bluuebunny/crossref_metadata_2025_split huggingface dataset:
Definable equivalence relations on algebraically closed fields
Authors: van den Dries Lou , Marker David , Martin Gary 
Published: May 2006

This article was inspired by the question: is there a definable equivalence relation on the field of complex numbers, each of whose equivalence classes has exactly two elements? The answer turned out to be no, as we now explain in greater detail.Let Κ be an algebraically closed field and let E be a definable equivalence relation on Κ. [Note: By “definable” we will always mean “definable with parameters”.] Either E has one cofinite class, or all classes are finite and there is a number d such that all but a finite set of classes have cardinality d. In the latter case let B be the finite set of elements of Κ which are not in a class of size d. We prove the following result.Theorem 1. a) If char(Κ) = 0 or char(Κ) = p > d, then ∣B∣ ≡ 1 (mod d).b) If char(Κ) = 2 and d = 2, then ∣B∣ ≡ 0 (mod 2).c) If char(Κ) = p > 2 and d = p + s, where 1 ≤ s ≤ p/2, then ∣B∣ ≡ p + 1 (mod d).Furthermore, a)−c) are the only restrictions on ≡B≡.If one is in the right mood, one can view this theorem as saying that the “algebraic cardinality” of the complex numbers is congruent to 1 (mod n) for every n.§1 contains a reduction of the problem to the special case where E is induced by a rational function in one variable. §2 contains the main calculations and the proofs of a)−c). §3 contains eight families of examples showing that all else is possible. In §4 we prove an analogous result for real closed fields.
Creativeness and completeness in recursion categories of partial recursive operators
Authors: Montagna Franco , Sorbi Andrea 
Published: May 2006

Recursion categories have been proposed by Di Paola and Heller in [DPH] as the basis for a category-theoretic approach to recursion theory, in the context of a more general and ambitious project of a purely algebraic treatment of incompleteness phenomena. The way in which the classical notion of creative set is rendered in this new category-theoretic framework plays, therefore, a central role. This is done in [DPH] (Definition 8.1) by defining the notion of creative domains or, rather, domains which are creative relative to some criterion: thus, in a recursion category, every criterion provides a notion of creativeness.A basic result on creative domains (cf. [DPH, Theorem 8.13]) is that, under certain assumptions, a version of the classical result, due to Myhill [MYH], stating that every creative set is complete, holds: in a recursion category with equality (i.e. exists for every object X) and having enough atoms, every domain which is creative with respect to atoms is also complete.
The number of pairwise non-elementarily-embeddable models
Authors: Shelah Saharon 
Published: May 2006

AbstractWe get consistency results on I(λ, T, T) under the assumption that D(T) has cardinality > ∣T∣. We get positive results and consistency results on IE(λ, T1, T).The interest is model-theoretic, but the content is mostly set-theoretic: in Theorems 1–3, combinatorial; in Theorems 4–7 and 11(2), to prove consistency of counterexamples we concentrate on forcing arguments; and in Theorems 8–10 and 11(1), combinatorics for counterexamples; the rest are discussion and problems. In particular:(A) By Theorems 1 and 2, if T ⊆ T1 are first order countable, T complete stable but ℵ0-unstable, λ > ℵ0, and ∣D(T)∣ > ℵ0, then IE(λ, T1, T) > Min{2λ, ℶ2}.(B) By Theorems 4,5,6 of this paper, if e.g. V = L, then in some generic extension of V not collapsing cardinals, for some first order T ⊆ T1, ∣T∣ = ℵ0, ∣T1∣ = ℵ1, ∣D(T)∣ = ℵ2 and IE(ℵ2, T1, T) = 1.This paper (specifically the ZFC results) is continued in the very interesting work of Baldwin on diversity classes [Bl]. Some more advances can be found in the new version of [Sh300] (see Chapter III, mainly §7); they confirm 0.1, 0.2 and 14(1), 14(2).
On ideals and stationary reflection
Authors: Johnson C. A. 
Published: May 2006

