bluuebunny/crossref_metadata_2025_split huggingface dataset:
Model theory under the axiom of determinateness
Authors: Spector Mitchell 
Published: May 2006

AbstractWe initiate the study of model theory in the absence of the Axiom of Choice, using the Axiom of Determinateness as a powerful substitute. We first show that, in this context, is no more powerful than first-order logic. The emphasis then turns to upward Löwenhein-Skolem theorems; ℵ1 is the Hanf number of first-order logic, of , and of a strong fragment of , The main technical innovation is the development of iterated ultrapowers using infinite supports; this requires an application of infinite-exponent partition relations. All our theorems can be proven from hypotheses weaker than AD.
Omitting types for stable ccc theories
Authors: Newelski Ludomir 
Published: May 2006

In this paper we investigate omitting types for a certain kind of stable theories which we call stable ccc theories. In Theorem 2.1 we improve Steinhorn's result from [St]. We prove also some independence results concerning omitting types. The main results presented in this paper were part of the author's Ph.D. thesis [N1].Throughout, we use the standard set-theoretic and model-theoretic notation, such as can be found for example in [Sh] or [M]. So in particular T is always a countable complete theory in the language L. We consider all models of T and all sets of parameters subsets of the monster model ℭ, which is very saturated. Ln(A) denotes the Lindenbaum-Tarski algebra of formulas with parameters from A and n free variables. We omit n in Ln(A) when n = 1 or when it is clear from the context what n is. If φ, ψ ∈ L(A) are consistent then we say that φ is below ψ if ψ⊢ψ. For a type p and a set A ⊆ ℭ, p(A) is the set of tuples of elements of A which satisfy p. Formulas are special cases of types. We say that a type p is isolated over A if, for some φ() ∈ L(A), φ() ⊢ p(x), i.e. φ isolates p. For a formula φ, [φ] denotes the class of types which contain φ. We assume that the reader is familiar with some basic knowledge of forking, as presented in [Sh, III] or [M].Throughout, we work in ZFC. and denote (countable) transitive models of ZFC. cov K is the minimal number of meager sets covering the real line R. In this paper we prove theorems showing connections between omitting types and the combinatorics of the real line. More results in this direction are presented in [N2] and [N3].
Hilbert's program relativized; Proof-theoretical and foundational reductions
Authors: Feferman Solomon 
Published: May 2006

In the Symposium on Hilbert's Program to which the following was contributed, I was asked to talk about the metamathematical aspects of Hilbert's Program (H.P.), while the other two speakers (Simpson and Prawitz) were to deal with the mathematical and philosophical aspects respectively. However, more so than for other foundational schemes, these three aspects of H.P., both as originally conceived and in its subsequent developments, are intimately linked.Here I shall survey a body of proof-theoretical results stemming from H.P., but organized in a way that is closely tied to various reductive foundational aims, albeit going beyond those advanced by Hilbert. I believe this view of reductive proof-theory (not original with me) helps one to better understand what has been achieved thereby than other, more familiar accounts.
Some remarks on the algebraic structure of the Medvedev Lattice
Authors: Sorbi Andrea 
Published: May 2006

AbstractThis paper investigates the algebraic structure of the Medvedev lattice . We prove that is not a Heyting algebra. We point out some relations between and the Dyment lattice and the Mučnik lattice. Some properties of the degrees of enumerability are considered. We give also a result on embedding countable distributive lattices in the Medvedev lattice.
The McKinsey axiom is not canonical
Authors: Goldblatt Robert 
Published: May 2006

The logic KM is the smallest normal modal logic that includes the McKinsey axiomIt is shown here that this axiom is not valid in the canonical frame for KM, answering a question first posed in the Lemmon-Scott manuscript [Lemmon, 1966].The result is not just an esoteric counterexample: apart from interest generated by the long delay in a solution being found, the problem has been of historical importance in the development of our understanding of intensional model theory, and is of some conceptual significance, as will now be explained.The relational semantics for normal modal logics first appeared in [Kripke, 1963], where a number of well-known systems were shown to be characterised by simple first-order conditions on binary relations (frames). This phenomenon was systematically investigated in [Lemmon, 1966], which introduced the technique of associating with each logic L a canonical frame which invalidates every nontheorem of L. If, in addition, each L-theorem is valid in , then L is said to be canonical. The problem of showing that L is determined by some validating condition C, meaning that the L-theorems are precisely those formulae valid in all frames satisfying C, can be solved by showing that satisfies C—in which case canonicity is also established. Numerous cases were studied, leading to the definition of a first-order condition Cφ associated with each formula φ of the formwhere Ψ is a positive modal formula.
On adding (ξ) to weak equality in combinatory logic
Authors: Bunder Martin W. , Hindley J. Roger , Seldin Jonathan P. 
Published: May 2006

