bluuebunny/crossref_metadata_2025_split huggingface dataset:
A partial functions version of Church's simple theory of types
Authors: Farmer William M. 
Published: May 2006

AbstractChurch's simple theory of types is a system of higher-order logic in which functions are assumed to be total. We present in this paper a version of Church's system called PF in which functions may be partial. The semantics of PF, which is based on Henkin's general-models semantics, allows terms to be nondenoting but requires formulas to always denote a standard truth value. We prove that PF is complete with respect to its semantics. The reasoning mechanism in PF for partial functions corresponds closely to mathematical practice, and the formulation of PF adheres tightly to the framework of Church's system.
Uncountable theories that are categorical in a higher power
Authors: Laskowski Michael Chris 
Published: May 2006

AbstractIn this paper we prove three theorems about first-order theories that are categorical in a higher power. The first theorem asserts that such a theory either is totally categorical or there exist prime and minimal models over arbitrary base sets. The second theorem shows that such theories have a natural notion of dimension that determines the models of the theory up to isomorphism. From this we conclude that I(T,ℵα,) = ℵ0 + ∣α∣ where ℵα = the number of formulas modulo T-equivalence provided that T is not totally categorical. The third theorem gives a new characterization of these theories.
Complete local rings as domains
Authors: Stoltenberg-Hansen V. , Tucker J. V. 
Published: May 2006

Contents: Introduction. §1: Computable rings and modules. §2: Ideal membership relation. §3: Effective structured domains. §4: Completion of a local ring as a domain. §5: The recursive completion. Epilogue. References.Introduction. Completion is an important general mathematical device. Often, but not always, a completion takes the following form. Let A be a topological algebraic structure whose topology is derived from a metric. For A, a topological algebra Â and an embedding i: A → Â are constructed such that Â is a complete metric space in which A is densely embedded by i. The long list of structures for which such completions exist begins with Cantor's construction of the real number field and includes objects like the p-adic integers, Baire space, and Boolean algebras. In Bourbaki [6] a careful and thorough account of completions for arbitrary topological groups and fields is given, for which it is important to note that the topological structures need not be metrizable, but must possess a uniformity.The effectiveness of the completion process of a computable structure A cannot be readily studied using the tools of computable algebra, simply because the resulting structure Â is almost invariably uncountable. However, in particular cases, it has been possible to define and study the substructure Ak of computable elements of Â; this has been done for the structures mentioned above, starting with the field
of recursive real numbers.In this paper we analyse the effectivity of the completion of a local ring R. We do this using structured Scott-Ershov domains. Our study may be considered as a prototype containing methods applicable to a broad class of completions, including all the examples mentioned above, except for the real number field, which needs a generalisation of the domain concept.A Scott-Ershov domain D formalises how a set Dt of possibly “infinite” elements, called total elements, is constructed from a set Dc of “finite” elements, called compact elements. This is achieved by means of an approximation ordering which determines a topology on D and, in particular, on Dt. Our methodology is to associate to a given topological algebra A a structured domain D(A) such that the total elements D(A)t form a topological algebra topologically isomorphic to A. In such circumstances A is said to be domain definable by D(A). The theory of computability for domains is now applied to study the effectivity of the topological algebra A.
On the elementary theory of restricted elementary functions
Authors: Dries Lou van den 
Published: May 2006

As a contribution to definability theory in the spirit of Tarski's classical work on (R, <, 0, 1, +, ·) we extend here part of his results to the structureHere exp ∣[0, 1] and sin ∣[0, π] are the restrictions of the exponential and sine function to the closed intervals indicated; formally we identify these restricted functions with their graphs and regard these as binary relations on R. The superscript “RE” stands for “restricted elementary” since, given any elementary function, one can in general only define certain restrictions of it in RRE.Let (RRE, constants) be the expansion of RRE obtained by adding a name for each real number to the language. We can now formulate our main result as follows.Theorem. (RRE, constants) is strongly model-complete.This means that every formula ϕ(X1, …, Xm) in the natural language of (RRE, constants) is equivalent to an existential formulawith the extra property that for each x ∈ Rm such that ϕ(x) is true in RRE there is exactly one y ∈ Rn such that ψ(x, y) is true in RRE. (Here ψ is quantifier free.)
A system of logic for partial functions under existence-dependent kleene equality
Authors: Andréka H. , Craig W. , Németi I. 
Published: May 2006

