bluuebunny/crossref_metadata_2025_split huggingface dataset:
Recursively presented Abelian groups: Effective p-Group theory. I
Authors: Lin Charlotte 
Published: May 2006

The study of effectiveness in classical mathematics is rapidly expanding, through recent research in algebra, topology, model theory, and functional analysis. Well-known contributors are Barwise (Wisconsin), Crossley (Monash), Dekker (Rutgers), Ershoff (Novosibirsk), Feferman (Stanford), Harrington (Berkeley), Mal′cev (Novosibirsk), Morley (Cornell), Nerode (Cornell), Rabin (Hebrew University), Shore (Cornell). Further interesting work is due to Kalantari (University of California, Santa Barbara), Metakides (Rochester), Millar (Wisconsin), Remmel (University of California, San Diego), Nurtazin (Novosibirsk). Areas investigated include enumerated algebras, models of complete theories, vector spaces, fields, orderings, Hilbert spaces, and boolean algebras.We investigate the effective content of the structure theory of p-groups. Recall that a p-group is a torsion abelian group in which the (finite) order of each element is some power of a fixed prime p. (In the sequel, “group” = “additively written abelian group”.)The structure theory of p-groups is based on the two elementary notions of order and height. Recall that the order of x is the least integer n such that nx = 0. The height of x is the number of times p divides x, that is, the least n such that x = pny for some y in the group but x ≠ pn+1y for any y. If for each n ∈ ω there is a “pnth-root” yn, so that x = pnyn, then we say that x has infinite height. In 1923, Prüfer related the two notions as criteria for direct sum decomposition, provingTheorem. Every group of bounded order is a direct sum of cyclic groups, andTheorem. Every countable primary group with no (nonzero) elements of infinite height is a direct sum of cyclic groups.
A recursive model for arithmetic with weak induction
Authors: Adamowicz Zofia , Morales-Luna Guillermo 
Published: May 2006

Before stating the results we would like to thank the referee for reorganizing the whole paper and changing its original logical terminology to an algebraical terminology.Let Z* be a proper elementary extension of the integral domain Z. For any b ∈ Z*∖Z we define Z〈b〉 as the following subring of Z*:It is easy to prove that the natural divisibility relation on Z〈b〉 is the restriction to Z〈b〉 of the divisibility on Z* (see Lemma 1.1 below). The much stronger statement that Z〈b〉 is algebraically closed in Z* does not hold for all b. Similarly, there are many b's such that Z〈b〉 is not a recursive ring, e.g. if the set of standard primes dividing b is not recursive then Z〈b〉 cannot be a recursive ring under any bijection with N.Our main result is the following.Theorem. There exists an element b ∈ Z*∖Z such that(1) Z〈b〉 is algebraically closed in Z*,(2) Z〈b〉 is a recursive ring, and(3) each infinite element of Z〈b〉 is divisible by a standard integer > 1 (so the only “primes” of Z〈b〉 are the standard ones).An easy consequence of (1) is that certain induction axioms are satisfied by Z〈b〉. This explains the title of the paper. In the last section of the paper, §3, we shall say more about this.
Elementary equivalence of Cσ(K) spaces for totally disconnected, compact Hausdorff K
Authors: Heinrich S. , Henson C. Ward , Moore L. C. 
Published: May 2006

This paper is a continuation of the authors' paper [7]; in particular, we give a sharper and more useful criterion for the approximate elementary equivalence of Cσ(K) spaces, where K is a totally disconnected compact Hausdorff space. (See Theorem 2 below.) As an application, we obtain a complete description of the Banach spaces X which are approximately equivalent to c0. Namely, X ≡ Ac0 iff X = Cσ(K) where K is a totally disconnected compact Hausdorff space which has a dense set of isolated points and σ is an involutory homeomorphism on K which has a unique fixed point t, and t is not an isolated point. (See Theorem 7.)These results are derived from an analysis which we give of the elementary theories of structures (B, σ), where B is a Boolean algebra and σ is an involutory automorphism of B which leaves at most one nontrivial ultrafilter invariant. Let U(σ) denote this ultrafilter, if it exists; let U(σ) = B in case σ leaves no nontrivial ultrafilter invariant. We show that the elementary theory of (B, σ) is completely determined by the theory of (B, U(σ)) (and conversely, because U(σ) is definable in (B, σ)). This makes possible the use of the explicit invariants given by Éršov [4] for structures (B, U) where U is an ultrafilter on B. (These generalize the Tarski invariants for Boolean algebras [15].) We also use the Éršov invariants in the proof of our main result.
Finest partitions for ultrafilters
Authors: Kanamori Akihiro 
Published: May 2006

