bluuebunny/crossref_metadata_2025_split huggingface dataset:
A theory of properties
Authors: Turner Ray 
Published: May 2006

Frege's attempts to formulate a theory of properties to serve as a foundation for logic, mathematics and semantics all dissolved under the weight of the logicial paradoxes. The language of Frege's theory permitted the representation of the property which holds of everything which does not hold of itself. Minimal logic, plus Frege's principle of abstraction, leads immediately to a contradiction. The subsequent history of foundational studies was dominated by attempts to formulate theories of properties and sets which would not succumb to the Russell argument. Among such are Russell's simple theory of types and the development of various iterative conceptions of set. All of these theories ban, in one way or another, the self-reference responsible for the paradoxes; in this sense they are all “typed” theories. The semantical paradoxes, involving the concept of truth, induced similar nightmares among philosophers and logicians involved in semantic theory. The early work of Tarski demonstrated that no language that contained enough formal machinery to respresent the various versions of the Liar could contain a truth-predicate satisfying all the Tarski biconditionals. However, recent work in both disciplines has led to a re-evaluation of the limitations imposed by the paradoxes.In the foundations of set theory, the work of Gilmore [1974], Feferman [1975], [1979], [1984], and Aczel [1980] has clearly demonstrated that elegant and useful type-free theories of classes are feasible. Work on the semantic paradoxes was given new life by Kripke's contribution (Kripke [1975]). This inspired the recent work of Gupta [1982] and Herzberger [1982]. These papers demonstrate that much room is available for the development of theories of truth which meet almost all of Tarski's desiderata.
The special model axiom in nonstandard analysis
Authors: Ross David 
Published: May 2006

κ-saturation [SL] is probably the single most useful property of nonstandard models of analysis. For some applications, however, stronger saturation hypotheses seem necessary. Henson formulated his elegant κ-isomorphism property (see [H1], and §2 below) to address this need. This property, though well-suited to certain situations (notably those arising in Banach space theory), is often difficult to apply in practice (see [SL, §7.7]).In this paper I describe an alternative to κ-isomorphism which is much easier to use; in particular, a proof which assumes that the nonstandard model is fully saturated can usually be converted directly to one using this special model axiom.Precise definitions of both these axioms appear in §2. In §3 I prove some simple properties of the special model axiom, one of which is that it is at least as strong as κ-isomorphism. In §§4, 5, and 6 the axiom is used to construct a few examples, many of which are pathological, or at the very least counterintuitive. (These examples are given primarily to illustrate use of the axiom; the only one of independent interest is Theorem 5.5.) In §7 some alternative axioms and open problems are discussed.Many of the results in this paper grew out of discussion and correspondence with C. Ward Henson, to whom I am consequently most grateful.
Large discrete parts of the E-tree
Authors: Simmons Harold 
Published: May 2006

Let PA be first order Peano arithmetic, let Λ be the lattice of Π1 sentences modulo PA, and let S be the poset of prime filters of Λ ordered by reverse inclusion. We show there are large convex discrete parts of S; in particular there are convex parts which form a completed Baire tree or an Aronszajn tree.The elements of S, which we call nodes, correspond to the extensions of PA which are complete for sentences. Equivalently, for each model of PA the Π1-theory ∀() of is a node, and every node occurs in this form. Note that the Π1-theory ∀() of the standard model (i.e. the filter of true Π1 sentences) is the unique root of S.This poset S, which is sometimes called the E-tree, was first studied in [1] where it is shown that:(1) The poset is tree-like, i.e. the set of predecessors of any node is linearly ordered.(2) The poset has branches, each of which is closed under unions and intersections; in particular each branch has a maximum member.(3) There are branches on which ∀() does not have an immediate successor. Further properties of the E-tree are given in [2]−[7]. In particular in [4] Misercque shows that:(4) There are branches on which ∀() does have an immediate successor.(5) There are nodes with both an immediate predecessor and an immediate successor.The two results (3) and (4) show that there are fundamentally different branches of S, and (5) shows that parts of branches may be discrete.
Weakly semirecursive sets
Authors: Jockusch Carl G. , Owings James C. 
Published: May 2006