It is a theorem of Prikry [7] that ifκcarries a uniformη-descendingly complete ultrafilter then the stationary reflection propertyfails. In this paper we will derive similar results, but here from properties of filters (or ideals) rather than ultrafilters.Throughoutκandηwill denote regular cardinals withη<κ(in particularκwill be uncountable), andIwill denote an ideal onκ, by which we mean a setI⊆P(κ) such that (i)Iis closed under taking subsets and finite unions and (ii)αЄIfor eachα<κ, butκ∉I.Iis said to beμ-complete if it is closed under taking unions of size <μ,I* = {X⊆κ∣κ−XЄI} is the filter dual toIand ifAЄI+(=P(κ) −I), thenI∣Ais the ideal onκgiven byI∣A= {X⊆κ∣X∩AЄI}. Ifh: A→κthenhis said to be (i) unbounded modIif for eachα<κ,h−1(α) = {ξЄA∣h(ξ)<α} ЄIand (ii) a least function forIifhis unbounded modIand wheneverg: A→κis a function, unbounded modI, then {ξЄA∣g{ξ) <h{ξ)} ЄI.
Some uses of dilators in combinatorial problems. II
Authors: Abrusci V. Michele , Girard Jean-Yves , van de Wiele Jacques 
Published: May 2006

AbstractWe study increasing F-sequences, where F is a dilator: an increasing F-sequence is a sequence (indexed by ordinal numbers) of ordinal numbers, starting with 0 and terminating at the first step x where F(x) is reached (at every step x + 1 we use the same process as in decreasing F-sequences, cf. [2], but with “+ 1” instead of “−1”). By induction on dilators, we shall prove that every increasing F-sequence terminates and moreover we can determine for every dilator F the point where the increasing F-sequence terminates.We apply these results to inverse Goodstein sequences, i.e. increasing (1 + Id)(ω)-sequences. We show that the theoremevery inverse Goodstein sequence terminates(a combinatorial theorem about ordinal numbers) is not provable in ID1.For a general presentation of the results stated in this paper, see [1].We use notions and results concerning the category ON (ordinal numbers), dilators and bilators, summarized in [2, pp. 25–31].
L'axiome de normalité pour les espaces totalement ordonnés
Authors: Haddad Labib , Morillon Marianne 
Published: May 2006

AbstractWe show that the following property (LN) holds in the basic Cohen model as sketched by Jech: The order topology of any linearly ordered set is normal. This proves the independence of the axiom of choice from LN in ZF, and thus settles a question raised by G. Birkhoff (1940) which was partly answered by van Douwen (1985).
Sequential theories and infinite distributivity in the lattice of chapters
Authors: Stern Alan S. 
Published: May 2006

AbstractWe introduce a notion of complexity for interpretations, which is used to prove some new results about interpretations of sequential theories. In particular, we give a new, elementary proof of Pudlák's theorem that sequential theories are connected. We also demonstrate a counterexample to the infinitary distributive lawin the lattice of chapters, in which the chapters a and bi are compact. (Counterexamples in which a is not compact have been found previously.)
Positive ∑ operations on ordinals and normal filters on greatly mahlo cardinals
Authors: Jech Thomas 
Published: May 2006

If ℱ is a normal filter on a regular uncountable cardinal κ, let ║f║ be the ℱ-norm of an ordinal function f. We introduce the class of positive ordinal operations and prove that if F is a positive operation then ║F(f)║ ≥ F(║f║). For each η < κ+ let fη be the canonical ηth function. We show that if F is a ∑ operation then F(fη) = fF(η).As an application we show that if κ is greatly Mahlo then there are normal filters on κ of order greater than κ+.
Normality and
Authors: Zrotowski R. 
Published: May 2006

AbstractThe main result of this paper is that if κ is not a weakly Mahlo cardinal, then the following two conditions are equivalent:1. is κ+-complete.2. is a prenormal ideal.Our result is a generalization of an announcement made in [Z]. We say that is selective iff for every -function f: κ → κ there is a set X ∈ such that f∣(κ − X) is one-to-one. Our theorem provides a positive partial answer to a question of B. Wȩglorz from [BTW, p. 90], viz.: is every selective ideal with κ+-complete, isomorphic to a normal ideal?The theorem is also true for fine ideals on [λ]<κ for any κ ≤ λ, i.e. if κ is not a weakly Mahlo cardinal then the Boolean algebra is λ+-complete iff is a prenormal ideal (in the sense of [λ/<κ).
Forcing isomorphism
Authors: Baldwin J. T. , Laskowski M. C. , Shelah S. 
Published: May 2006

