bluuebunny/crossref_metadata_2025_split huggingface dataset:
On partitioning the infinite subsets of large cardinals
Authors: Watro R. J. 
Published: May 2006

Let λ be an ordinal less than or equal to an infinite cardinal κ. For S ⊂ κ, [S]λ denotes the collection of all order type λ subsets of S. A set X ⊂ [κ]λ will be called Ramsey iff there exists p ∈ [κ]κ such that either [p]λ ⊂ X or [p]λ ∩ X = ∅. The set p is called homogeneous for X.The infinite Ramsey theorem implies that all subsets of [ω]n are Ramsey for n < ω. Using the axiom of choice, one can define a non-Ramsey subset of [ω]ω. In [GP], Galvin and Prikry showed that all Borel subsets of [ω]ω are Ramsey, where one topologizes [ω]ω as a subspace of Baire space. Silver [S] proved that analytic sets are Ramsey, and observed that this is best possible in ZFC.When κ > ω, the assertion that all subsets of [κ]n are Ramsey is a large cardinal hypothesis equivalent to κ being weakly compact (and strongly inaccessible). Again, is not possible in ZFC to have all subsets of [κ]ω Ramsey. The analogy to the Galvin-Prikry theorem mentioned above was established by Kleinberg, extending work by Kleinberg and Shore in [KS]. The set [κ]ω is given a topology as a subspace of κω, which has the usual product topology, κ taken as discrete. It was shown that all open subsets of [κ]ω are Ramsey iff κ is a Ramsey cardinal (that is, κ → (κ)<ω).In this note we examine the spaces [κ]λ for κ ≥ λ ≥ ω. We show that κ Ramsey implies all open subsets of [κ]λ are Ramsey for λ < κ, and that if κ is measurable, then all open subsets of [κ]κ are Ramsey. Let us remark here that we can with the same methods prove these results with “κ-Borel” in the place of “open”, where the κ-Borel sets are the smallest collection containing the opens and closed under complementation and intersections of length less than κ. Also, although here we consider just subsets of [κ]λ, it is no more difficult to show that partitions of [κ]λ into less than κ many κ-Borel sets have, under the appropriate hypothesis, size κ homogeneous sets.
Propositional logic based on the dynamics of belief
Authors: Gärdenfors Peter 
Published: May 2006

In this article propositions will be identified with a certain kind of changes of belief. The intended interpretation is that a proposition is characterised by the change it would induce if added to a state of belief. Propositions will thus be defined as functions from states of belief to states of belief. A set of postulates concerning the properties and existence of propositions will be formulated. A proposition will be said to be a tautology iff it is the identity function on states of belief. The main result is that the logic determined by the set of postulates is intuitionistic propositional logic.The basic epistemic concept is that of a belief model, which is defined as a pair 〈, 〉, where is a nonempty set and is a class of functions from to . The elements in will be called states of belief and they will be denoted K, K′,…. A discussion of the epistemological interpretation of the states of belief can be found in Gärdenfors [2]. Here, no assumptions about the structure of the elements in will be made.The elements in will be called propositions, and A, B, C, … will be used as variables over . Functions from states of belief to states of belief can be characterised as epistemic inputs. The intended interpretation of the functions in is that they correspond to changes of belief where the new evidence is accepted as “certain” or “known” in the resulting state of belief. This means that not all functions defined on can properly be called propositions.
The strength of nonstandard methods in arithmetic
Authors: Henson C. Ward , Kaufmann Matt , Keisler H. Jerome 
Published: May 2006

AbstractWe consider extensions of Peano arithmetic suitable for doing some of nonstandard analysis, in which there is a predicate N(x) for an elementary initial segment, along with axiom schemes approximating ω1-saturation. We prove that such systems have the same proof-theoretic strength as their natural analogues in second order arithmetic. We close by presenting an even stronger extension of Peano arithmetic, which is equivalent to ZF for arithmetic statements.
Generalisation of first-order logic to nonatomic domains
Authors: Roeper P. 
Published: May 2006