AbstractWe introduce the notion of “semi-r.e.” for subsets of ω, a generalization of “semirecursive” and of “r.e.”, and the notion of “weakly semirecursive”, a generalization of “semi-r.e.”. We show that A is weakly semirecursive iff, for any n numbers x1, …,xn, knowing how many of these numbers belong to A is equivalent to knowing which of these numbers belong to A. It is shown that there exist weakly semirecursive sets that are neither semi-r.e. nor co-semi-r.e. On the other hand, we exhibit nonzero Turing degrees in which every weakly semirecursive set is semirecursive. We characterize the notion “A is weakly semirecursive and recursive in K” in terms of recursive approximations to A. We also show that if a finite Boolean combination of r.e. sets is semirecursive then it must be r.e. or co-r.e. Several open questions are raised.
Relevance and paraconsistency—a new approach
Authors: Avron Arnon 
Published: May 2006

In this work we describe a new approach to the notions of relevance and paraconsistency. Unlike the works of Anderson and Belnap or da Costa (see [2], [8] and [7]) we shall mainly be guided in it by semantical intuitions. In the first two sections we introduce and investigate the algebraic structures that reflect those intuitions. The corresponding formal systems are briefly described in the third section (a more detailed treatment of these systems, including full proofs, will be given in another paper).Our basic intuitive idea is that of “domains of discourse” or “relevance domains”. Classical logic, so we think, is valid in as much as sentences get values inside one domain; limitations on its use can be imposed only with respect to inferences in which more than one domain is involved. There are two basic binary relations over the collection of domains. One is relevance. It is reflexive and symmetric (but not necessarily transitive). Under a given interpretation two sentences are relevant to each other when their values are in relevant domains. Another basic relation between domains, no less important, is that of grading according to “degrees of reality”. The idea behind it is not new. Gentzen, for example, divided in [9] the world of mathematics into three grades, representing three “levels of reality”. The elementary theory of numbers has the highest degree or level of reality; set theory has the smallest degree and mathematical analysis occupies the intermediate level. In the theory of types, or in the accumulative von Neumann universe for set theory, we can find indication of a richer hierarchy.
The elementary theory of free pseudo p-adically closed fields of finite corank
Authors: Efrat Ido 
Published: May 2006

Let be p-adic closures of a countable Hilbertian field K. The main result of [EJ] asserts that the field has the following properties for almost all σ1,…,σe + m ϵ G(K) (in the sense of the unique Haar measure on G(K)e+m):(a) Kσ is pseudo p-adically closed (abbreviation: PpC), i.e., each nonempty absolutely irreducible variety defined over Kσ has a Kσ-rational point, provided that it has a simple rational point in each p-adic closure of Kσ.(b) G(Kσ) ≅ De,m, where De,m is the free profinite product of e copies Γ1,…, Γe of G(ℚp) and a free profinite group of rank m.(c) Kσ has exactly e nonequivalent p-adic valuation rings. They are the restrictions Oσ1,…, Oσe of the unique p-adic valuation rings on , respectively.In this paper we show that this result is in a certain sense the best possible. More precisely, we first show that the class of fields which satisfy (a)–(c) above is elementary in the appropriate language e(K), which is the ordinary first-order language of rings augmented by constant symbols for the elements of K and by e new unary relation symbols (interpreted as e p-adic valuation rings).
A limit on relative genericity in the recursively enumerable sets
Authors: Lempp Steffen , Slaman Theodore A. 
Published: May 2006

AbstractWork in the setting of the recursively enumerable sets and their Turing degrees. A set X is low if X′, its Turing jump, is recursive in ∅′ and high if X′ computes ∅″. Attempting to find a property between being low and being recursive, Bickford and Mills produced the following definition. W is deep, if for each recursively enumerable set A, the jump of A ⊕ W is recursive in the jump of A. We prove that there are no deep degrees other than the recursive one.Given a set W, we enumerate a set A and approximate its jump. The construction of A is governed by strategies, indexed by the Turing functionals Φ. Simplifying the situation, a typical strategy converts a failure to recursively compute W into a constraint on the enumeration of A, so that (W ⊕ A)′ is forced to disagree with Φ(−;A′). The conversion has some ambiguity; in particular, A cannot be found uniformly from W.We also show that there is a “moderately” deep degree: There is a low nonzero degree whose join with any other low degree is not high.
The structure of graphs all of whose Y-sections are internal sets
Authors: Živaljević Boško 
Published: May 2006

