bluuebunny/crossref_metadata_2025_split huggingface dataset:
Tarski and geometry
Authors: Szczerba L. W. 
Published: May 2006

Tarski published his first geometry paper, [24b], in 1924. As is well known, the area of the union of two disjoint figures is the sum of the areas of these two figures. This observation is the basis of a method for proving that two figures, say A and B, have the same area: if we can divide each of the two figures A and B into a finite number of pairwise disjoint subfigures A1,…,An and B1,…,Bn such that for every i, figures Ai and Bi are congruent (we say that two such figures are equivalent by finite decomposition), then figures A and B have the same area. The method is by no means universal. For example a disc and a rectangle can never be equivalent by finite decomposition, even if they have the same area. Hilbert [1922, Kapitel IV] proved from his axiom system the so-called De Zolt axiom:If a polygon V is a proper subset of a polygon W then they are not equivalent by a finite decomposition.Hilbert's proof is elementary but difficult. In [24b] Tarski gave an easy but nonelementary proof of a stronger version of the De Zolt axiom:If a polygon V is a proper subset of a polygon W then they are not equivalent by finite decomposition into any figures.
Back and forth constructions in modal logic: An interpolation theorem for a family of modal logics
Authors: Weaver George , Welaish Jeffrey 
Published: May 2006

The following is a contribution to the abstract study of the model theory of modal logics. Historically, individual modal logics have been specified deductively; and, as a result, it has seemed natural to view modal logics as sets of sentences provable in some deductive system. This proof theoretic view has influenced the abstract study of modal logics. For example, Fine [1975] defines a modal logic to be any set of sentences in the modal language L□ which contains all tautologies, all instances of the schema (□(ϕ ⊃ Ψ) ⊃ (□ϕ ⊃ □Ψ)), and which is closed under modus ponens, necessitation and substitution.Here, however, modal logics are equated with classes of “possible world” interpretations. “Worlds” are taken to be ordered pairs (a, λ), where a is a sentential interpretation and λ is an ordinal. Properties of the accessibility relation are identified with those classes of binary relational systems closed under isomorphisms. The origin of this approach is the study of alternative Kripke semantics for the normal modal logics (cf. Weaver [1973]). It is motivated by the desire that modal logics provide accounts of both logical truth and logical consequence (cf. Corcoran and Weaver [1969]) and the realization that there are alternative Kripke semantics for S4, B and M which give identical accounts of logical truth, but different accounts of logical consequence (cf. Weaver [1973]). It is shown that the Craig interpolation theorem holds for any modal logic which has characteristic modal axioms and whose associated class of binary relational systems is closed under subsystems and finite direct products. The argument uses a back and forth construction to establish a modal analogue of Robinson's theorem. The argument for the interpolation theorem from Robinson's theorem employs modal analogues of the Ehrenfeucht-Fraïssé characterization of elementary equivalence and Hintikka's distributive normal form, and is itself a modal analogue of a first order argument (cf. Weaver [1982]).
Jensen's ⃞ principles and the Novák number of partially ordered sets
Authors: Veličković Boban 
Published: May 2006

In this paper we consider various properties of Jensen's □ principles and use them to construct several examples concerning the so-called Novák number of partially ordered sets.In §1 we give the relevant definitions and review some facts about □ principles. Apart from some simple observations most of the results in this section are known.In §2 we consider the Novák number of partially ordered sets and, using □ principles, give counterexamples to the productivity of this cardinal function. We also formulate a principle, show by forcing that it is consistent and use it to construct an ℵ2-Suslin tree T such that forcing with T × T collapses ℵ1.In §3 we briefly consider games played on partially ordered sets and relate them to the problems of the previous section. Using a version of □ we give an example of a proper partial order such that the game of length ω played on is undetermined.In §4 we raise the question of whether the Novák number of a homogenous partial order can be singular, and show that in some cases the answer is no.We assume familiarity with the basic techniques of forcing. In §1 some facts about large cardinals (e.g. weakly compact cardinals are -indescribable) and elementary properties of the constructible hierarchy are used. For this and all undefined terms we refer the reader to Jech [10].
Omitting types in -minimal theories
Authors: Marker David 
Published: May 2006