AbstractBecause the main difference between combinatory weak equality and λβ-equality is that the ruleis valid for the latter but not the former, it is easy to assume that another way of defining combinatory β-equality is to add rule (ξ) to the postulates for weak equality. However, to make this true, one must choose the definition of combinatory abstraction in (ξ) very carefully. If one tries to use one of the more common abstraction algorithms, the result will be an equality, =ξ, that is either equivalent to βη-equality (and so strictly stronger than β-equality) or else strictly weaker than β-equality. This paper will study the relations =ξ for several commonly used abstraction-algorithms, distinguish between them, and axiomatize them.
Reducts of the random graph
Authors: Thomas Simon 
Published: May 2006

Let Γ be the unique (up to isomorphism) countable graph with the following property: (*) Given any two finite disjoint subsets U and V of Γ, there exists a vertex z ∈ Γ joined to every vertex in U and to none in V.Thus Γ is the countable, universal, homogeneous graph; also known as the random graph. In this paper, we shall study the reducts of Γ Here a reduct of Γ is defined to be a permutation group (G, Γ) such that:(i) Aut(Γ) ≤ G; and(ii) G is a closed subgroup of Sym(Γ).Equivalently, there exists a structure for some language L such that:(iii) has universe Γ;(iv) for each R ∈ L, is definable without parameters in Γ; and(v) G = Aut().
Belief contraction in the context of the general theory of rational choice
Authors: Rott Hans 
Published: May 2006

AbstractThis paper reorganizes and further develops the theory of partial meet contraction which was introduced in a classic paper by Alchourrón, Gärdenfors, and Makinson. Our purpose is threefold. First, we put the theory in a broader perspective by decomposing it into two layers which can respectively be treated by the general theory of choice and preference and elementary model theory. Second, we reprove the two main representation theorems of AGM and present two more representation results for the finite case that “lie between” the former, thereby partially answering an open question of AGM. Our method of proof is uniform insofar as it uses only one form of “revealed preference”, and it explains where and why the finiteness assumption is needed. Third, as an application, we explore the logic characterizing theory contractions in the finite case which are governed by the structure of simple and prioritized belief bases.
On an interpretation of second order quantification in first order intuitionistic propositional logic
Authors: Pitts Andrew M. 
Published: May 2006

AbstractWe prove the following surprising property of Heyting's intuitionistic propositional calculus, IpC. Consider the collection of formulas, ϕ, built up from propositional variables (p, q, r, …) and falsity (⊥) using conjunction (∧), disjunction (∨) and implication (→). Write ⊢ϕ to indicate that such a formula is intuitionistically valid. We show that for each variablepand formula ϕ there exists a formulaApϕ (effectively computable from ϕ), containing only variables not equal topwhich occur in ϕ, and such that for all formulas ψ not involvingp, ⊢ψ →Apϕ if and only if ⊢ψ → ϕ. Consequently quantification over propositional variables can be modelled in IpC, and there is an interpretation of the second order propositional calculus, IpC2, in IpC which restricts to the identity on first order propositions.An immediate corollary is the strengthening of the usual interpolation theorem for IpC to the statement that there are least and greatest interpolant formulas for any given pair of formulas. The result also has a number of interesting consequences for the algebraic counterpart of IpC, the theory of Heyting algebras. In particular we show that a model of IpC2can be constructed whose algebra of truth-values is equal to any given Heyting algebra.
The strength of the isomorphism property
Authors: Jin Renling , Shelah Saharon 
Published: May 2006

AbstractIn § 1 of this paper, we characterize the isomorphism property of nonstandard universes in terms of the realization of some second-order types in model theory. In §2, several applications are given. One of the applications answers a question of D. Ross in [this Journal, vol. 55 (1990), pp. 1233–1242] about infinite Loeb measure spaces.
U-monad topologies of hyperfinite time lines
Authors: Jin Renling 
Published: May 2006