If a uniform ultrafilter U over an uncountable cardinal κ is not outright countably complete, probably the next best thing is that it have a finest partition: a master function f:κ → ω with ƒ−({n}) ∉ U for each n ϵ ω such that for any g: κ → κ, either (a) it is one-to-one on a set in U, or (b) it factors through ƒ (mod U), i.e. for some function h, {α < κ ∣ h(f(α)) = g(α)} ϵ U. In this paper, it is shown that recent contructions of irregular ultrafilters over ω1 can be amplified to incorporate a finest partition.Henceforth, let us assume that all ultrafilters are uniform.There has been an extensive study of substantial hypotheses, which are nonetheless weaker than countable completeness, on ultrafilters over uncountable cardinals. To survey some results and to establish a context, let us first recall the Rudin-Keisler (RK) ordering on ultrafilters: If Ui is an ultrafilter over Iii for i = 1, 2, then U1 ≤RKU2 iff there is a projecting function Ψ:I2 → I1 such that U1 = Ψ*(U2) = {X ⊆, I1∣ Ψ−1(X) ϵ U2}· U1, =RKU2 iff U1, ≤RK and U2 and U2≤RKU1; and U1<RKU2 iff U1≤RKU2 yet U1 ≠RKU2. In terms of this ordering, if an ultrafilter U has a finest partition ƒ, then ƒ*(U) over ω is maximum amongst all RK predecessors of U: for any g:κ → κ, if g*(U) <RKU, then g is not one-to-one on a set in U, so since g factors through ƒ with some h,g*(U) = h*(ƒ*(U)). Say now that an ultrafilter U over κ > ω is indecomposable iff whenever ω < λ < κ, there is no V ≤RKU such that V is a (uniform) ultrafilter over λ.
The order indiscernibles of divisible ordered abelian groups
Authors: Rosenthal David 
Published: May 2006

There has been much work in developing the interconnections between model theory and algebra. Here we look at a particular example, the divisible ordered abelian groups, and show how the indiscernibles are related to the algebraic structure. Now a divisible ordered abelian group is a model of Th (Q, +, 0, <) and so is linearly ordered by <. Thus the theory is unstable and has a large number of models. It is therefore unrealistic to expect that a simple condition will completely determine a model. Instead we would just like to obtain nice algebraic invariants.Definition. A subset C of a divisible ordered abelian group is a set of (order) indiscernibles iff for every sequence of integers n1,…,nk and for every c1 < … < ck and d1 < … < dk in CNote that this simplified form of indiscernibility is an immediate consequence of quantifier elimination for the theory. The above definition could be formulated in the language of +, 0, < but we have used subtraction as a matter of convenience. Similarly we may also use rational coefficients. Also note that a set of order indiscernibles is usually defined with respect to some external order. But in this case there are only two possibilities: the ordering inherited from or its reverse. So we will always assume that a set of order indiscernibles has the ordering inherited from . We may sometimes refer to a set of indiscernibles as a sequence of indiscernibles if we want to explicitly mention the ordering associated with the set.
Self-reference with negative types
Authors: Hiller A. P. , Zimbarg J. 
Published: May 2006

The universe of sets, V, is usually seen as an entity structured in successive levels, each level being made up of objects and collections of objects belonging to the previous levels. This process of obtaining sets and axioms for set theory can be seen in Scott [74] and Shoenfield [77].The approach we want to take differs from the previous one very strongly: the seeds from which we want to generate our universe of classes are to be the one-variable predicates (given by one-free-variable formulas) of the formal language we shall be using. In other words, any one-variable predicate of the language is to be represented as a class in our universe. In this sense, we view our theory as being about a self-referential language, a language whose predicates refer to objects which are predicates of the language itself.We want, in short, a system such that: (i) any predicate may be represented by an object to be studied by the theory itself; (ii) the axioms for the theory may be derived from the general principle that we are dealing with a language that aims at describing its own predicates; and (iii) the theory should be strong enough to derive ZFC and suggest answers to the existence of large cardinals and to the continuum hypothesis.An objection to such a project arises immediately: in view of the Russell-Zermelo paradox, how is it possible to have all predicates of the language as elements of the universe? This objection will be easy to deal with: we shall provide our language with a type structure to avoid paradox.
Idempotent ideals on abelian groups
Authors: Pelc Andrzej 
Published: May 2006

