa + b, where (a, b) belongs to S × A. We show that AS and σ(As) are interpretable in each other, and then, that we can transfer some properties between classes (or theories) of “reduced” domains and corresponding groups, such as being elementary, axiomatisability (for classes), decidability, completeness, or, in some cases, existence of a model-completion (for theories).We study the extensions of the additive group of A by the group S, acting by right multiplication, and show that sometimes σ(AS) is the unique extension of this type. We also give conditions allowing us to eliminate parameters appearing in interpretations.We emphasize the case where the domain is a division ring K and S is its multiplicative group K×. Here, the interpretations can always be done without parameters. If the centre of K contains more than two elements, then σ(K) is the only extension of the additive group of K by its multiplicative group acting by right multiplication, and the class of all such σ(K)'s is elementary and finitely axiomatisable. We give, in particular, an axiomatisation for this class and for the class of σ(K)'s where K is an algebraically closed field of characteristic 0. From these results it follows that some classical model-companion results about theories of fields can be translated and restated as results about theories of solvable groups of class 2.
Identities on cardinals less than ℵω
Authors: Gilchrist M. , Shelah S. 
Published: May 2006

Let κ be an uncountable cardinal and the edges of a complete graph with κ vertices be colored with ℵ0 colors. For the Erdős-Rado theorem implies that there is an infinite monochromatic subgraph. However, if , then it may be impossible to find a monochromatic triangle. This paper is concerned with the latter situation. We consider the types of colorings of finite subgraphs that must occur when the edges of the complete graph on vertices are colored with ℵ0 colors. In particular, we are concerned with the case ℵ1 ≤ κ ≤ ℵω.The study of these color patterns (known as identities) has a history that involves the existence of compactness theorems for two cardinal models [2]. When the graph being colored has size ℵ1, the identities that must occur have been classified by Shelah [4]. If the graph has size greater than or equal to ℵω the identities have also been classified in [3]. The number of colors is fixed at ℵ0 as it is the natural place to start and the results here can be generalized to situations where more colors are used.There is one difference that we now make explicit. When countably many colors are used we can define the following coloring of the complete graph on vertices. First consider the branches in the complete binary tree of height ω to be vertices of a complete graph.
Relational and partial variable sets and basic predicate logic
Authors: Ghilardi Silvio , Meloni Giancarlo 
Published: May 2006

AbstractIn this paper we study the logic of relational and partial variable sets, seen as a generalization of set-valued presheaves, allowing transition functions to be arbitrary relations or arbitrary partial functions. We find that such a logic is the usual intuitionistic and co-intuitionistic first order logic without Beck and Frobenius conditions relative to quantifiers along arbitrary terms. The important case of partial variable sets is axiomatizable by means of the substitutivity schema for equality. Furthermore, completeness, incompleteness and independence results are obtained for different kinds of Beck and Frobenius conditions.
A functorial property of the Aczel-Buchholz-Feferman function
Authors: Weiermann Andreas 
Published: May 2006

AbstractLet Ω be the least uncountable ordinal. Let be the category where the objects are the countable ordinals and where the morphisms are the strictly monotonic increasing functions. A dilator is a functor on which preserves direct limits and pullbacks. Let τ < ΩE ≔ min{ξ > Ω: ξ = ωξ}. Then τ has a unique “term”-representation in Ω. λξη.ωξ + η and countable ordinals called the constituents of τ. Let δ < Ω and K(τ) be the set of the constituents of τ. Let β = max K(τ). Let [β] be an occurrence of β in τ such that τ[β] = τ. Let be the fixed point-free version of the binary Aczel-Buchholz-Feferman-function (which is defined explicitly in the text below) which generates the Bachman-hierarchy of ordinals. It is shown by elementary calculations that is a dilator for every γ > max{β.δ.ω}.
Rational Pavelka predicate logic is a conservative extension of Łukasiewicz predicate logic
Authors: Hájek Petr , Paris Jeff , Shepherdson John 
Published: May 2006