Ordinary equational logic is a connective-free fragment of first-order logic which is concerned with total functions under the relation of ordinary equality. In [AN] (see also [AN1]) and in [Cr] it has been extended in two equivalent ways into a near-equational system of logic for partial functions. The extension given in [Cr] deals with partial functions under two relationships: a relationship of existence-dependent existence and one of existence-dependent Kleene equality. For the language that involves both relationships a set of rules was given that is complete. Those rules in the set that involve only existence-dependent existence turned out to be complete for the sublanguage that involves this relationship only. In the present paper we give a set of rules that is complete for the other sublanguage, namely the language of partial functions under existence-dependent Kleene equality.This language lacks a certain, often needed, power of expressing existence and fails, in particular, to be an extension of the language that underlies ordinary equational logic. That it possesses a fairly simple complete set of rules is therefore perhaps more of theoretical than of practical interest. The present paper is thus intended to serve as a supplement to [Cr] and, less directly, to [AN]. The subject is further rounded out, and some contrast is provided, by [Rob]. The systems of logic treated there are based on the weaker language in which partial functions are considered under the more basic relation of Kleene equality.
A note on definable Skolem functions
Authors: Scowcroft Philip 
Published: May 2006

This note arose out of my efforts to understand results of van den Dries, Denef, and Weispfenning on definable Skolem functions in the elementary theory of Qp. The first person to prove their existence was van den Dries, who devised and applied a model-theoretic criterion for theories, admitting elimination of quantifiers, which also admit definable Skolem functions [3]. The proof, though elegant, does not describe how one defines the Skolem functions. In the particular case of Qp, Denef found an ingenious, easily described method for writing out the definitions [2, pp. 14–15]. Unfortunately, his argument directly applies only in the following special case: ifand there is a fixed m ≥ 1 such thatfor all , then can be given as a definable function of . While this special case includes many of interest, van den Dries' theorem seems more general. Weispfenning suggested how his results on primitive-recursive quantifier elimination could produce algorithms yielding definitions of Skolem functions in the specific theories van den Dries considered [10, pp. 470–471]. Though these algorithms provide a more concrete version of van den Dries' theorem, and do not suffer the lack of generality of Denef's result, Weispfenning's argument is extremely subtle and applies only to certain theories of valued fields.
On the use of inaccessible numbers and order indiscernibles in lower bound arguments for random access machines
Authors: Maass Wolfgang 
Published: May 2006

AbstractWe prove optimal lower bounds on the computation time for several well-known test problems on a quite realistic computational model: the random access machine. These lower bound arguments may be of special interest for logicians because they rely on finitary analogues of two important concepts from mathematical logic: inaccessible numbers and order indiscernibles.
On universal semiregular invariant measures
Authors: Zakrzewski Piotr 
Published: May 2006

AbstractWe consider countably additive, nonnegative, extended real-valued measures which vanish on singletons. Such a measure is universal on a set X iff it is defined on all subsets of X and is semiregular iff every set of positive measure contains a subset of positive finite measure. We study the problem of existence of a universal semiregular measure on X which is invariant under a given group of bijections of X. Moreover we discuss some properties of universal, semiregular, invariant measures on groups.
On the validity of hilbert's nullstellensatz, artin's theorem, and related results in grothendieck toposes
Authors: MacCaull W. A. 
Published: May 2006

Using formally intuitionistic logic coupled with infinitary logic and the completeness theorem for coherent logic, we establish the validity, in Grothendieck toposes, of a number of well-known, classically valid theorems about fields and ordered fields. Classically, these theorems have proofs by contradiction and most involve higher order notions. Here, the theorems are each given a first-order formulation, and this form of the theorem is then deduced using coherent or formally intuitionistic logic. This immediately implies their validity in arbitrary Grothendieck toposes. The main idea throughout is to use coherent theories and, whenever possible, find coherent formulations of formulas which then allow us to call upon the completeness theorem of coherent logic. In one place, the positive model-completeness of the relevant theory is used to find the necessary coherent formulas.The theorems here deal with polynomials or rational functions (in s indeterminates) over fields. A polynomial over a field can, of course, be represented by a finite string of field elements, and a rational function can be represented by a pair of strings of field elements. We chose the approach whereby results on polynomial rings are reduced to results about the base field, because the theory of polynomial rings in s indeterminates over fields, although coherent, is less desirable from a model-theoretic point of view. Ultimately we are interested in the models.This research was originally motivated by the works of Saracino and Weispfenning [SW], van den Dries [Dr], and Bunge [Bu], each of whom generalized some theorems from algebraic geometry or ordered fields to (commutative, von Neumann) regular rings (with unity).
Ultrapowers without the axiom of choice
Authors: Spector Mitchell 
Published: May 2006

AbstractA new method is presented for constructing models of set theory, using a technique of forming pseudo-ultrapowers. In the presence of the axiom of choice, the traditional ultrapower construction has proven to be extremely powerful in set theory and model theory; if the axiom of choice is not assumed, the fundamental theorem of ultrapowers may fail, causing the ultrapower to lose almost all of its utility. The pseudo-ultrapower is designed so that the fundamental theorem holds even if choice fails; this is arranged by means of an application of the omitting types theorem. The general theory of pseudo-ultrapowers is developed. Following that, we study supercompactness in the absence of choice, and we analyze pseudo-ultrapowers of models of the axiom of determinateness and various infinite exponent partition relations. Relationships between pseudo-ultrapowers and forcing are also discussed.
The cofinality of cardinal invariants related to measure and category
Authors: Bartoszynski Tomek , Ihoda Jaime I. , Shelah Saharon 
Published: May 2006