The purpose of this paper is to give structural results on graphs lying in the product of two hyperfinite sets X and Y, whose Y-sections are either all internal sets or all of “small” cardinality with respect to the saturation assumption imposed on our nonstandard universe. These results generalize those of [KKML] and [HeRo]. In [KKML] Keisler, Kunen, Miller and Leth proved, among other results, that any countably determined function in the product of two internal sets X and Y can be covered by countably many internal functions provided that the nonstandard universe is at least ℵ-saturated. This shows that any countably determined function can be represented as a union of countably many restrictions of internal functions to countably determined sets. On the other hand, Henson and Ross use in [HeRo] Choquet's capacitability theorem to prove that any Souslin function in the product of two internal sets X and Y is a.e. equal to an internal function. (Here “a.e.” refers to an arbitrary but fixed bounded Loeb measure.) Therefore, in our terminology, every Souslin function possesses an internal a.e. lifting.After the introductory §0, where all the necessary terminology is introduced, we continue by presenting the structural result for graphs all of whose Y-sections are of cardinality ≤κ (provided that the nonstandard universe is ≤κ+-saturated) in §1. We show that, under the above saturation assumption, a κ-determined graph with all of the Y-sections of cardinality ≤κ is covered by κ-many internal functions. Therefore, any such graph is a union of κ-many κ-determined functions. In particular if the graph in question is Borel, Souslin, κ-Borel or κ-Souslin (or a member of one of the Borel, κ-Borel or projective hierarchies) then the corresponding constituting functions are of the same “complexity”. Thus, any Borel graph all of whose Y-sections are at most countable is a union of countably many Borel functions and, consequently, has Borel domain. In the setting of Polish topological spaces this was proved by Novikov (see [De]).
Amenable equivalence relations and Turing degrees
Authors: Kechris Alexander S. 
Published: May 2006

In [12] Slaman and Steel posed the following problem:Assume ZF + DC + AD. Suppose we have a function assigning to each Turing degree d a linear order <d of d. Then must the rationals embed order preservingly in <d for a cone of d's?They had already obtained a partial answer to this question by showing that there is no such d ↦ <d with <d of order type ζ = ω* + ω on a cone. Already the possibility that <d has order type ζ · ζ was left open.We use here, ideas and methods associated with the concept of amenability (of groups, actions, equivalence relations, etc.) to prove some general results from which one can obtain a positive answer to the above problem.
End-extensions preserving power set
Authors: Forster Thomas , Kaye Richard 
Published: May 2006

AbstractWe consider the quantifier hierarchy of Takahashi [1972] and show how it gives rise to reflection theorems for some large cardinals in ZF, a new natural subtheory of Zermelo's set theory, a potentially useful new reduction of the consistency problem for Quine's NF, and a sharpening of another reduction of this problem due to Boffa.
Models with second order properties in successors of singulars
Authors: Grossberg Rami 
Published: May 2006

AbstractLet L(Q) be first order logic with Keisler's quantifier, in the λ+ interpretation (= the satisfaction is defined as follows: M ⊨ (Qx)φ(x) means there are λ+ many elements in M satisfying the formula φ(x)).Theorem 1. Let λ be a singular cardinal; assume □λ and GCH. If T is a complete theory in L(Q) of cardinality at most λ, and p is an L(Q) 1-type so that T strongly omits p( = p has no support, to be defined in §1), then T has a model of cardinality λ+ in the λ+ interpretation which omits p.Theorem 2. Let λ be a singular cardinal, and let T be a complete first order theory of cardinality λ at most. Assume □λ and GCH. If Γ is a smallness notion then T has a model of cardinality λ+ such that a formula φ(x) is realized by λ+ elements of M iff φ(x) is not Γ-small. The theorem is proved also when λ is regular assuming λ = λ<λ. It is new when λ is singular or when ∣T∣ = λ is regular.Theorem 3. Let λ be singular. If Con(ZFC + GCH + ∃κ) [κ is a strongly compact cardinal]), then the following is consistent: ZFC + GCH + the conclusions of all above theorems are false.
S-homogeneity and automorphism groups
Authors: Bouscaren Elisabeth , Laskowski Michael C. 
Published: May 2006