AbstractAn ideal I defined on a group G is called idempotent if for every A ∈ I, {g ∈ G:Ag−1 ∉ ∈ I} ∈ I. We show that a countably complete idempotent ideal on an abelian group cannot be prime but may have strong saturation properties.
Intuitionistic fuzzy logic and intuitionistic fuzzy set theory
Authors: Takeuti Gaisi , Titani Satoko 
Published: May 2006

In 1965 Zadeh introduced the concept of fuzzy sets. The characteristic of fuzzy sets is that the range of truth value of the membership relation is the closed interval [0, 1] of real numbers. The logical operations ⊃, ∼ on [0, 1] which are used for Zadeh's fuzzy sets seem to be Łukasiewciz's logic, where p ⊃ q = min(1, 1 − p + q), ∼ p = 1 − p. L. S. Hay extended in [4] Łukasiewicz's logic to a predicate logic and proved its weak completeness theorem: if P is valid then P + Pn is provable for each positive integer n. She also showed that one can without losing consistency obtain completeness of the system by use of additional infinitary rule.Now, from a logical standpoint, each logic has its corresponding set theory in which each logical operation is translated into a basic operation for set theory; namely, the relation ⊆ and = on sets are translation of the logical operations → and ↔. For Łukasiewicz's logic, P Λ (P ⊃ Q). ⊃ Q is not valid. Translating it to the set version, it follows that the axiom of extensionality does not hold. Thus this very basic principle of set theory is not valid in the corresponding set theory.
A generalization of Tennenbaum's theorem on effectively finite recursive linear orderings
Authors: Watnick Richard 
Published: May 2006

An effective translation of the fact that any infinite ordered set contains an infinite ascending or descending sequence is that any infinite recursive set A ⊆ Q has a recursive subset with order type ω or ω*. Tennenbaum's theorem states that this translation is false, and there is a counterexample A with order type ω + ω*. Tennenbaum suggested that this counterexample is an infinite recursive linearly ordered set which is effectively finite, and that the collection of all such counterexamples could provide a concrete model of nonstandard arithmetic. The purpose of this paper is to determine the collection of order types for which there is a counterexample.It is readily seen that any counterexample to the effective translation must have order type ω + Z · α + ω* for some α [2], [3]. Let be the collection of order types α for which there is a counterexample. As a test case, we have previously shown that contains the constructive scattered orderings [3], [4]. In this paper we determine exactly which order types are in . We easily show that if ω+ Z · α + ω* is recursive, then α is . The main result is that . Consequently, .
Ordinal definability in Jensen's model
Authors: Zadrożny Włodzimierz 
Published: May 2006

In [J] R. Jensen proved that any model of ZFC can be generically extended to a model of ZFC + (∃a ⊆ ω)(V = L[a]). This extension was made via a class P of forcing conditions. One can ask what can be said about the class HOD in such extensions. In particular one can ask whether V = HOD holds in Jensen's model. A full answer to this question is given by the following assertion:Theorem. Let N = L[a] be given by forcing with Jensen's conditions (the classP).Then:1. N ⊨ V ≠ HOD,2. N ⊨ (∀n < ω)(HODn + 1 ⊈ HODω).3. N ⊨ (∀α ≥ ω)(HODα = HODω), or equivalently HODω + 1 = HODω.The result is established by investigating homogeneity properties of certain complete Boolean algebras that are associated with the partial orderings of forcing conditions for coding using almost disjoint sets.We explain shortly the first of our two technical results: It is rather well known that if we have a subset then by performing threefold almost disjoint coding we come to a set a0 ⊆ ω such that ; cf. [JS]. This is achieved by iterating over almost disjoint forcing . That means we first code by a subset , of [ω1ω2) using the almost disjoint forcing conditions . Then we code by aω, contained in [ω, ω1), using forcing with over .
Effective discontinuity and a characterisation of the superjump
Authors: Hartley John P. 
Published: May 2006

One of the most fundamental results of higher-type recursion theory was first proved by Thomas Grilliot [3]. He showed that the functional embodying number quantification, 2E, is computable from a type-2 object F if and only if F has an effective discontinuity. For the background to this work we recommend Normann's book [9] or his summary [10], and for an example of its many applications Normann-Wainer [11].The purpose of §1 of this paper is to give an effective proof of the “only if” part of this result: i.e. a construction of a discontinuity of F from the information 2E = {e}F. This contrasts with Grilliot's argument which is topological and nonconstructive, and shows that “Grilliot's trick” for computing 2E from a discontinuity of F is essentially the only method.We then apply the method to two more general situations in which it gives new results. In §2 we treat type-3 objects in which 3E is computable (assuming AC and CH), and obtain a strengthening of the result of Hartley [5, §5]. In §3 we consider the superjump and characterise it as the least among a class of functionals with a particular sort of family of discontinuities.
More on proper forcing
Authors: Shelah Saharon 
Published: May 2006