The quantifiers of standard predicate logic are interpreted as ranging over domains of individuals, and interpreted formulae beginning with a quantifier make claims to the effect that something is true of every individual, i.e. of the whole domain, or of some individuals, i.e. of part of the domain. To state that something is true of all or part of a totality seems to be the basic significance of universal and existential quantification, and this by itself does not involve a specification of the structure of the totality. This means that the notion of quantification by itself does not demand totalities of individuals, i.e. atomic totalities, as domains of quantification. Nonatomic domains, such as volumes of space, or surfaces, are equally in order. So one might say that a certain predicate applies “everywhere” or “somewhere” in such a domain. All that the concept of quantification requires is a totality which is structured in terms of a part-to-whole relation, and appropriate properties that apply to part or all of the totality. Quantification does not demand that the totality have smallest parts, or atoms. There is no conflict with the sense of universal or existential quantification if the domain is nonatomic, if every one of its parts has itself proper parts.The most general kind of quantification theory must then deal with totalities of any kind, atomic or not. The relationships among the parts of a domain are described by the theory of Boolean algebras, which we can regard as the most general characterisation of a totality, of a domain of quantification.In this paper I shall be concerned with this generalised theory of quantification, which encompasses nonatomic domains as well as atomic and mixed domains, i.e. totalities consisting entirely or partly of individuals.
Dominical categories: recursion theory without elements
Authors: di Paola Robert A. , Heller Alex 
Published: May 2006

Dominical categories are categories in which the notions of partial morphisms and their domains become explicit, with the latter being endomorphisms rather than subobjects of their sources. These categories form the basis for a novel abstract formulation of recursion theory, to which the present paper is devoted. The abstractness has of course its usual concomitant advantage of generality: it is interesting to see that many of the fundamental results of recursion theory remain valid in contexts far removed from their classic manifestations. A principal reason for introducing this new formulation is to achieve an algebraization of the generalized incompleteness theorem, by providing a category-theoretic development of the concepts and tools of elementary recursion theory that are inherent in demonstrating the theorem.Dominical recursion theory avoids the commitment to sets and partial functions which is characteristic of other formulations, and thus allows for an intrinsic recursion theory within such structures as polyadic algebras. It is worthy of notice that much of elementary recursion theory can be developed without reference to elements.By Gödel's generalized incompleteness theorem for consistent arithmetical system T we mean any statement of the following sort:(1) if every recursive set is definable in T, then T is essentially undecidable [41]; or(2) if all recursive functions are definable in T, then T is essentially undecidable [41]; or(3) if every recursive set is definable in T, then T0 and R0 (the sets of Gödel numbers of the theorems and refutables of T) are recursively inseparable [39]; or(4) if all re sets are representable in T, then T0 is creative [28], [39]; or(5) if T is a Rosser theory (i.e., all disjoint re sets are strongly separable in T), then T0 and R0 are effectively inseparable [39].
On nontrivial types of U-rank 1
Authors: Buechler Steven 
Published: May 2006

AbstractTheorem A. Suppose that T is superstable and p is a nontrivial type of U-rank 1. Then R(p, L, ∞)= 1.Theorem B. Suppose that T is totally transcendental and p is a nontrivial type of U-rank 1. Then p has Morley rank 1.
Reducts of (C, +, ·) which contain +
Authors: Marker D. , Pillay A. 
Published: May 2006

AbstractWe show that the structure (C, +, ·) has no proper non locally modular reducts which contain +. In other words, if X ⊂ Cn is constructible and not definable in the module structure (C, +, λa)a Є C (where λa denotes multiplication by a) then multiplication is definable in (C, +, X).
Full reflection of stationary sets below ℵω
Authors: Jech Thomas , Shelah Saharon 
Published: May 2006

AbstractIt is consistent that, for every n ≥ 2, every stationary subset of ωn consisting of ordinals of cofinality ωκ, where κ = 0 or κ ≤ n − 3, reflects fully in the set of ordinals of cofinality ωn−1. We also show that this result is best possible.
Decision problems for differential equations
Authors: Denef J. , Lipshitz L. 
Published: May 2006

