bluuebunny/crossref_metadata_2025_split huggingface dataset:
Some local definability results on countable topological structures
Authors: Eisenmenger Holger 
Published: May 2006

L denotes a fixed finitary similarity type, B, respectively P a new relation symbol, an L-structure in the usual sense, and (, σ) a topological L-structure, where σ is a topology on A. (, σ) is countable if is countable and σ has a countable base. The formal language for our study of topological structures is . is the least fragment of the (monadic) second-order, infinitary language closed under negation (⇁), countable disjunction (∨), countable conjunction (∧), quantification over individual variables (∃ν, ∀ν), and quantification over set variables in the form ∃V(t ∈ V → φ) [respectively ∃V(t ∈ V → φ] where t is an L-term and each free occurrence of V in φ is negative [respectively positive]. We abbreviate ∃V(t ∈ V ∧ φ) and ∀V(t ∈ V → φ) by ∃V ∈ ν φ respectively ∀V ∈ ν φ. (For detailed information on we reIer to [1].)i, j, … m, n range over ω. a, x, etc. denote finite tuples; a ∈ A means that all members of a are in A. IdA denotes the identity on A, Perm(A) the set of all permutations of A, and Aut() (respectively Aut(, σ)) the set of all automorphisms of (respectively (σ)). Let F ⊆ Perm(A), B ⊆ Am(m ≥ 1), and μ be a system of subsets of A. B (respectively μ) is called invariant under F if for all ƒ ∈ F, ƒ(B) = B (respectively ƒ(μ) = μ). denotes the least system of subsets of A which contains μ and which is closed under arbitrary union, .For the rest of this paragraph let A be a countable nonempty set.
The perfect set theorem and definable wellorderings of the continuum
Authors: Kechris Alexander S. 
Published: May 2006

AbstractLet Γ be a collection of relations on the reals and let M be a set of reals. We call M a perfect set basis for Γ if every set in Γ with parameters from M which is not totally included in M contains a perfect subset with code in M. A simple elementary proof is given of the following result (assuming mild regularity conditions on Γ and M): If M is a perfect set basis for Γ, the field of every wellordering in Γ is contained in M. An immediate corollary is Mansfield's Theorem that the existence of a Σ21 wellordering of the reals implies that every real is constructible. Other applications and extensions of the main result are also given.
A characterization of companionable, universal theories
Authors: Wheeler William H. 
Published: May 2006

A first-order theory is companionable if it is mutually model-consistent with a model-complete theory. The latter theory is then called a model-companion for the former theory. For example, the theory of formally real fields is a companionable theory; its model-companion is the theory of real closed fields. If a companionable, inductive theory has the amalgamation property, then its model-companion is actually a model-completion. For example, the theory of fields is a companionable, inductive theory with the amalgamation property; its model-completion is the theory of algebraically closed fields.The goal of this paper is the characterization, by “algebraic” or “structural” properties, of the companionable, universal theories which satisfy a certain finiteness condition. A theory is companionable precisely when the theory consisting of its universal consequences is companionable. Both theories have the same model-companion if either has one. Accordingly, nought is lost by the restriction to universal theories. The finiteness condition, finite presentation decompositions, is an analogue for an arbitrary theory of the decomposition of a radical ideal in a Noetherian, commutative ring into a finite intersection of prime ideals for the theory of integral domains. The companionable theories with finite presentation decompositions are characterized by two properties: a coherence property for finitely generated submodels of finitely presented models and a homomorphism lifting property for homomorphisms from submodels of finitely presented models.
A continuous functional with noncollapsing hierarchy
Authors: Normann Dag 
Published: May 2006

In [5] S. S. Wainer introduces a hierarchy for arbitrary type-2-functionals. Given F, he defines a set of ordinal notations OF, and for each a ∈ OF, a function fa recursive in F and an ordinal ∣a∣F < For any f recursive in F there is an a ∈ OF such that f is primitive recursive in fa.Let ρF be the least ordinal α such that for any f recursive in F there is an α ∈ OF with ∣a∣F ≤ α such that f is primitive recursive in fa. If ρF < the hierarchy breaks down. In Bergstra and Wainer [2] ρF is described as “the real ordinal of the 1-section of F”.Using standard methods (originally due to Kleene) one may prove that if F is normal, then ρF = Feferman has proved that if F is recursive, then ρF = ω2.Let 1-section (F) = l-sc(F) = {f; f is recursive in F} where f is a total object of type 1.Grilliot [4] proved that F ↾ 1-sc(F) is continuous if and only if F is not normal.Let h be an associate for a given functional F, and assume that h is recursive in the jump of an element of 1-sc(F).
Uniform upper bounds on ideals of turing degrees
Authors: Hodes Harold T. 
Published: May 2006