§1. A counterexample and preservation of “proper + X”.Theorem. Suppose V satisfies, , and for some A ⊆ ω1, every B ⊆ ω1, belongs to L[A].Then we can define a countable support iterationsuch that the following conditions hold:a) EachQiis proper and ⊩Pi “Qi, has power ℵ1”.b) Each Qi is -complete for some simple ℵ1-completeness system.c) Forcing with Pα = Lim adds reals.Proof. We shall define Qi by induction on i so that conditions a) and b) are satisfied, and Ci, is a Qi-name of a closed unbounded subset of ω1. Let : ξ < ω1› ∈ L[A] be a list of all functions f which are from δ to δ for some δ < ω1 and let h: ω1 → ω1, h ∈ L[A], be defined by h(α) = Min{β: β > α and Lβ[A]⊨ “∣α∣ = ℵ0”}.Suppose we have defined Qj for every j < i; then Pi is defined, is proper (as each Qj, j < i, is proper, and by III 3.2) and has a dense subset of power ℵ (by III 4.1). Let Gi ⊆ Pi be generic so clearly there is B ⊆ ω1, such that in V[Gi] every subset of ω1 belongs to L[A, B], The following now follows:Fact. In V[Gi], every countableN ⥽(H(ℵ2), ∈, A, B) is isomorphic toLβ[A ∩ δ, B ∩ δ] for some β < h(δ), where δ = δ(N) = ω1, ∩ N.
Logics containing K4. Part II
Authors: Fine Kit 
Published: May 2006

This paper establishes another very general completeness result for the logics within the field of K4. With each finite transitive frame ℭ we may associate a formula — Bℭ which validates just those frames ℑ in which ℭ is not in a certain sense embeddable (to be exact, ℭ is not the p-morphic image of any subframe of ℑ. By a subframe logic we mean the result of adding such formulas as axioms to K4. The general result is that each subframe logic has the finite model property.There are a continuum of subframe logics and they include many of the standard ones, such as T, S4, S4.3, S5 and G. It turns out that the subframe logics are exactly those complete for a condition that is closed under subframes (any subframe of a frame satisfying the condition also satisfies the condition). As a consequence, every logic complete for a condition closed under subframes has the finite model property.It is ascertained which of the subframe logics are compact. It turns out that the compact logics are just those whose axioms express an elementary condition. Tests are given for determining whether a given axiom expresses an elementary condition and for determining what it is in case it does.In one respect the present general completeness result differs from most of the others in the literature. The others have usually either been what one might call logic based or formula based. They have usually either been to the effect that all of the logics containing a given logic are complete or to the effect that all logics whose axioms come from a given syntactically characterized class of formulas are complete. The present result is, by contrast, what one might call frame based. The axioms of the logics to be proved complete are characterized most directly in terms of their connection with certain frames.
A simple relationship between Buchholz's new system of ordinal notations and Takeuti's system of ordinal diagrams
Authors: Okada Mitsuhiro 
Published: May 2006

Buchholz [4] simplified the system of ordinal notations of the Schütte school (cf. [12]), by using the notion of collapsing functions (cf. [5]). In this paper we give a simple relationship between Buchholz's new system of ordinal notations and Takeuti's system of ordinal diagrams. From this simple relationship it turns out that the structures of these two systems are very close.We give two systems OT(I) (§1) and OT(I, A) (§2) of ordinal notations which are considered generalizations of Buchholz's original system, where I and A are well-ordered sets. The original system OT of Buchholz [4] is OT(ω + 1, {0}) in our sense. Here the set OT(I) of ordinal notations is defined as a subset of the set Od(I) of ordinal diagrams in [6], and the set OT(I, A) of ordinal notations as a subset of the set O(I, A) of ordinal diagrams in [14].
Term models for weak set theories with a universal set
Authors: Forster T. E. 
Published: May 2006