In this paper we shall consider some decision problems for ordinary differential equations. All differential equations will be algebraic differential equations, i.e. equations of the form P(x, y, y′, …, y(n)) = 0 (or P(x, y1, …, ym, y′1 …, y′m, …) = 0 in the case of several dependent variables), where P is a polynomial in all its variables with rational coefficients. (We call P a differential polynomial.)Jáskowski [6] showed that there is no algorithm to determine if a system of algebraic differential equations (in several dependent variables) has a solution on [0, 1]. An easier proof of this using results on Hilbert's tenth problem is given in [2]. It is natural to restrict the problem in the hope of finding something which is decidable. Two ways to do this are the following: (a) One can ask only for the existence of solutions locally, say around x = 0. Here solution may mean (i) analytic functions, (ii) germs of C∞ functions, or (iii) formal power series, (b) One can consider only one equation in one dependent variable, but ask for the existence of a solution on an interval. In [4] we gave an algorithm for deciding (a)(ii) and (a)(iii) and showed that (a)(i) is undecidable. In §4 (Theorem 4.1) we shall show that (b) (for real analytic solutions) is undecidable. This partially answers Problem 9 of Rubel [13], which stimulated this investigation. We shall also prove (Theorem 4.2) that, given an analytic function f(x) which is the solution of an algebraic differential equation, determined by some initial conditions at x = 0, one cannot in general determine if the radius of convergence of f is < 1 or ≥ 1.
An ideal characterization of Mahlo cardinals
Authors: Feng Qi 
Published: May 2006

AbstractWe show that a cardinal κ is a (strongly) Mahlo cardinal if and only if there exists a nontrivial κ-complete κ-normal ideal on κ. Also we show that if κ is Mahlo and λ ≧ κ and λ<κ = λ then there is a nontrivial κ-complete κ-normal fine ideal on Pκ(λ). If κ is the successor of a cardinal, we consider weak κ-normality and prove that if κ = μ+ and μ is a regular cardinal then (1) μ< μ = μ if and only if there is a nontrivial κ-complete weakly κ-normal ideal on κ, and (2) if μ< μ = μ < λ<μ = λ then there is a nontrivial κ-complete weakly κ-normal fine ideal on Pκ(λ).
On chains of relatively saturated submodels of a model without the order property
Authors: Grossberg Rami 
Published: May 2006

AbstractLet M be a given model with similarity type L = L(M), and let L′ be any fragment of L∣L(M∣+,ω of cardinality ∣L(M)∣. We call N ≺ ML′-relatively saturated iff for every B ⊆ N of cardinality less than ∥N∥ every L′-type over B which is realized in M is realized in N. We discuss the existence of such submodels.The following are corollaries of the existence theorems.(1) If M is of cardinality at least ℶω1, and fails to have the ω order property, then there exists N ≺ M which is relatively saturated in M of cardinality ℶω1.(2) Assume GCH. Let ψ ∈ Lω1, ω, and let L′ ⊆ Lω1, ω be a countable fragment containing ψ. If ∃χ > ℵ0 such that I(χ, ψ) < 2χ, then for every M ⊨ ψ and every cardinal λ < ∥M∥ of uncountable cofinality, M has an L′-relatively saturated submodel of cardinality λ.
Reachability is harder for directed than for undirected finite graphs
Authors: Ajtai Miklos , Fagin Ronald 
Published: May 2006

AbstractAlthough it is known that reachability in undirected finite graphs can be expressed by an existential monadic second-order sentence, our main result is that this is not the case for directed finite graphs (even in the presence of certain “built-in” relations, such as the successor relation). The proof makes use of Ehrenfeucht-Fraïssé games, along with probabilistic arguments. However, we show that for directed finite graphs with degree at mostk, reachability is expressible by an existential monadic second-order sentence.
Patching ideal families and enforcing reflection
Authors: Leary Christopher C. 
Published: May 2006