AbstractRational Pavelka logic extends Łukasiewicz infinitely valued logic by adding truth constants r̄ for rationals in [0. 1]. We show that this is a conservative extension. We note that this shows that provability degree can be defined in Łukasiewicz logic. We also give a counterexample to a soundness theorem of Belluce and Chang published in 1963.
Multiplicative conjunction and an algebraic meaning of contraction and weakening
Authors: Avron A. 
Published: May 2006

AbstractWe show that the elimination rule for the multiplicative (or intensional) conjunction Λ is admissible in many important multiplicative substructural logics. These include LLm (the multiplicative fragment of Linear Logic) and RMIm (the system obtained from LLm by adding the contraction axiom and its converse, the mingle axiom.) An exception is Rm (the intensional fragment of the relevance logic R, which is LLm together with the contraction axiom). Let SLLm and SRm be, respectively, the systems which are obtained from LLm and Rm by adding this rule as a new rule of inference. The set of theorems of SRm is a proper extension of that of Rm, but a proper subset of the set of theorems of RMIm. Hence it still has the variable-sharing property. SRm has also the interesting property that classical logic has a strong translation into it. We next introduce general algebraic structures, called strong multiplicative structures, and prove strong soundness and completeness of SLLm relative to them. We show that in the framework of these structures, the addition of the weakening axiom to SLLm corresponds to the condition that there will be exactly one designated element, while the addition of the contraction axiom corresponds to the condition that there will be exactly one nondesignated element (in the first case we get the system BCKm, in the second - the system SRm). Various other systems in which multiplicative conjunction functions as a true conjunction are studied, together with their algebraic counterparts.
Dominating and unbounded free sets
Authors: Solecki Slawomir , Spinas Otmar 
Published: May 2006

AbstractWe prove that every analytic set in ωω × ωω with σ-bounded sections has a not σ-bounded closed free set. We show that this result is sharp. There exists a closed set with bounded sections which has no dominating analytic free set. and there exists a closed set with non-dominating sections which does not have a not σ-bounded analytic free set. Under projective determinacy analytic can be replaced in the above results by projective.
Cardinal preserving ideals
Authors: Gitik Moti , Shelah Saharon 
Published: May 2006

AbstractWe give some general criteria, when κ-complete forcing preserves largeness properties—like κ-presaturation of normal ideals on λ (even when they concentrate on small cofinalities). Then we quite accurately obtain the consistency strength “NSλ is αi-preserving”, for λ > α2.
Displaying the modal logic of consistency
Authors: Wansing Heinrich 
Published: May 2006

AbstractIt is shown that the constructive four-valued logic N4 can be faithfully embedded into the modal logic S4. This embedding is used to obtain complete, cut-free display sequent calculi for N4 and C4, the modal logic of consistency over N4. C4 is a natural monotonic base system for semantics-based non-monotonic reasoning.
A minimal counterexample to universal baireness
Authors: Hauser Kai 
Published: May 2006

AbstractFor a canonical model of set theory whose projective theory of the real numbers is stable under set forcing extensions, a set of reals of minimal complexity is constructed which fails to be universally Baire. The construction uses a general method for generating non-universally Baire sets via the Levy collapse of a cardinal, as well as core model techniques. Along the way it is shown (extending previous results of Steel) how sufficiently iterable fine structure models recognize themselves as global core models.
Standardization principle of nonstandard universes
Authors: Murakami Masahiko 
Published: May 2006

AbstractA bounded ultrasheaf is a nonstandard universe constructed from a superstructure in a Boolean valued model of set theory. We consider the bounded elementary embeddings between bounded ultrasheaves. Then the standardization principle is true if and only if the ultrafilters are comparable by the Rudin-Frolik order. The base concept is that the bounded elementary embeddings correspond to the complete Boolean homomorphisms. We represent this by the Rudin-Keisler order of ultrafilters of Boolean algebras.
Sequentially continuous linear mappings in constructive analysis
Authors: Bridges Douglas , Mines Ray 
Published: May 2006