Let L be a first order language containing a binary relation symbol <.Definition. Suppose ℳ is an L-structure and < is a total ordering of the domain of ℳ. ℳ is ordered minimal (-minimal) if and only if any parametrically definable X ⊆ ℳ can be represented as a finite union of points and intervals with endpoints in ℳ.In any ordered structure every finite union of points and intervals is definable. Thus the -minimal structures are the ones with no unnecessary definable sets. If T is a complete L-theory we say that T is strongly (-minimal if and only if every model of T is -minimal.The theory of real closed fields is the canonical example of a strongly -minimal theory. Strongly -minimal theories were introduced (in a less general guise which we discuss in §6) by van den Dries in [1]. Extending van den Dries' work, Pillay and Steinhorn (see [3], [4] and [2]) developed an extensive structure theory for definable sets in strongly -minimal theories, generalizing the results for real closed fields. They also established several striking analogies between strongly -minimal theories and ω-stable theories (most notably the existence and uniqueness of prime models). In this paper we will examine the construction of models of strongly -minimal theories emphasizing the problems involved in realizing and omitting types. Among other things we will prove that the Hanf number for omitting types for a strongly -minimal theory T is at most (2∣T∣)+, and characterize the strongly -minimal theories with models order isomorphic to (R, <).
Finitely generated free Heyting algebras
Authors: Bellissima Fabio 
Published: May 2006

AbstractThe aim of this paper is to give, using the Kripke semantics for intuitionism, a representation of finitely generated free Heyting algebras. By means of the representation we determine in a constructive way some set of “special elements” of such algebras. Furthermore, we show that many algebraic properties which are satisfied by the free algebra on one generator are not satisfied by free algebras on more than one generator.
The geometry of weakly minimal types
Authors: Buechler Steven 
Published: May 2006

AbstractLet T be superstable. We say a type p is weakly minimal if R(p, L, ∞) = 1. Let M ⊨ T be uncountable and saturated, H = p(M). We say D ⊂ H is locally modular if for all X, Y ⊂ D with X = acl(X) ∩ D, Y = acl(Y) ∩ D and X ∩ Y ≠ ∅,Theorem 1. Let p ∈ S(A) be weakly minimal and D the realizations of stp(a/A) for some a realizing p. Then D is locally modular or p has Morley rank 1.Theorem 2. Let H, G be definable over some finite A, weakly minimal, locally modular and nonorthogonal. Then for all a ∈ H∖acl(A), b ∈ G∖acl(A) there area′ ∈ H, b′ ∈ G such that a′ ∈ acl(abb′A)∖acl(aA). Similarly when H and G are the realizations of complete types or strong types over A.
A saturation property of ideals and weakly compact cardinals
Authors: Takahashi Joji 
Published: May 2006

Suppose κ is a regular uncountable cardinal, λ is a cardinal > 1, and is a κ-complete uniform ideal on κ. This paper deals with a saturation property Sat(κ, λ, ) of , which is a weakening of usual λ-saturatedness. Roughly speaking, Sat(κ, λ, ) means that can be densely extended to λ-saturated ideals on small fields of subsets of κ. We will show that some consequences of the existence of a λ-saturated ideal on κ follow from weaker ∃: Sat(κ, λ, ), and that ∃: Sat(κ, λ, ) is connected with weak compactness and complete ineffability of κ in much the same way as the existence of a saturated ideal on κ is connected with measurability of κ.In §2, we define Sat(κ, λ, ), mention a few results that can be proved by straightforward adaptation of known methods, and discuss generic ultrapowers of ZFC−, which will be used repeatedly in the subsequent sections as a main technical tool. A related concept Sat(κ, λ) is also defined and shown to be equivalent to ∃: Sat(κ, λ, ) under a certain condition.In §3, we show that ∃: Sat(κ, κ, ) implies that κ is highly Mahlo, improving results in [KT] and [So].
Some properties of epistemic set theory with collection
Authors: Scedrov Andre 
Published: May 2006