AbstractIn an ω1-saturated nonstandard universe a cut is an initial segment of the hyperintegers which is closed under addition. Keisler and Leth in [KL] introduced, for each given cut U, a corresponding U-topology on the hyperintegers by letting O be U-open if for any x ϵ O there is a y greater than all the elements in U such that the interval [x − y, x + y] ⊆ O. Let U be a cut in a hyperfinite time line , which is a hyperfinite initial segment of the hyperintegers. The U-monad topology of is the quotient topology of the U-topological space modulo U. In this paper we answer a question of Keisler and Leth about the U-monad topologies by showing that when is κ-saturated and has cardinality κ,(1) if the coinitiality of U1, is uncountable, then the U1,-monad topology and the U2-monad topology are homcomorphic iff both U1, and U2 have the same coinitiality; and (2) can produce exactly three different U-monad topologies (up to homeomorphism) for those U's with countable coinitiality. As a corollary can produce exactly four different U-monad topologies if the cardinality of is ω1.
On strong provability predicates and the associated modal logics
Authors: Ignatiev Konstantin N. 
Published: May 2006

AbstractPA is Peano Arithmetic. Pr(x) is the usual Σ1,-formula representing provability in PA. A strong provability predicate is a formula which has the same properties as Pr(·) but is not Σ1. An example: Q is ω-provable if PA + ¬Q is ω-inconsistent (Boolos [4]). In [5] Dzhaparidze introduced a joint provability logic for iterated ω-provability and obtained its arithmetical completeness.In this paper we prove some further modal properties of Dzhaparidze's logic, e.g., the fixed point property and the Craig interpolation lemma. We also consider other examples of the strong provability predicates and their applications.
Interpolation in fragments of classical linear logic
Authors: Roorda Dirk 
Published: May 2006

AbstractWe study interpolation for elementary fragments of classical linear logic. Unlike in intuitionistic logic (see [Renardel de Lavalette, 1989]) there are fragments in linear logic for which interpolation does not hold. We prove interpolation for a lot of fragments and refute it for the multiplicative fragment (→, +), using proof nets and quantum graphs. We give a separate proof for the fragment with implication and product, but without the structural rule of permutation. This is nearly the Lambek calculus. There is an appendix explaining what quantum graphs are and how they relate to proof nets.
Contraction-free sequent calculi for intuitionistic logic
Authors: Dyckhoff Roy 
Published: May 2006

Gentzen's sequent calculus LJ, and its variants such as G3 [21], are (as is well known) convenient as a basis for automating proof search for IPC (intuitionistic propositional calculus). But a problem arises: that of detecting loops, arising from the use (in reverse) of the rule ⊃⇒ for implication introduction on the left. We describe below an equivalent calculus, yet another variant on these systems, where the problem no longer arises: this gives a simple but effective decision procedure for IPC.The underlying method can be traced back forty years to Vorob′ev [33], [34]. It has been rediscovered recently by several authors (the present author in August 1990, Hudelmaier [18], [19], Paulson [27], and Lincoln et al. [23]). Since the main idea is not plainly apparent in Vorob′ev's work, and there are mathematical applications [28], it is desirable to have a simple proof. We present such a proof, exploiting the Dershowtiz-Manna theorem [4] on multiset orderings.Consider the task of constructing proofs in Gentzen's sequent calculus LJ of intuitionistic sequents Γ⇒ G, where Γ is a set of assumption formulae and G is a formula (in the language of zero-order logic, using the nullary constant f [absurdity], the unary constant ¬ [negation, with ¬A =defA ⊃ f] and the binary constants &, ∨, and ⊃ [conjunction, disjunction, and implication respectively]). By the Hauptsatz [15], there is an apparently simple algorithm which breaks up the sequent, growing the proof tree until one reaches axioms (of the form Γ⇒ A where A is in Γ), or can make no further progress and must backtrack or even abandon the search. (Gentzen's argument in fact was to use the subformula property derived from the Hauptsatz to limit the size of the search tree. Došen [5] improves on this argument.)
Interpolation theorems, lower bounds for proof systems, and independence results for bounded arithmetic
Authors: Krajíček Jan 
Published: May 2006