If two models of a first-order theory are isomorphic, then they remain isomorphic in any forcing extension of the universe of sets. In general however, such a forcing extension may create new isomorphisms. For example, any forcing that collapses cardinals may easily make formerly nonisomorphic models isomorphic. However, if we place restrictions on the partially-ordered set to ensure that the forcing extension preserves certain invariants, then the ability to force nonisomorphic models of some theory T to be isomorphic implies that the invariants are not sufficient to characterize the models of T.A countable first-order theory is said to be classifiable if it is superstable and does not have either the dimensional order property (DOP) or the omitting types order property (OTOP). If T is not classifiable, Shelah has shown in [5] that sentences in L∞,λ do not characterize models of T of power λ. By contrast, in [8] Shelah showed that if a theory T is classifiable, then each model of cardinality λ is described by a sentence of L∞,λ. In fact, this sentence can be chosen in the . ( is the result of enriching the language by adding for each μ < λ a quantifier saying the dimension of a dependence structure is greater than μ) Further work ([3], [2]) shows that ⊐+ can be replaced by ℵ1.
An existence theorem for a special ultrafilter when 𝔡 = 𝔠
Authors: Moloney James J. 
Published: May 2006

For an ultrafilter , consider the ultrapower NN/. 〈an〉/ is in the top sky of NN/ if there exists a sequence 〈bn〉 ∈ NN such thatandIn [M2] we showed, assuming the Continuum Hypothesis, that there are exactly 10 c/p's (where c is the ring of real convergent sequences and p is a prime ideal of c). To get the lower bound we showed that there will be at least 10 c/p's in any model of ZFC where there exist both of the following kinds of ultrafilter:(i) nonprincipal P-points,(ii) non-P-points such that when the top sky is removed from NN/, the remaining model has countable cofinality.In [M2] we showed that the Continuum Hypothesis implies the existence of the ultrafilter in (ii). In this paper we show that its existence is implied by an axiom weaker than the Continuum Hypothesis, in fact weaker than Martin's Axiom, namely,(*) If is a subset of NN such that for any f: N → N there exists g ∈ such that g(n) > f(n) for all n, then ∣∣ = .
On the number of nonisomorphic models of size |T|
Authors: Chowdhury Ambar 
Published: May 2006

AbstractLet T be an uncountable, superstable theory. In this paper we proveTheorem A. If T has finite rank, then I(|T|, T) ≥ ℵ0.Theorem B. If T is trivial, then I(|T|, T) ≥ ℵ0.
The degrees of conditional problems
Authors: Gao Su 
Published: May 2006

AbstractIn this paper we define and study conditional problems and their degrees. The main result is that the class of conditional degrees is a lattice extending the ordinary Turing degrees and it is dense. These properties are not shared by ordinary Turing degrees. We show that the class of conditional many-one degrees is a distributive lattice. We also consider properties of semidecidable problems and their degrees, which are analogous to r.e. sets and degrees.
The Beth-closure of ℒ(Qα) is not finitely generated
Authors: Hella Lauri , Luosto Kerkko 
Published: May 2006

AbstractWe prove that if ℵα is uncountable and regular, then the Beth-closure of ℒωω(Qα) is not a sublogic of ℒαω(Qn), where Qn is the class of all n-ary generalized quantifiers. In particular, B(ℒωω(Qα)) is not a sublogic of any finitely generated logic; i.e., there does not exist a finite set Q of Lindström quantifiers such that B(ℒωω(Qα)) ≤ ℒωω(Q).
Adding dominating reals with ωω of bounding posets
Authors: Pawlikowski Janusz 
Published: May 2006

Let B be the random real forcing. Miller [Mi] asked if there are ZFC models M ⊆ N such that forcing with BM over N adds a dominating real. A YES answer was provided by Judah and Shelah in [JS], where in a long and sophisticated construction they built such models. In this paper we prove that forcing with BV over VI, where I is the infinitely often equal real forcing of [Mi], adds a dominating real over VI. This greatly simplifies the YES answer to Miller's question. Moreover it turns out that B may be replaced here by E, the eventually different real forcing of [Mi]. This answers the second part of Miller's question. We also prove that both side by side products I × B and I × E add a Hechler dominating real over V.In this section we establish the main result of the paper; namely, we prove that forcing over VI with either of the posets BV or EV adds a dominating real over VI.First we recall the definitions of I and E from [Mi]. The infinitely often equal real forcing is the setordered by extension. Miller [Mi] proves that I is ωω bounding.
Four concepts from “geometrical” stability theory in modules
Authors: Kucera T. G. , Prest M. 
Published: May 2006