Myhill [12] extended the ideas of Shapiro [15], and proposed a system of epistemic set theory IST (based on modal S4 logic) in which the meaning of the necessity operator is taken to be the intuitive provability, as formalized in the system itself. In this setting one works in classical logic, and yet it is possible to make distinctions usually associated with intuitionism, e.g. a constructive existential quantifier can be expressed as (∃x) □ …. This was first confirmed when Goodman [7] proved that Shapiro's epistemic first order arithmetic is conservative over intuitionistic first order arithmetic via an extension of Gödel's modal interpretation [6] of intuitionistic logic.Myhill showed that whenever a sentence □A ∨ □B is provable in IST, then A is provable in IST or B is provable in IST (the disjunction property), and that whenever a sentence ∃x.□A(x) is provable in IST, then so is A(t) for some closed term t (the existence property). He adapted the Friedman slash [4] to epistemic systems.Goodman [8] used Epistemic Replacement to formulate a ZF-like strengthening of IST, and proved that it was a conservative extension of ZF and that it had the disjunction and existence properties. It was then shown in [13] that a slight extension of Goodman's system with the Epistemic Foundation (ZFER, cf. §1) suffices to interpret intuitionistic ZF set theory with Replacement (ZFIR, [10]). This is obtained by extending Gödel's modal interpretation [6] of intuitionistic logic. ZFER still had the properties of Goodman's system mentioned above.
Combinatorial and recursive aspects of the automorphism group of the countable atomless Boolean algebra
Authors: Madison E. W. , Zimmermann-Huisgen B. 
Published: May 2006

AbstractGiven an admissible indexing φ of the countable atomless Boolean algebra ℬ, an automorphism F of ℬ is said to be recursively presented (relative to φ) if there exists a recursive function p ϵ Sym(ω) such that F ∘ φ = φ ∘ p. Our key result on recursiveness: Both the subset of Aut(ℬ) consisting of all those automorphisms which are recursively presented relative to some indexing, and its complement, the set of all “totally nonrecursive” automorphisms, are uncountable.This arises as a consequence of the following combinatorial investigations: (1) A comparison of the cycle structures of ƒ and , where ƒ is a permutation of some free basis of ℬ and is the automorphism of ℬ induced by ƒ.(2) An explicit description of the permutations of ω whose conjugacy classes in Sym(ω) are (a) uncountable, (b) countably infinite, and (c) finite.
Maximal chains in the fundamental order
Authors: Buechler Steven 
Published: May 2006

AbstractSuppose T is superstable. Let ≤ denote the fundamental order on complete types, [p] the class of the bound of p, and U(—) Lascar's foundation rank (see [LP]). We proveTheorem 1. If q < p and there is no r such that q < r < p, then U(q) + 1 = U(p).Theorem 2. Suppose U(p) < ω and ξ1 < … < ξk is a maximal descending chain in the fundamental order with ξk = [p]. Then k = U(p).That the finiteness of U(p) in Theorem 2 is necessary follows fromTheorem 3. There is an ω-stable theory with a type p ϵ S1(ϕ) such that(1) U(p) = ω + 1, and(2) there is a maximal descending chain of proper extensions of [p] which has order type ω.
On the strength of nonstandard analysis
Authors: Henson C. Ward , Keisler H. Jerome 
Published: May 2006

It is often asserted in the literature that any theorem which can be proved using nonstandard analysis can also be proved without it. The purpose of this paper is to show that this assertion is wrong, and in fact there are theorems which can be proved with nonstandard analysis but cannot be proved without it. There is currently a great deal of confusion among mathematicians because the above assertion can be interpreted in two different ways. First, there is the following correct statement: any theorem which can be proved using nonstandard analysis can be proved in Zermelo-Fraenkel set theory with choice, ZFC, and thus is acceptable by contemporary standards as a theorem in mathematics. Second, there is the erroneous conclusion drawn by skeptics: any theorem which can be proved using nonstandard analysis can be proved without it, and thus there is no need for nonstandard analysis.The reason for this confusion is that the set of principles which are accepted by current mathematics, namely ZFC, is much stronger than the set of principles which are actually used in mathematical practice. It has been observed (see [F] and [S]) that almost all results in classical mathematics use methods available in second order arithmetic with appropriate comprehension and choice axiom schemes.
Prime models and almost decidability
Authors: Millar Terrence 
Published: May 2006

This paper introduces and investigates a notion that approximates decidability with respect to countable structures. The paper demonstrates that there exists a decidable first order theory with a prime model that is not almost decidable. On the other hand it is proved that if a decidable complete first order theory has only countably many complete types, then it has a prime model that is almost decidable. It is not true that every decidable complete theory with only countably many complete types has a decidable prime model. It is not known whether a complete decidable theory with only countably many countable models up to isomorphism must have a decidable prime model. In [1] a weaker result was proven—if every complete extension, in finitely many additional constant symbols, of a theory T fails to have a decidable prime model, then T has 2ω nonisomorphic countable models. The corresponding statement for saturated models is false, even if all the complete types are recursive, as was shown in [2]. This paper investigates a variation of the open question via a different notion of effectiveness—almost decidable.A tree Tr will be a subset of ω<ω that is closed under predecessor. For elements f, g in ω<ω ∪ ωω, ƒ ⊲ g iffdf ∀i < lh(ƒ)[ƒ(i) = g(i)].
ω1-like recursively saturated models of Presburger's arithmetic
Authors: Harnik Victor 
Published: May 2006