Given I, a reasonable countable set of Turing degrees, can we find some sort of canonical strict upper bound on I? If I = {a ∣ a ≤ b}, the upper bound on I which springs to mind is b′. But what if I is closed under jump? This question arises naturally out of the question which motivates a large part of hierarchy theory: Is there a canonical increasing function from a countable ordinal, preferably a large one, into D, the set of Turing degrees? If d is to be such a function, it is natural to require that d(α + 1) = d(α)′; but how should d(λ) depend on d ↾ λ, where λ is a limit ordinal?For any I ⊆ D, let MI, = ⋃I. Towards making the above questions precise, we introduce ideals of Turing degrees.Definition 1. I ⊆ D is an ideal iff I is closed under jump and join, and I is downward-closed, i.e., if a ≤ b & b ϵ I then a ϵ I.The following definition reflects the hierarchy-theoretic motivation for this paper.Definition 2. For I ⊆ D and A ⊆ ω, I is an A-hierarchy ideal iff for some countable ordinal α, MI = Lα[A]∩ ωω.All hierarchy ideals are ideals, but not conversely.Early in the game Spector knocked out the best sort of canonicity for upper bounds on ideals, proving that no set of degrees closed under jump has a least upper bound.
Compactness of a supervaluational language
Authors: Bencivenga E. 
Published: May 2006

Compactness of supervaluational semantics has long been an open problem. Recently, Peter Woodruff [2] showed that the full quantificational language is not compact. In the present note, I will show that the quantifier-free fragment of the language is compact. Since this result can be easily extended to the monadic quantificational language, and since Woodruff's result only requires the presence of one binary predicate, the two results combined give a complete solution of the problem.My language L contains infinitely many individual constants, infinitely many n-ary predicates (for every n > 0), the usual connectives, the symbols E! and =, and parentheses. (Atomic) sentences are defined as usual, and Ab/a is the result of uniformly substituting b for a in the sentence A.An interpretation of L is an ordered pair I = 〈D, φ〉, where D is a set (possibly empty) and φ is a unary function, total from the set of n-ary predicates into the power set of Dn, and partial from the set of constants into D.A classical valuation for an interpretation I = 〈D, φ〉 is a total unary function V from the set of sentences into {T, F} such that(a) if φ is defined for all of a1, …, am, then V(Pa1 … an) = T iff 〈φ(a1), …, φ(an)〉 ∈ φ(P);(b) if φ(a) and φ(b) are both defined, then V(a = b) = T iff φ(a) = φ(b;(c) if exactly one of φ(a) and φ(b) is defined, then V(a = b) = F;(d) V(a = a) = T;(e) if V(a = b) = T and A is atomic, then V(A) = V(Ab/a);(f)V(E!a) = T iff φ(a) is defined;(g) V(~ A) = T iff V(A) = F, and similarly for the other connectives.
State-strategies for games in Fσδ ∩ Gδσ
Authors: Büchi J. Richard 
Published: May 2006

The Cantor-Bendixson theorem says: For every closed set , ∣∣ ≤ ω0 ∨ ⊇ perfect (and therefore is countable or has the power of the continuum). W. H. Young proved the alternative for Gδ-sets, and Hausdorff extended it to Souslin-sets. Cantor used his ordinals to prove the theorem (I like to think he invented them for this purpose). In Hausdorff's Mengenlehre ordinals do not officially enter in either of the three proofs. You should try to put them back. It is very important never to hide away ω1 when it actually is there, and it is there when the continuum problem is the subject.In Davis [6] you find that Cantor's alternative is equivalent to another alternative: (I wins ) ∨ (J wins ). Here means a lopsided game in which one of the players makes long moves on the game tree. So, the classical theorems can be restated as determinacy results for lopsided games, and now I make this observation: The original CB-proof “actually presents” a winning strategy. The HD-proofs do no such thing; all you know at the end is existence of a winning strategy (and Davis' remark is needed to gain this knowledge).You understand now why I like CB-proofs, particularly for determinacy of games. They do exist also for games in which both players make short moves. Many years ago I tried to publish such a proof for determinacy of Fσ-games. The referee said it was not worth the trouble. I say you should remake this proof (it is a special case of the one presented in this paper), because it is the ideal of a determinacy proof.
Corrigendum: “Rings which admit elimination of quantifiers”
Authors: Rose Bruce I. 
Published: May 2006