AbstractWe consider the question of when, given a subset A of M, the setwise stabilizer of the group of automorphisms induces a closed subgroup on Sym(A). We define s-homogeneity to be the analogue of homogeneity relative to strong embeddings and show that any subset of a countable, s-homogeneous, ω-stable structure induces a closed subgroup and contrast this with a number of negative results. We also show that for ω-stable structures s-homogeneity is preserved under naming countably many constants, but under slightly weaker conditions it can be lost by naming a single point.
Hyper-regular lattice-ordered groups
Authors: Gluschankof Daniel , Lucas François 
Published: May 2006

It is a well-known fact that the notion of an archimedean order cannot be formalized in the first-order calculus. In [12] and [18], A. Robinson and E. Zakon characterized the elementary class generated by all the archimedean, totally-ordered abelian groups (o-groups) in the language 〈+,<〉, calling it the class of regularly ordered or generalized archimedean abelian groups. Since difference (−) and 0 are definable in that language, it is immediate that in the expanded language 〈 +, −, 0, < 〉 the definable expansion of the class of regular groups is also the elementary class generated by the archimedean ones. In the more general context of lattice-ordered groups (l-groups), the notion of being archimedean splits into two different notions: a strong one (being hyperarchimedean) and a weak one (being archimedean). Using the representation theorem of K. Keimel for hyperarchimedean l-groups, we extend in this paper the Robinson and Zakon characterization to the elementary class generated by the prime-projectable, hyperarchimedean l-groups. This characterization is also extended here to the elementary class generated by the prime-projectable and projectable archimedean l-groups (including all complete l-groups). Finally, transferring a result of A. Touraille on the model theory of Boolean algebras with distinguished ideals, we give the classification up to elementary equivalence of the characterized class.We recall that a lattice-ordered group, l-group for short, is a structure
Pointwise compact and stable sets of measurable functions
Authors: Shelah S. , Fremlin D. H. 
Published: May 2006

In a series of papers culminating in [9], M. Talagrand, the second author, and others investigated at length the properties and structure of pointwise compact sets of measurable functions. A number of problems, interesting in themselves and important for the theory of Pettis integration, were solved subject to various special axioms. It was left unclear just how far the special axioms were necessary. In particular, several results depended on the fact that it is consistent to suppose that every countable relatively pointwise compact set of Lebesgue measurable functions is ‘stable’ in Talagrand's sense, the point being that stable sets are known to have a variety of properties not shared by all pointwise compact sets. In the present paper we present a model of set theory in which there is a countable relatively pointwise compact set of Lebesgue measurable functions which is not stable and discuss the significance of this model in relation to the original questions. A feature of our model which may be of independent interest is the following: in it, there is a closed negligible set Q ⊆ [0, 1]2 such that whenever D ⊆ [0,1] has outer measure 1, thenhas inner measure 1 (see 2G below).We embark immediately on the central ideas of this paper, setting out a construction of a partially ordered set which forces a fairly technical proposition in measure theory (IS below); the relevance of this proposition to pointwise compact sets will be discussed in §2.
On solvable centerless groups of Morley rank 3
Authors: Davis Mark Kelly , Nesin Ali 
Published: May 2006