For several years much research has been directed at ideals associated with large cardinals. In this paper we discuss ideal families and use this tool to investigate the saturation properties of various large cardinal ideals.Much of the work in this paper is a direct outgrowth of material presented in my doctoral dissertation written at the University of Michigan under the patient guidance of Andreas Blass. I would also like to thank the referee for several helpful questions and comments on the original version of this paper.
A construction of non-well-founded sets within Martin-Löf's type theory
Authors: Lindström Ingrid 
Published: May 2006

AbstractIn this paper, we show that non-well-founded sets can be defined constructively by formalizing Hallnäs' limit definition of these within Martin-Löfs theory of types. A system is a type W together with an assignment of and to each ∝ ∈ W. We show that for any system W we can define an equivalence relation =w such that ∝ =w ß ∈ U and =w is the maximal bisimulation. Aczel's proof that CZF can be interpreted in the type V of iterative sets shows that if the system W satisfies an additional condition (*), then we can interpret CZF minus the set induction scheme in W. W is then extended to a complete system W* by taking limits of approximation chains. We show that in W* the antifoundation axiom AFA holds as well as the axioms of CFZ−.
Trees and -subsets of ω1ω1
Authors: Mekler Alan , Väänänen Jouko 
Published: May 2006

AbstractWe study descriptive set theory in the space by letting trees with no uncountable branches play a similar role as countable ordinals in traditional descriptive set theory. By using such trees, we get, for example, a covering property for the class of -sets of .We call a family of trees universal for a class of trees if ⊆ and every tree in can be order-preservingly mapped into a tree in . It is well known that the class of countable trees with no infinite branches has a universal family of size ℵ1. We shall study the smallest cardinality of a universal family for the class of trees of cardinality ≤ ℵ1 with no uncountable branches. We prove that this cardinality can be 1 (under ¬CH) and any regular cardinal κ which satisfies (under CH). This bears immediately on the covering property of the -subsets of the space .We also study the possible cardinalities of definable subsets of . We show that the statement that every definable subset of has cardinality <ωn or cardinality is equiconsistent with ZFC (if n ≥ 3) and with ZFC plus an inaccessible (if n = 2).Finally, we define an analogue of the notion of a Borel set for the space and prove a Souslin-Kleene type theorem for this notion.
On the number of automorphisms of uncountable models
Authors: Shelah Saharon , Tuuri Heikki , Väänänen Jouko 
Published: May 2006

AbstractLet σ() denote the number of automorphisms of a model of power ω1. We derive a necessary and sufficient condition in terms of trees for the existence of an with . We study the sufficiency of some conditions for . These conditions are analogous to conditions studied by D. Kueker in connection with countable models.
Sous-groupes periodiques d'un groupe stable
Authors: Poizat Bruno , Wagner Frank 
Published: May 2006

AbstractWe develop a Sylow theory for stable groups satisfying certain additional conditions (2- finiteness, solvability or smallness) and show that their maximal p-subgroups are locally finite and conjugate. Furthermore, we generalize a theorem of Baer-Suzuki on subgroups generated by a conjugacy class of p-elements.
Consistency of Suslin's hypothesis, a nonspecial Aronszajn tree, and GCH
Authors: Schlindwein Chaz 
Published: May 2006

Introduction. In [Sh, Chapter IX], Shelah constructs a model of set theory in which Suslin's hypothesis is true, yet there is an Aronszajn tree which is not special. In his model, we have . He asks whether the same result could be obtained consistently with CH. In this paper, we answer his question in the affirmative.Let us say that a tree T is S-st-special iff there is a function ƒ with dom(f) = {t ∈ T: rank(t) ∈ S} and for every t1 < t2 both in dom(t) we have f(t2) ≠ f(t1) < rank(t1). In Shelah's model, every tree is S-st-special for some fixed stationary costationary set S. Also, there is some tree T such that T is not S′-st-special whenever S′ – S is stationary. These properties, which are sufficient to ensure that Suslin's hypothesis holds and that T is not special, also hold in the model constructed in this paper. These properties also ensure that every Aronszajn tree has a stationary antichain (i.e., an antichain A such that {rank(t): t ∈ A} is stationary). Hence, it is natural to ask whether there is a model of Suslin's hypothesis in which some Aronszajn tree has no stationary antichain. We answer in the affirmative in [S].The construction we use owes much to Shelah's approach to the theorem, due to Jensen (see [DJ]), that CH is consistent with Suslin's hypothesis. This is given in [Sh, Chapter V].
Stratified languages
Authors: Pétry A. 
Published: May 2006