Matatyahu Rubin pointed out that the proof of Lemma 6.1 [2] works only for rings of prime or zero characteristic. This invalidates the characterization of semiprime rings with the descending chain condition on right or left ideals which admit elimination of quantifiers given in [2] and cited in the abstract [1]. Although the correct characterization is easy to derive, it is complex to state.Let be the class of finite fields. Let be the class of 2 × 2 matrix rings over a field with a prime number of elements. Let be the class of rings of the form GF(pn) ⊕ GF(pk) such that either n = k or g.c.d.(n, k) = 1 and p is a prime. Let ′ be the class of algebraically closed fields. Let P denote the set of all prime numbers together with zero. Let be the set of all ordered pairs (f, Q) where Q is a finite subset of P and f: Q → ⋃ ⋃ ⋃ such that the characteristic of the ring f(q) is q. Finally, let be the class of rings of the form ⊕q∈Qf(q) for some (f,Q) in .A corrected version of Theorem 6.2 [2] isTheorem 1. Let R be a ring with the descending chain condition on left or right ideals and without nonzero trivial ideals. Then R admits elimination of quantifiers if only if R belong to.
Concerning ultrafilters on ultrapowers
Authors: Henle J. M. 
Published: May 2006

This paper concerns ultrafilters on a cardinal γ extending the filter of λ-closed, unbounded sets, λ < γ. The history of these ultrafilters is closely connected with that of the axiom of determinacy (AD). Solovay noticed first that, under AD, there was such an ultrafilter for γ = ℵ1; λ = ω. Later, Kleinberg found that the existence of such ultrafilters followed from the partition relation γ → (γ)λ+λ. Specific instances of this and more powerful relations on cardinals were then proved from AD by Martin, Kunen, Paris, and others. The axiom of determinacy was recently shown consistent with ZF relative to something less than a supercompact cardinal by Martin and Steel. Solovay's and Kleinberg's results were actually stronger, and we discuss this at the end of the paper. Good references for these results include [K2] and [KM].We are interested here in the case where γ is the ultrapower of a strong partition cardinal κ (a cardinal satisfying for all α < κ). Such cardinals exist in great abundance assuming AD, and in fact, if sufficiently many cardinals are strong, then AD holds in L[R] [KKMW].
A constructive interpretation of the full set theory
Authors: Turchin Valentin F. 
Published: May 2006

The interpretation of the ZF set theory reported in this paper is, actually, part of a wider effort, namely, a new approach to the foundation of mathematics, which is referred to as The Cybernetic Foundation. A detailed exposition of the Cybernetic Foundation will be published elsewhere. Our approach leads to a full acceptance of the formalism of the classical set theory, but interprets it using only the idea of potential, but not actual (completed) infinity, and dealing only with finite objects that can actually be constructed. Thus we have a finitist proof of the consistency of ZF. This becomes possible because we set forth a metatheory of mathematics which goes beyond the classical logic and set theory and, of course, cannot be formalized in ZF, yet yields proofs which are as convincing—at least, from the author's viewpoint—as any mathematical proof can be.Our metatheory is based on the following two ideas. Firstly, we define the semantics of the mathematical language using the cybernetical concept of knowledge. According to this concept, to say that a cybernetic system (a human being, in particular) has some knowledge is to say that it has some models of reality. In the Cybernetic Foundation we consider mathematics as the art of constructing linguistic models of reality.
Saturation of homogeneous resplendent models
Authors: Knight Julia F. 
Published: May 2006

The complete diagram of a structure , denoted by Dc(), is the set of all sentences true in the structure (, a)a∈. A structure is said to be resplendent if for every sentence θ involving a new relation symbol R in addition to symbols occurring in Dc(), if θ is consistent with Dc(), then there is a relation P on such that (see[1]).Baldwin asked whether a homogeneous recursively saturated structure is necessarily resplendent. Here it is shown that this need not be the case. It is shown that if is an uncountable homogeneous resplendent model of an unstable theory, then must be saturated. The proof is related to the proof in [5] that an uncountable homogeneous recursively saturated model of first order Peano arithmetic must be saturated. The example for Baldwin's question is an uncountable homogeneous model for a particular unstable theory, such that is recursively saturated and omits some type. (The continuum hypothesis is needed to show the existence of such a model in power ℵ1.)The proof of the main result requires two lemmas.
Epistemic set theory is a conservative extension of intuitionistic set theory
Authors: Flagg R. C. 
Published: May 2006