In [H1] Hrushovski introduced a number of ideas concerning the relations between types which have proved to be of importance in stability theory. These relations allow the geometries associated to various types to be connected. In this paper we consider the meaning of these concepts in modules (and more generally in abelian structures). In particular, we provide algebraic characterisations of notions such as hereditary orthogonality, “p -internal” and “p-simple”. These characterisations are in the same spirit as the algebraic characterisations of such concepts as orthogonality and regularity, that have already proved so useful. Of the concepts that we consider, p-simplicity is dealt with in [H3] and the other three concepts in [H2].The descriptions arose out of our desire to develop some intuition for these ideas. We think that our characterisations may well be useful in the same way to others, particularly since our examples are algebraically uncomplicated and so understanding them does not require expertise in the model theory of modules. Furthermore, in view of the increasing importance of these notions, the results themselves are likely to be directly useful in the model-theoretic study of modules and, via abelian structures, in more general stability-theoretic contexts. Finally, some of our characterisations suggest that these ideas may be relevant to the algebraic problem of understanding the structure of indecomposable injective modules.
Weakly normal closures of filters on Pκλ
Authors: Shioya Masahiro 
Published: May 2006

The study of filters on Pκλ started by Jech [5] as a natural generalization of that of filters on an uncountable regular cardinal κ. Several notions including weak normality have been generalized. However, there are two versions proposed as weak normality for filters on Pκλ. One is due to Abe [1] as a straightforward generalization of weak normality for filters on κ due to Kanamori [6] and the other is due to Mignone [8]. While Mignone's version is weaker than normality, Kanamori-Abe's version is not in general. In fact, Abe [2] has proved, generalizing Kanamori [6], that a filter is weakly normal in the sense of Abe iff it is weakly normal in the sense of Mignone and there exists no disjoint family of cfλ-many positive sets. Therefore Kanamori-Abe's version is essentially a large cardinal property and Mignone's version seems to be the most natural formulation of “weak” normality.In this paper, we study weak normality in the sense of Mignone. In [8], Mignone studies weak normality of canonically defined filters. We complement his chart and try to find the weakly normal closures of these filters (i.e., the minimal weakly normal filters extending them). Therefore our result is a natural refinement of Carr [4].It is now well known that combinatorics on Pκλ is not a naive generalization of that on κ. For example, Menas [7] showed that stationarity on Pκλ can be characterized by 2-dimensional regressive functions, but not by 1-dimensional ones when λ is strictly larger than κ. We show in terms of weak normality that combinatorics on Pκλ vary drastically with respect to cfλ.
The McKinsey axiom is not compact
Authors: Wang Xiaoping 
Published: May 2006

The canonicity and compactness of the KM system are problems historically important in the development of our understanding of intensional logic (as explained in Goldblatt's paper, The McKinsey axiom is not canonical). The problems, however, were unsolved for years in modal logic. In the beginning of 1990, Goldblatt showed that KM is not canonical in The McKinsey axiom is not canonical. The remaining task is to solve the problem of the compactness of KM. In this paper we present a proof showing that the KM system is not compact.The symbols of the language of propositional modal logic are as follows:1. A denumerably infinite set of sentence letters, for example, {p0, P1, p2, …};2. The Boolean connectives &, ⋁, ¬, →, ↔ and parentheses;3. The modal operators L and M where M is defined as ¬L¬.The formation rules of well-formed propositional modal formulae are the formation rules of formulae in classic propositional logic plus the following rule:If A is a well-formed formula, so is LA.A normal modal system is a set of formulae that contains all tautologies and the formulaand is closed under the following transformation rules:Uniform substitution. If A is a theorem, so is every substitution-instance of A.Modus ponens. If A and A → B are theorems, so is B.Necessitation. If A is a theorem, so is LA.Let L be a normal system. Then a set S of formulae is L-consistent if and only if for any formula B, which is the conjunction of some formulae in S, B is not included in L. A set S of formulae is maximal if and only if for every formula A, S either contains A or contains ¬A. A set S of formulae is maximal L-consistent if and only if it is both maximal and L-consistent.
Natural internal forcing schemata extending ZFC: Truth in the universe?
Authors: Melles Garvin 
Published: May 2006