AbstractWe consider arbitrary stratified languages. We study structures which satisfy the same stratified sentences and we obtain an extension of Keisler's Isomorphism Theorem to this situation. Then we consider operations which are definable by a stratified formula and modify the ‘type’ of their argument by one; we prove that for such an operationFthe sentencec=F(c)and the schemeφ(c) ↔ (F(c)), where↔(x) varies among all the stratified formulas with no variable other thanxfree, imply the same stratified {c}-sentences.
Abacus logic: The lattice of quantum propositions as the poset of a theory
Authors: Malhas Othman Qasim 
Published: May 2006

AbstractWith a certain graphic interpretation in mind, we say that a function whose value at every point in its domain is a nonempty set of real numbers is an Abacus. It is shown that to every collection C of abaci there corresponds a logic, called an abacus logic, i.e.. a certain set of propositions partially ordered by generalized implication. It is also shown that to every collection C of abaci there corresponds a theory Jc in a classical propositional calculus such that the abacus logic determined by C is isomorphic to the poset of Jc. Two examples are given. In both examples abacus logic is a lattice in which there happens to be an operation of orthocomplementation. In the first example abacus logic turns out to be the Lindenbaum algebra of Jc. In the second example abacus logic is a lattice isomorphic to the ortholattice of subspaces of a Hilbert space. Thus quantum logic can be regarded as an abacus logic. Without suggesting “hidden variables” it is finally shown that the Lindenbaum algebra of the theory in the second example is a subalgebra of the abacus logic B of the kind studied in example 1. It turns out that the “classical observables” associated with B and the “quantum observables” associated with quantum logic are not unrelated. The value of a classical observable contains, in coded form, information about the “uncertainty” of a quantum observable. This information is retrieved by decoding the value of the corresponding classical observable.
Équivalence élémentaire et isomorphisme des corps de courbe sur un corps algébriquement clos
Authors: Duret Jean-Louis 
Published: May 2006

Cet article fait suite à [D] dont il utilise les résultats.Soit if ℒ langage constitué de deux constantes 0 et 1, et de deux fonctions + et ·; si A est un sous-ensemble d'une structure de ℒ, nous noterons if ℒ (A) le langage obtenu en ajoutant à ℒ if les éléments de A comme constantes.Tous les corps considérés seront commutatifs. Nous appellerons corps de courbe sur un corps k une extension K de k finiment engendrée de degré de transcendance 1 sur k telle que k soit relativement algébriquement clos dans K.Nous appellerons ensemble de coefficients d'une courbe sur un corps k un sous-ensemble A de K tel qu'il existe un système de générateurs de l'idéal de cette courbe dont les coefficient sont des éléments de A.Nous nous proposons d'étudier les conjectures suivantes:1. Conjecture. Si K est un corps de courbe sur un corps algébriquement clos k, il existe un sous-ensemble fini A de k tel que tout corps de courbe sur kélémentairement équivalent à K dans le langage ℒ (A), lui est k-isomorphe.2. Conjecture. Deux corps de courbe sur un corps algébriquement clos K élémentairement équivalents dans le langage ℒ sont isomorphes.Nous démontrerons ces conjectures lorsque le genre est différent de 1, et si le genre est 1, lorsque la caractéristique est nulle et le corps de courbe sans multiplication complexe.Une version plus simple de cet article est parue dans [D′].
The theory of the recursively enumerable weak truth-table degrees is undecidable
Authors: Ambos-Spies Klaus , Nies André , Shore Richard A. 
Published: May 2006

AbstractWe show that the partial order of -sets under inclusion is elementarily definable with parameters in the semilattice of r.e. wtt-degrees. Using a result of E. Herrmann, we can deduce that this semilattice has an undecidable theory, thereby solving an open problem of P. Odifreddi.