In [6] Gödel observed that intuitionistic propositional logic can be interpreted in Lewis's modal logic (S4). The idea behind this interpretation is to regard the modal operator □ as expressing the epistemic notion of “informal provability”. With the work of Shapiro [12], Myhill [10], Goodman [7], [8], and Ščedrov [11] this simple idea has developed into a successful program of integrating classical and intuitionistic mathematics.There is one question quite central to the above program that has remained open. Namely:Does Ščedrov's extension of the Gödel translation to set theory provide a faithful interpretation of intuitionistic set theory into epistemic set theory?In the present paper we give an affirmative answer to this question.The main ingredient in our proof is the construction of an interpretation of epistemic set theory into intuitionistic set theory which is inverse to the Gödel translation. This is accomplished in two steps. First we observe that Funayama's theorem is constructively provable and apply it to the power set of 1. This provides an embedding of the set of propositions into a complete topological Boolean algebra . Second, in a fashion completely analogous to the construction of Boolean-valued models of classical set theory, we define the -valued universe V(). V() gives a model of epistemic set theory and, since we use a constructive metatheory, this provides an interpretation of epistemic set theory into intuitionistic set theory.
Small filter forcing
Authors: Canjar R. Michael 
Published: May 2006

Qλ is the set of nonprincipal filters on ω which are generated by fewer than λ sets, for λ a fixed uncountable, regular cardinal ≤ c. We analyze forcing with Qλ, where Qλ is partially ordered in such a way that a filter F1 is more informative than F2 iff F1 includes F2. Qλ-forcing adjoins an ultrafilter on ω but adds no new reals. We analyze Qλ-forcing from a forcing-theoretic viewpoint. We also analyze the properties of Qλ-generic ultrafilters. These properties are independent of ZFC and depend very much on the ground model. In particular, we study Qλ-forcing over ground models which are Cohen real extensions, random real extensions, and models which satisfy Martin's Axiom.In §2 we give notations and definitions, and review some of the basic facts about forcing and ultrafilters which we will use. In §3 we introduce Qλ-forcing and prove some basic lemmas about it. §4 studies Qc-forcing. §§5, 6, and 7 analyze Qλ-forcing over ground models of Martin's Axiom, ground models which are generated by Cohen reals, and ground models which are generated by random reals, respectively. Qλ-forcing over Cohen real and random real models is isomorphic to the notion of forcing which adjoins a Cohen generic subset of λ; this is proved in §8.
Lattice embeddings into the recursively enumerable degrees
Authors: Ambos-Spies K. , Lerman M. 
Published: May 2006

The classification of algebraic structures which can be embedded into ℛ, the uppersemilattice of recursively enumerable degrees, is the key to answering certain questions about Th(ℛ), the elementary theory of ℛ. In particular, these classification problems are important for answering decidability questions about fragments of Th(ℛ). Thus the solutions of Fried berg [F] and Mučnik [M] to Post's problem were easily extended to show that all finite partially ordered sets are embeddable into ℛ, and hence that ∃1 ∩ Th(ℛ), the existential theory of ℛ, is decidable. (The language used is ℒ′, the pure predicate calculus together with a binary relation symbol ≤ to be interpreted as the ordering of ℛ) The problem of determining which finite lattices are embeddable into ℛ has been a long-standing open problem, and is one of the major obstacles to determining whether ∀2 ∩ Th(ℛ), the universal-existential theory of ℛ, is decidable. Shore has obtained some nice partial results in this direction. Embeddings also played a central role in showing that Th(ℛ) is not ℵ0-categorical (Lerman, Shore and Soare [LeShSo]), thus resolving a problem posed by Jockusch. Harrington and Shelah [HS] embedded all 0′-presentable partially ordered sets into ℛ in such a way that the partially ordered sets can be uniformly recovered from four parameters. They used these embeddings to show that Th(ℛ) is undecidable.The first nontrivial extension of the embeddings of Friedberg and Mučnik to lattice embeddings was obtained independently by Lachlan [La1] and Yates [Y] who showed that the four-element Boolean algebra can be embedded into ℛ. Thomason [T] and Lerman independently extended this result to include all finite distributive lattices. The nondistributive case, however, was much more difficult. Lachlan [La2] embedded the two five-element nondistributive lattices M5 and N5 (see Figures 1 and 2) into ℛ, and his proof could easily have been extended to include a larger class of lattices.
Natural deduction based set theories: a new resolution of the old paradoxes
Authors: Gilmore Paul C. 
Published: May 2006