AbstractA proof of the (propositional) Craig interpolation theorem for cut-free sequent calculus yields that a sequent with a cut-free proof (or with a proof with cut-formulas of restricted form; in particular, with only analytic cuts) withkinferences has an interpolant whose circuit-size is at mostk. We give a new proof of the interpolation theorem based on a communication complexity approach which allows a similar estimate for a larger class of proofs. We derive from it several corollaries:(1)Feasible interpolation theorems for the following proof systems:(a)resolution(b)a subsystem ofLKcorresponding to the bounded arithmetic theory(α)(c)linear equational calculus(d)cutting planes.(2)New proofs of the exponential lower bounds (for new formulas)(a)for resolution ([15])(b)for the cutting planes proof system with coefficients written in unary ([4]).(3)An alternative proof of the independence result of [43] concerning the provability of circuit-size lower bounds in the bounded arithmetic theory(α).In the other direction we show that a depth 2 subsystem ofLKdoes not admit feasible monotone interpolation theorem (the so called Lyndon theorem), and that a feasible monotone interpolation theorem for the depth 1 subsystem ofLKwould yield new exponential lower bounds for resolution proofs of the weak pigeonhole principle.
Decidability of Scott's model as an ordered ℚ-vectorspace
Authors: Erdélyi-Szabó Miklós 
Published: May 2006

AbstractLet L = 〈<, +, hq, 1〉q∈ℚ where ℚ is the set of rational numbers and hq is a one-place function symbol corresponding to multiplication by q. Then the L-theory of Scott's model for intuitionistic analysis is decidable.
The finite model property for various fragments of linear logic
Authors: Lafont Yves 
Published: May 2006

To show that a formula A is not provable in propositional classical logic, it suffices to exhibit a finite boolean model which does not satisfy A. A similar property holds in the intuitionistic case, with Kripke models instead of boolean models (see for instance [11]). One says that the propositional classical logic and the propositional intuitionistic logic satisfy a finite model property. In particular, they are decidable: there is a semi-algorithm for provability (proof search) and a semi-algorithm for non provability (model search). For that reason, a logic which is undecidable, such as first order logic, cannot satisfy a finite model property.The case of linear logic is more complicated. The full propositional fragment LL has a complete semantics in terms of phase spaces [2, 3], but it is undecidable [9]. The multiplicative additive fragment MALL is decidable, in fact PSPACE-complete [9], but the decidability of the multiplicative exponential fragment MELL is still an open problem. For affine logic, that is, linear logic with weakening, the situation is somewhat better: the full propositional fragment LLW is decidable [5].Here, we show that the finite phase semantics is complete for MALL and for LLW, but not for MELL. In particular, this gives a new proof of the decidability of LLW. The noncommutative case is mentioned, but not handled in detail.
Splitting number at uncountable cardinals
Authors: Zapletal Jindřich 
Published: May 2006

AbstractWe study a generalization of the splitting number s to uncountable cardinals. We prove that 𝔰(κ) > κ+ for a regular uncountable cardinal κ implies the existence of inner models with measurables of high Mitchell order. We prove that the assumption 𝔰(ℵω) > ℵω+1 has a considerable large cardinal strength as well.
Cardinal functions on ultra products of Boolean algebras
Authors: Peterson Douglas 
Published: May 2006

This article is concerned with functions k assigning a cardinal number to each infinite Boolean algebra (BA), and the behaviour of such functions under ultraproducts. For some common functions k we havefor others we have ≤ instead, under suitable assumptions. For the function π character we go into more detail. More specifically, ≥ holds when F is regular, for cellularity, length, irredundance, spread, and incomparability. ≤ holds for π. ≥ holds under GCH for F regular, for depth, π, πχ, χ, h-cof, tightness, hL, and hd. These results show that ≥ can consistently hold in ZFC since if V = L holds then all uniform ultrafilters are regular. For π-character we prove two more results: (1) If F is regular and ess , then(2) It is relatively consistent to have , where A is the denumerable atomless BA.A thorough analysis of what happens without the assumption that F is regular can be found in Rosłanowski, Shelah [8] and Magidor, Shelah [5]. Those papers also mention open problems concerning the above two possible inequalities.
Pretopologies and completeness proofs
Authors: Sambin Giovanni 
Published: May 2006