We know quite a lot about the general structure of ω-stable solvable centerless groups of finite Morley rank. Abelian groups of finite Morley rank are also well-understood. By comparison, nonabelian nilpotent groups are a mystery except for the following general results:• An ω1-categorical torsion-free nonabelian nilpotent group is an algebraic group over an algebraically closed field of characteristic 0 [Z3].• A nilpotent group of finite Morley rank is the central product of a definable subgroup of finite exponent and of a definable divisible subgroup [N3].• A divisible nilpotent group of finite Morley rank is the direct product of its torsion part (which is central) and of a torsion-free subgroup [N3].However, we do not understand nilpotent groups of bounded exponent. It seems that the classification of nilpotent (but nonabelian)p-groups of finite Morley rank is impossible. Even the nilpotent groups of Morley rank 2 contain insurmountable difficulties [C], [T] . At first glance, this may seem to be an obstacle to proving the Cherlin-Zil'ber conjecture (“simple groups of finite Morley rank are algebraic groups”). Our purpose in this article is to show that if such a group is a definable subgroup of a nonnilpotent group, then it is possible to obtain a classification within the boundaries of our present knowledge. In this respect, our article may be considered as a relief to those who are trying to classify simple groups of finite Morley rank.Before explicitly stating our result, we need the following definition.
Lower bounds to the size of constant-depth propositional proofs
Authors: Krajíček Jan 
Published: May 2006

AbstractLK is a natural modification of Gentzen sequent calculus for propositional logic with connectives ¬ and ∧,∨ (both of bounded arity). Then for every d ≥ 0 and n ≥ 2, there is a set of depth d sequents of total size O(n3+d) which are refutable in LK by depth d + 1 proof of size exp(O(log2n)) but such that every depth d refutation must have the size at least exp(nΩ(1)). The sets express a weaker form of the pigeonhole principle.
Nonarithmetical ℵ0-categorical theories with recursive models
Authors: Knight Julia F. 
Published: May 2006

In what follows, L is a recursive language. The structures to be considered are L-structures with universe named by constants from ω. A structure is recursive A if the open diagram D() is recursive. Lerman and Schmerl [L-S] proved the following result.Let T be an ℵ0-categorical elementary first-order theory. Suppose that for all n, , and T is arithmetical. Then T has a recursive model.The aim of this paper is to extend Theorem 0.1. Stating the extension requires some terminology. Consider finitary formulas with symbols from L and sometimes extra constants from ω. For each n ∈ ω, the Σn and Πn formulas are as usual. Then Bnformulas are Boolean combinations of Σn formulas. For an L-structure , Dn() denotes the set of Bn sentences in the complete diagram Dc(). A complete Σn theory is a maximal consistent set of ΣnL-sentences. We may write φ(x), or Γ(x), to indicate that the free variables of the formula φ, or the set Γ, are among those in x. A complete Bn type for x is a maximal consistent set Γ(x) of Bn formulas with just the free variables x.If T is ℵ0-categorical, then for each x only finitely many complete types Γ(x) are consistent with T. While Lerman and Schmerl stated their result just for ℵ0-categorical theories, essentially the same proof yields the following.Theorem 0.2. Let T be a consistent, complete theory such that for all n andx, only finitely many complete Bn types Γ(x) are consistent with T.
Schur-Zassenhaus theorem revisited
Authors: Borovik Alexandre V. , Nesin Ali 
Published: May 2006

One of the purposes of this paper is to prove a partial Schur-Zassenhaus Theorem for groups of finite Morley rank.Theorem 2.Let G be a solvable group of finite Morley rank. Let π be a set of primes, and let H ⊲ G a normal π-Hall subgroup. Then H has a complement in G.This result has been proved in [1] with the additional assumption thatGis connected, and thought to be generalized in [2] by the authors of the present article. Unfortunately in the last section of the latter paper there is an irrepairable mistake. Here we give a new proof of the Schur-Zassenhaus Theorem using the results of [2] up to the last section and a new result that we are going to state below.The second author has shown in [11] that a nilpotentω-stable group is the central product of a divisible subgroup and a subgroup of bounded exponent, generalizing a well-known result of Angus Macintyre about abelian groups [8]. One could ask a similar question for solvable groups: are they a product of two subgroups, one divisible, one of bounded exponent? One is allowed to be hopeful because of the well-known decomposition of the connected solvable algebraic groups over algebraically closed fields as the product of the unipotent radical and a torus.
Theories of modules closed under direct products
Authors: Villemaire Roger 
Published: May 2006

AbstractWe generalize to theories of modules (complete or not) a result of U. Felgner stating that a complete theory of abelian groups is a Horn theory if and only if it is closed under products. To prove this we show that a reduced product of modules ΠFMi (i ϵ I) is elementarily equivalent to a direct product of ultraproducts of the modules Mi(i ϵ I).