Let Pr be Presburger's arithmetic, i.e., the complete theory of the structure (ω, +). Lipshitz and Nadel showed in [4] that a countable model of Pr is recursively saturated iff it can be expanded to a model of Peano arithmetic, PA. As a starting point for an introductory discussion, let us mention one more fact about countable recursively saturated models of Pr. If is such a model then the following is readily seen (as explained also in §1):Any two countable recursively saturated elementary endextensions of are isomorphic.If we drop “countable” from the assumption of this statement, we can still say that the two models are ∞ω-equivalent. Must they be isomorphic if both have cardinality ℵ1? Certainly not, since one of the models can be ω1-like while the other is not. Once we realized this much, let us concentrate on ω1-like structures. We prove in §3:Theorem. Any countable recursively saturated modelof Pr hasisomorphism types of ω1-like recursively saturated elementary endextensions. Only one of these is the isomorphism type of a structure that can be expanded to a model of PA.The key technical result is proven in §2. It says that an as above has precisely two countable recursively saturated elementary endextensions which are nonisomorphic over .
The axiom of determinacy implies dependent choices in L(R)
Authors: Kechris Alexander S. 
Published: May 2006

AbstractWe prove the following Main Theorem: ZF + AD + V ⇒ L(R) ⇒ DC. As a corollary we have that Con(ZF + AD) ⇒ Con(ZF + AD + DC). Combined with the result of Woodin that Con(ZF + AD) ⇒ Con(ZF + AD + ¬ ACω) it follows that DC (as well as ACω) is independent relative to ZF + AD. It is finally shown (jointly with H. Woodin) that ZF + AD + ¬DCR, where DCR is DC restricted to reals, implies the consistency of ZF + AD + DC, in fact implies R# (i.e. the sharp of L(R)) exists.
Deux remarques à propos de la propriété de recouvrement fini
Authors: Poizat Bruno 
Published: May 2006

Cette note est consacrée à quelques compléments à mon article [6], que le lecteur est supposé avoir sous les yeux, et qui traite de la propriété de recouvrement fini, de Keisler. Je rappelle seulement ici que:a) La définition originale est la suivante: une théorie complète T n'a pas la p.r.f. si à toute formule f(x, ȳ) de son langage est associé un entier n, de sorte que tout ensemble (fini!) {f(x, ā1), …, f(x, ās)}, où les āi; sont des paramètres extraits d'un modèle M de T, soit consistant dès que ses parties à n éléments le sont. Une théorie sans p.r.f. satisfait done un renforcement du théorème de compacité; elle est en ce sens plus simple qu'une théorie avec p.r.f.b) T n'a pas la p.r.f. si et seulement si elle est stable (i.e. tout type sur un modèle M de T est définissable; à toute formule f(x, ȳ) est associée une formule g(ȳ, ā) à paramètres dans M, telle que les de M pour lesquels f(x, ) est dans le type en question soient précisément ceux qui vérifient g(ȳ, ā)), et si en outre on peut exprimer par une formule sur le paramètre ā que g(ȳ, ā) définit un f-type consistant: à f1(x, ȳ), …, fs(x, ȳ), g1(ȳ, ), …, gs(ȳ, ) est associé une formule h(), telle que pour tout ā de tout modèle M de T, ā satisfasse h si et seulement si l'ensemble de tous les fi(x, ), où est dans M et satisfait gi(ȳ, ā), est consistant.
Characteristically simple ℵ0-categorical groups
Authors: Gilman Robert H. 
Published: May 2006