A mapping u: X → Y between metric spaces is sequentially continuous if for each sequence (xn) converging to x ∈ X, (u(xn)) converges to u(x). It is well known in classical mathematics that a sequentially continuous mapping between metric spaces is continuous; but, as all proofs of this result involve the law of excluded middle, there appears to be a constructive distinction between sequential continuity and continuity. Although this distinction is worth exploring in its own right, there is another reason why sequential continuity is interesting to the constructive mathematician: Ishihara [8] has a version of Banach's inverse mapping theorem in functional analysis that involves the sequential continuity, rather than continuity, of the linear mappings; if this result could be upgraded by deleting the word “sequential”, then we could prove constructively the standard versions of the inverse mapping theorem and the closed graph theorem.Troelstra [9] showed that in Brouwer's intuitionistic mathematics (INT) a sequentially continuous mapping on a separable metric space is continuous. On the other hand, Ishihara [6, 7] proved constructively that the continuity of sequentially continuous mappings on a separable metric space is equivalent to a certain boundedness principle for subsets of ℕ; in the same paper, he showed that the latter principle holds within the recursive constructive mathematics (RUSS) of the Markov School. Since it is not known whether that principle holds within Bishop's constructive mathematics (BISH), of which INT and RUSS are models and which can be regarded as the constructive core of mathematics, the exploration of sequential continuity within BISH holds some interest.
A remark on Martin's Conjecture
Authors: Gao Su 
Published: May 2006

AbstractWe prove that the strong Martin conjecture is false. The counterexample is the first-order theory of infinite atomic Boolean algebras. We show that for this class of Boolean algebras, the classification of their (ω + ω)-elementary theories can be reduced to the classification of the elementary theories of their quotient algrbras modulo the Frechét ideals.
Home Rule in New York State
Authors: McGoldrick Joseph 
Published: May 2006

Home rule in the United States has for many years been confined to the states west of the Mississippi. As late as 1920 Ohio and Michigan were the only eastern states giving their cities autonomy. Since that date, however, there has appeared a tendency in the larger urban industrial states to treat cities more generously. New York in 1923 and Pennsylvania in 1924 have now joined the ranks of home-rule states. But many of the largest states, for example, Illinois, still give no such power to their municipalities, while in Massachusetts it can still be said that cities have the same legal status as “an infant, an idiot or a lunatic.”The addition of New York to this group has just been rendered certain. On September 2, the court of appeals of the state rendered a unanimous decision declaring the home-rule amendment to have been validly adopted. This ended two months of uncertainty caused by the decision of the appellate division of the supreme court in New York City that the home-rule amendment was not validly a part of the constitution and that all state and local legislation passed under its aegis was void.
First and Second Sessions of the Sixty-Eighth Congress: December 3, 1923 to June 7, 1924; December 1, 1924 to March 4, 1925
Authors: Rogers Lindsay 
Published: May 2006

The Sixty-eighth Congress had only two sessions. Not since the Sixty-fourth Congress (1915–17) had members enjoyed such lengthy vacations, for, of the Congresses since 1900, seven have had special sessions. This now seems to be the rule rather than the exception, so numerous and technical are the problems that confront the national legislature.The President and Congress. Four months before the Congress met, Mr. Harding died and Mr. Coolidge succeeded to the Presidency. The first session ran its course under the shadow of the approaching Presidential election.
Inferential equivalence and natural deduction
Authors: Hiż H. 
Published: May 2006

There are axiomatizations of the sentential calculus which, in effect, assert the inferential equivalence (mutual deducibility) of classes of sentences. A well known axiomatization of this sort consists ofAnother one, closely connected with A1–A3, is comprised ofB1–B3 jointly assert the inferential equivalence of a sentence of the form (p ⊃ q) ⊃ r and two sentences of the forms q ⊃ r and ∼p ⊃ r respectively. This axiomatization requires the rule of substitution and the rule of detachment for conditionals.The existence of such axiomatizations suggests a possibility of formulating natural deduction by means of inferential equivalence instead of the usual one-sided inference, customarily represented by ‘⊦’. ‘H’ will stand here for inferential equivalence. It may be looked upon as a combination of signs ‘⊦’ and ‘⊣’. ‘H’ is thought of as a metasystematic constant functor of two arguments each of which is a class of names of sentences. For purposes of this paper the first argument may be just a single name of a sentence, the second a pair (not necessarily ordered) of names of sentences. The role ‘H’ plays is determined by **1 –*5 below. Besides that ‘H’ remains here uninterpreted. In particular, it is not directly postulated that it is an identity relation.