AbstractThe comprehension principle of set theory asserts that a set can be formed from the objects satisfying any given property. The principle leads to immediate contradictions if it is formalized as an axiom scheme within classical first order logic. A resolution of the set paradoxes results if the principle is formalized instead as two rules of deduction in a natural deduction presentation of logic. This presentation of the comprehension principle for sets as semantic rules, instead of as a comprehension axiom scheme, can be viewed as an extension of classical logic, in contrast to the assertion of extra-logical axioms expressing truths about a pre-existing or constructed universe of sets. The paradoxes are disarmed in the extended classical semantics because truth values are only assigned to those sentences that can be grounded in atomic sentences.
An interpolation theorem in many-valued logic
Authors: Hanazawa Masazumi , Takano Mitio 
Published: May 2006

A many-valued logic version of the Craig-Lyndon interpolation theorem has been given by Gill [1] and Miyama [2]. The former dealt with three-valued logic and the latter generalized it to M-valued logic with M ≥ 3. The purpose of this paper is to improve the form of Miyama's version of the interpolation theorem. The system used in this paper is a many-valued analogue (Takahashi [4], Rousseau [3]) of Gentzen's logical calculus LK. Let T = {1,…, M} be the set of truth values. An M-tuple (Γ1,…, ΓM) of sets of formulas is called a sequent, which is regarded as valid if for any valuation (in canonical sense) there is a truth value μ Є T such that the set Γμ contains a formula of the value μ with respect to the valuation. (In the next section, and thereafter, we change the format of the sequent for typographical reasons.) Miyama's result is as follows (in representative form):(I) If a sequent ({A}, ∅,…, ∅, {B}) is valid, then there is a formula D such that(i) every predicate or propositional variable occurring in D occurs in A and B, and(ii) the sequents {{A}, ∅,…, ∅, {D}) and (D}, ∅,…, ∅, {B}) are both valid.What shall be proved in this paper is the following (in representative form):(II) If a sequent ({A1}, {A2}, …, {AM}) is valid, then there is a formula D such that(i) every predicate or propositional variable occurring in D occurs in at least two of the formulas A1,…, AM, and(ii) the following M sequents are valid:({A1},{D},…,{D}),({D},{A2},…,{D}),…,({D},{D},…,{AM}).Clearly the former can be obtained as a corollary of the latter.
On nonstandard models in higher order logic
Authors: Hort Christian , Osswald Horst 
Published: May 2006

There are two concepts of standard/nonstandard models in simple type theory.The first concept—we might call it the pragmatical one—interprets type theory as a first order logic with countably many sorts of variables: the variables for the urelements of type 0,…, the n-ary relational variables of type (τ1, …, τn) with arguments of type (τ1,…,τn), respectively. If A ≠ ∅ then 〈Aτ〉 is called a model of type logic, if A0 = A and . 〈Aτ〉 is called full if, for every τ = (τ1,…,τn), . The variables for the urelements range over the elements of A and the variables of type (τ1,…, τn) range over those subsets of which are elements of . The theory Th(〈Aτ〉) is the set of all closed formulas in the language which hold in 〈Aτ〉 under natural interpretation of the constants. If 〈Bτ〉 is a model of Th(〈Aτ〉), then there exists a sequence 〈fτ〉 of functions fτ: Aτ → Bτ such that 〈fτ〉 is an elementary embedding from 〈Aτ〉 into 〈Bτ〉. 〈Bτ〉 is called a nonstandard model of 〈Aτ〉, if f0 is not surjective. Otherwise 〈Bτ〉 is called a standard model of 〈Aτ〉.This first concept of model theory in type logic seems to be preferable for applications in model theory, for example in nonstandard analysis, since all nice properties of first order model theory (completeness, compactness, and so on) are preserved.