A countable group is ℵ0-categorical if it is characterized up to isomorphism among all countable groups by its first order theory. We investigate countable ℵ0-categorical groups which are characteristically simple and have proper subgroups of finite index. We determine such groups up to isomorphism (Corollary 4 to Proposition 3), and we show that their theories are finitely axiomatizable (Proposition 2).Let K be a finite nonabelian simple group. In §3 we construct from K two denumberable groups, K* and K#, and use them to classify countable direct limits of finite Cartesian powers of K (Proposition 1). The results of §3 are used in §4 to show that K* and K# are ℵ0-categorical and to find axioms for their theories. By Corollary 2 to Proposition 2, K* and K# are counterexamples to the conjecture of U. Feigner [3, p. 309] that ℵ0-categorical groups are FC-solvable. Also in §4 the results mentioned in the first paragraph are proved. §2 is devoted to preliminary lemmas about finite Cartesian powers of K.A characteristic subgroup of a group G is a subgroup which is mapped onto itself by all automorphisms of G. G is characteristically simple if its only characteristic subgroups are itself and the identity subgroup. Each characteristic subgroup of a group is a union of orbits of the automorphism group of the group. If G is ℵ0-categorical, then as the automorphism group of G has only a finite number of orbits, G has only a finite number of characteristic subgroups.
Accessible sets and (Lω1ω)t-equivalence for T3 spaces
Authors: Martínez Juan Carlos 
Published: May 2006

Ziegler studies in [2] the expressive power of (Lωω)t for T3 topological spaces. He defines for every natural number n the set of n-types by induction: . If A is a T3 space, the n-type of a ∈ A is defined inductively by: : in every neighborhood of a there is an a′ ≠ a with tn(a′, A) = α}. These types are (Lωω)t-definable. Then, it is shown that two T3 spaces are (Lωω)t-equivalent precisely if for every n-type α they have the same number of points with n-type α (cf. [2]).In order to study the expressability of (Lω1ω)t, for T3 spaces, we introduce in this paper the notion of accessible set. Looking at the behaviour of convergence by means of this notion, we refine Ziegler's notion of n-type and introduce a new set Sn of n-types, which are (Lω1ω)t-definable. Then, we prove a characterization of (Lω1ω)t-equivalence for a wide class of T3 spaces. A T3 space A belongs to this class if there is a κ ∈ ω such that, for every n ∈ ω, there are at most κ n-types in Sn which are satisfiable in A. Such a space is said to be of a-finite type. Some relations between these spaces and the spaces of finite type in the sense of [2] are shown in the last section.The contents of the present paper are treated in more detail in [3].
An incomplete decidable modal logic
Authors: Cresswell M. J. 
Published: May 2006

The most common way of proving decidability in propositional modal logic is to shew that the system in question has the finite model property. This is not however the only way. Gabbay in [4] proves the decidability of many modal systems using Rabin's result in [8] on the decidability of the second-order theory of successor functions. In particular [4, pp. 258-265] he is able to prove the decidability of a system which lacks the finite model property. Gabbay's system is however complete, in the sense of being characterized by a class of frames, and the question arises whether there is a decidable modal logic which is not complete. Since no incomplete modal logic has the finite model property [9, p. 33], any proof of decidability must employ some such method as Gabbay's. In this paper I use the Gabbay/Rabin technique to prove the decidability of a finitely axiomatized normal modal propositional logic which is not characterized by any class of frames. I am grateful to the referee for suggesting improvements in substance and presentation.The terminology I am using is standard in modal logic. By aframeis understood a pair 〈W, R〉 in whichWis a class (of “possible worlds”) andR⊆W2. To avoid confusion in what follows, a frame will henceforth be referred to as aKripkeframe. By contrast, ageneral frameis a pair 〈,Π〉 in whichis a Kripke frame andΠis a collection of subsets ofWclosed under the Boolean operations and satisfying the condition that ifAis inΠthen so isR−1“A. A model on a frame (of either kind) is obtained by adding a functionVwhich assigns sets of worlds to propositional variables. In the case of a general frame we require thatV(p) ∈Π.
Arithmetic and the theory of types
Authors: Boffa M. 
Published: May 2006

A hundred years ago, Frege proposed a logical definition of the natural numbers based on the following idea:He replaced this circular definition by the following one:He tried afterwards to found his theory over a notion of class satisfying a general comprehension principle:Russell quickly derived a contradiction from this principle (the famous Russell's paradox) but saved Frege's arithmetic with his theory of types based on the following comprehension principle:In 1979, talking at the Claude Bernard University in Lyon, I remarked that 3 types suffice to provide Frege's arithmetic, showing in fact that PA2 (second order Peano arithmetic) holds in TT3 + AI (theory of types 0, 1, 2 plus a suitable axiom of infinity). I asked whether TT3 + AI was a conservative extension of PA2. Pabion [3] gave a positive answer by a subtle use of the Fraenkel-Moskowski method. This result will be improved in the present paper, with a view to getting models of NF3 + AI in which Frege's arithmetic forms a model isomorphic to a given countable model of PA2.