We shall be concerned here with weak axiomatic systems of set theory with a universal set. The language in which they are expressed is that of set theory—two primitive predicates, = and ϵ, and no function symbols (though some function symbols will be introduced by definitional abbreviation). All the theories will have stratified axioms only, and they will all have Ext (extensionality: (∀x)(∀y)(x = y· ↔ ·(∀z)(z ϵ x ↔ z ϵ y))). In fact, in addition to extensionality, they have only axioms saying that the universe is closed under certain set-theoretic operations, viz. all of the formand these will always include singleton, i.e., ι′x exists if x does (the iota notation for singleton, due to Russell and Whitehead, is used here to avoid confusion with {x: Φ}, set abstraction), and also x ∪ y, x ∩ y and − x (the complement of x). The system with these axioms is called NF2 in the literature (see [F]). The other axioms we consider will be those giving ⋃x, ⋂x, {y: y ⊆x} and {y: x ⊆ y}. We will frequently have occasion to bear in mind that 〈 V, ⊆ 〉 is a Boolean algebra in any theory extending NF2. There is no use of the axiom of choice at any point in this paper. Since the systems with which we will be concerned exhibit this feature of having, in addition to extensionality, only axioms stating that V is closed under certain operations, we will be very interested in terms of the theories in question. A T-term, for T such a theory, is a thing (with no free variables) built up from V or ∧ by means of the T-operations, which are of course the operations that the axioms of T say the universe is closed under.
Reduced coproducts of compact Hausdorff spaces
Authors: Bankston Paul 
Published: May 2006

AbstractBy analyzing how one obtains the Stone space of the reduced product of an indexed collection of Boolean algebras from the Stone spaces of those algebras, we derive a topological construction, the “reduced coproduct”, which makes sense for indexed collections of arbitrary Tichonov spaces. When the filter in question is an ultrafilter, we show how the “ultracoproduct” can be obtained from the usual topological ultraproduct via a compactification process in the style of Wallman and Frink. We prove theorems dealing with the topological structure of reduced coproducts (especially ultracoproducts) and show in addition how one may use this construction to gain information about the category of compact Hausdorff spaces.
Equalization of finite flowers
Authors: Berardi Stefano 
Published: May 2006

A dilator D is a functor from ON to itself commuting with direct limits and pull-backs. A dilator D is a flower iff D(x) is continuous. A flower F is regular iff F(x) is strictly increasing and F(f)(F(z)) = F(f(z)) (for f ϵ ON(x,y), z ϵ X).Equalization is the following axiom: if F, G ϵ Flr (class of regular flowers), then there is an H ϵ Flr such that F ° H = G ° H. From this we can deduce that if ℱ is a set ⊆ Flr, then there is an H ϵ Flr which is the smallest equalizer of ℱ (it can be said that H equalizes ℱ iff for every F, G ϵ ℱ we have F ° H = G ° H). Equalization is not provable in set theory because equalization for denumerable flowers is equivalent to -determinacy (see a forthcoming paper by Girard and Kechris).Therefore it is interesting to effectively find, by elementary means, equalizers even in the simplest cases. The aim of this paper is to prove Girard and Kechris's conjecture: “ is the (smallest) equalizer for Flr < ω” (where Flr < ω denotes the set of finite regular flowers). We will verify that is an equalizer of Flr < ω; we will sketch the proof that it is the smallest one at the end of the paper. We will denote by H.
Vaught's conjecture for o-minimal theories
Authors: Mayer Laura L. 
Published: May 2006

The notion of o-minimality was formulated by Pillay and Steinhorn [PS] building on work of van den Dries [D]. Roughly speaking, a theory T is o-minimal if whenever M is a model of T then M is linearly ordered and every definable subset of the universe of M consists of finitely many intervals and points. The theory of real closed fields is an example of an o-minimal theory.We examine the structure of the countable models for T, T an arbitrary o-minimal theory (in a countable language). We completely characterize these models, provided that T does not have 2ω countable models. This proviso (viz. that T has fewer than 2ω countable models) is in the tradition of classification theory: given a cardinal α, if T has the maximum possible number of models of size α, i.e. 2α, then no structure theorem is expected (cf. [Sh1]).O-minimality is introduced in §1. §1 also contains conventions and definitions, including the definitions of cut and noncut. Cuts and noncuts constitute the nonisolated types over a set.In §2 we study a notion of independence for sets of nonisolated types and the corresponding notion of dimension.In §3 we define what it means for a nonisolated type to be simple. Such types generalize the so-called “components” in Pillay and Steinhorn's analysis of ω-categorical o-minimal theories [PS]. We show that if there is a nonisolated type which is not simple then T has 2ω countable models.