Pretopologies were introduced in [S], and there shown to give a complete semantics for a propositional sequent calculus BL, here called basic linear logic, as well as for its extensions by structural rules,ex falso quodlibetor double negation. Immediately after Logic Colloquium '88, a conversation with Per Martin-Löf helped me to see how the pretopology semantics should be extended to predicate logic; the result now is a simple and fully constructive completeness proof for first order BL and virtually all its extensions, including the usual, or structured, intuitionistic and classical logic. Such a proof clearly illustrates the fact that stronger set-theoretic principles and classical metalogic are necessary only when completeness is sought with respect to a special class of models, such as the usual two-valued models.To make the paper self-contained, I briefly review in §1 the definition of pretopologies; §2 deals with syntax and §3 with semantics. The completeness proof in §4, though similar in structure, is sensibly simpler than that in [S], and this is why it is given in detail. In §5 it is shown how little is needed to obtain completeness for extensions of BL in the same language. Finally, in §6 connections with proofs with respect to more traditional semantics are briefly investigated, and some open problems are put forward.
Some applications of ordinal dimensions to the theory of differentially closed fields
Authors: Pong Wai Yan 
Published: May 2006

AbstractUsing the Lascar inequalities, we show that any finite rank δ-closed subset of a quasiprojective variety is definably isomorphic to an affine δ-closed set. Moreover, we show that if X is a finite rank subset of the projective space ℙn and a is a generic point of ℙn, then the projection from a is injective on X. Finally we prove that if RM = RC in DCF0, then RM = RU.
The model theory of differential fields with finitely many commuting derivations
Authors: McGrail Tracey 
Published: May 2006

AbstractIn this paper we set out the basic model theory of differential fields of characteristic 0, which have finitely many commuting derivations. We give axioms for the theory of differentially closed differential fields with m derivations and show that this theory is ω-stable, model complete, and quantifier-eliminable, and that it admits elimination of imaginaries. We give a characterization of forking and compute the rank of this theory to be ωm + 1.
A finite family weak square principle
Authors: Schimmerling Ernest 
Published: May 2006

Definition 1.1. Suppose that λ ≤ κ are cardinals and Γ is a subset of (κ, κ+). By , we mean the principle asserting that there is a sequence 〈Fν | ν ∈ lim(Γ)〉 such that for every ν ∈ lim(Γ), the following hold.(1) 1 ≤ card(Fν) < λ.(2) The following hold for every C ∈ Fν.(a) C ⊆ ν ∩ Γ,(b) C is club in ν,(c) o.t.(C) ≤ κ,By we mean . If Γ = (κ, κ+), then we write for and for .These weak square principles were introduced in [Sch2, 5.1]. They generalize Jensen's principles □κ and , which are equivalent to and respectively. Jensen's global □ principle implies □κ for all κ.Theorem 1.2. Suppose that is a core model. Assume that every countable premouse M which elementarily embeds into a level of is (ω1 + 1)-iterable. Then, for every κ, holds in .The minimal non-1-small mouse is essentially a sharp for an inner model with a Woodin cardinal. We originally proved Theorem 1.2 under the assumption that is 1-small, building on [MiSt] and [Sch2]. Some generalizations followed by combining our methods with those of [St2] and [SchSt2]. (For example, the tame countably certified core model Kc satisfies .) In order to eliminate the smallness assumption all together, one replaces our use of the Dodd-Jensen lemma in proofs of condensation properties for with the weak Dodd-Jensen lemma of [NSt].
On the role of implication in formal logic
Authors: Seldin Jonathan P. 
Published: May 2006

AbstractEvidence is given that implication (and its special case, negation) carry the logical strength of a system of formal logic. This is done by proving normalization and cut elimination for a system based on combinatory logic or λ-calculus with logical constants for and, or, all, and exists, but with none for either implication or negation. The proof is strictly finitary, showing that this system is very weak. The results can be extended to a “classical” version of the system. They can also be extended to a system with a restricted set of rules for implication: the result is a system of intuitionistic higher-order BCK logic with unrestricted comprehension and without restriction on the rules for disjunction elimination and existential elimination. The result does not extend to the classical version of the BCK logic.