Mathematicians have one over on the physicists in that they already have a unified theory of mathematics, namely, set theory. Unfortunately, the plethora of independence results since the invention of forcing has taken away some of the luster of set theory in the eyes of many mathematicians. Will man's knowledge of mathematical truth be forever limited to those theorems derivable from the standard axioms of set theory, ZFC? This author does not think so, he feels that set theorists' intuition about the universe of sets is stronger than ZFC. Here in this paper, using part of this intuition, we introduce some axiom schemata which we feel are very natural candidates for being considered as part of the axioms of set theory. These schemata assert the existence of many generics over simple inner models. The main purpose of this article is to present arguments for why the assertion of the existence of such generics belongs to the axioms of set theory.Our central guiding principle in justifying the axioms is what Maddy called the rule of thumb maximize in her survey article on the axioms of set theory, [8] and [9]. More specifically, our intuition conforms with that expressed by Mathias in his articleWhat is Maclane Missing? challenging Mac Lane's view of set theory.
About some symmetries of negation
Authors: Hösli Brigitte , Jäger Gerhard 
Published: May 2006

AbstractThis paper deals with some structural properties of the sequent calculus and describes strong symmetries between cut-free derivations and derivations, which do not make use of identity axioms. Both of them are discussed from a semantic and syntactic point of view.Identity axioms and cuts are closely related to the treatment of negation in the sequent calculus, so the results of this article explain some nice symmetries of negation.
A class of connected theories of order
Authors: Stern Alan S. , Świerczkowski Stanisław S. 
Published: May 2006

In 1962 Jan Mycielski proposed a very general notion of interpretability [M1]. This led to the question whether a given theory could be interpreted in the disjoint union of two theories, without being interpretable in any of them. He argued that in such a case it would be presumably simpler to study each of these theories separately, and hence conjectured that this situation can never occur for any of the well-known theories of mathematics. This conjecture has now been verified for the following theories (see [MPS], [P], [S1, 2]): ELO (endless, i.e., without maximal element, linear order), Th(〈ℚ, ≤〉), Th(〈ω, ≤〉) and all sequential theories (those which can code finite sequences of elements of their models). The latter include PA, ZF, GB and Th(〈ω,+,·〉). In view of these confirmations it became ever more plausible that the conjecture is valid also for RCF (real closed fields), i.e., for Th(〈ℝ,≤,+,·,0,1〉). In the present paper we show that Mycielski's conjecture is valid for a class of theories which includes RCF and OF (ordered fields).We consider only theories with equality and without function symbols. Interpretations will be meant local, multidimensional, and with parameters, as defined in [M1], [M2] and surveyed in [MPS] (for a recent definition see also [S2]). We shall write T0 ≪ T1 to say that T0 is interpretable in T1 (or that T1 interprets T0), and this will mean that for every theorem α of T0 there is an interpretation of α in T1.
A proof-theoretic characterization of the primitive recursive set functions
Authors: Rathjen Michael 
Published: May 2006

AbstractLet KP− be the theory resulting from Kripke-Platek set theory by restricting Foundation to Set Foundation. Let G: V → V (V ≔ universe of sets) be a Δ0-definable set function, i.e. there is a Δ0-formula φ(x, y) such that φ(x, G(x)) is true for all sets x, and V ⊨ ∀x∃!yφ(x, y). In this paper we shall verify (by elementary proof-theoretic methods) that the collection of set functions primitive recursive in G coincides with the collection of those functions which are Σ1-definable in KP− + Σ1-Foundation + ∀x∃!yφ(x, y). Moreover, we show that this is still true if one adds Π1-Foundation or a weak version of Δ0-Dependent Choices to the latter theory.
Generalized Fitting subgroup of a group of finite Morley rank
Authors: Nesin Ali 
Published: May 2006

AbstractWe define a characteristic and definable subgroup F*(G) of any group G of finite Morley rank that behaves very much like the generalized Fitting subgroup of a finite group. We also prove that semisimple subnormal subgroups of G are all definable and that there are finitely many of them.
The order types of termination orderings on monadic terms, strings and multisets
Authors: Martin Ursula , Scott Elizabeth 
Published: May 2006

AbstractWe consider total well-founded orderings on monadic terms satisfying the replacement and full invariance properties. We show that any such ordering on monadic terms in one variable and two unary function symbols must have order typeω,ω2orωω. We show that a familiar construction gives rise to continuum many such orderings of order typeω. We construct a new family of such orderings of order typeω2, and show that there are continuum many of these. We show that there are only four such orderings of order typeωω, the two familiar recursive path orderings and two closely related orderings. We consider also total well-founded orderings onNnwhich are preserved under vector addition. We show that any such ordering must have order typeωkfor some 1 ≤k≤n. We show that ifk<nthere are continuum many such orderings, and ifk=nthere are onlyn!, then! lexicographic orderings.