AbstractWe prove that the following are consistent with ZFC:1. 2ω = ℵω1 + #x039A;c = ℵω1 + ΚΒ = ΚU = ω2 (for measure and category simultaneously).2. .This concludes the discussion about the cofinality of Κc.
A Borel reductibility theory for classes of countable structures
Authors: Friedman Harvey , Stanley Lee 
Published: May 2006

AbstractWe introduce a reducibility preordering between classes of countable structures, each class containing only structures of a given similarity type (which is allowed to vary from class to class). Though we sometimes work in a slightly larger context, we are principally concerned with the case where each class is an invariant Borel class (i.e. the class of all models, with underlying set = ω, of an Lω1ω sentence; from this point of view, the reducibility can be thought of as a (rather weak) sort of Lω1ω-interpretability notion). We prove a number of general results about this notion, but our main thrust is to situate various mathematically natural classes with respect to the preordering, most notably classes of algebraic structures such as groups and fields.
Finite axiomatizability for equational theories of computable groupoids
Authors: Perkins Peter 
Published: May 2006

A computable groupoid is an algebra ‹N, g› where N is the set of natural numbers and g is a recursive (total) binary operation on N. A set L of natural numbers is a computable list of computable groupoids iff L is recursive, ‹N, ϕe› is a computable groupoid for each e ∈ L, and e ∈ L whenever e codes a primitive recursive description of a binary operation on N.Theorem 1. Let L be any computable list of computable groupoids. The set {e ∈ L: the equational theory of ‹N, ϕe› is finitely axiomatizable} is not recursive.Theorem 2. Let S be any recursive set of positive integers. A computable groupoid GS can be constructed so that S is inifinite iff GS has a finitely axiomatizable equational theory.The problem of deciding which finite algebras have finitely axiomatizable equational theories has remained open since it was first posed by Tarski in the early 1960's. Indeed, it is still not known whether the set of such finite algebras is recursively (or corecursively) enumerable. McKenzie [7] has shown that this question of finite axiomatizability for any (finite) algebra of finite type can be reduced to that for a (finite) groupoid.
The formal language of recursion
Authors: Moschovakis Yiannis N. 
Published: May 2006

This is the first of a sequence of papers in which we will develop a foundation for the theory of computation based on a precise, mathematical notion of abstract algorithm. To understand the aim of this program, one should keep in mind clearly the distinction between an algorithm and the object (typically a function) computed by that algorithm. The theory of computable functions (on the integers and on abstract structures) is obviously relevant to this work, but we will focus on making rigorous and identifying the mathematical properties of the finer (intensional) notion of algorithm.It is characteristic of this approach that we take recursion to be a fundamental (primitive) process for constructing algorithms, not a derived notion which must be reduced to others—e.g. iteration or application and abstraction, as in the classical λ-calculus. We will model algorithms by recursors, the set-theoretic objects one would naturally choose to represent (syntactically described) recursive definitions. Explicit and iterative algorithms are modelled by (appropriately degenerate) recursors.The main technical tool we will use is the formal language of recursion, FLR, a language of terms with two kinds of semantics: on each suitable structure, the denotation of a term t of FLR is a function, while the intension of t is a recursor (i.e. an algorithm) which computes the denotation of t. FLR is meant to be intensionally complete, in the sense that every (intuitively understood) “algorithm” should “be” (faithfully modelled, in all its essential properties by) the intension of some term of FLR on a suitably chosen structure.
A proof of morley's conjecture
Authors: Hart Bradd 
Published: May 2006

In the 1960's, it was conjectured that a complete first order theory in a countable language would have a nondecreasing spectrum on uncountable cardinals. This conjecture became known as Morley's conjecture. Shelah has proved this in [10]. The intent of this paper is to give a different proof which resembles a more naive way of approaching this theorem.Let I(T, λ) = the number of nonisomorphic models of T in cardinality λ. We prove:Theorem 0.1. If T is a complete countable first order theory then for ℵ0 < κ < λ, I(T,K) ≤ I(T, λ).In some sense, one can view Shelah's work on the classification of first order theories as an attack on Morley's conjecture. Over the years, he has shown that certain assumptions on a first order theory would lead to its having maximal spectrum in powers larger than the cardinality of its language (see §6 for precise references). At some point it must have seemed that Morley's conjecture would be a corollary to an exact calculation of all possible spectrums. In the end, this did not occur and, in fact, the exact spectrum functions are still not known (see [10]). Let us consider a naive approach to the proof.If we have two nonisomorphic models of the same cardinality and their cardinality is “large enough” then there should be some reason, irrespective of their cardinalities, which causes this nonisomorphism. If we could isolate this property and extend these models to a larger cardinality preserving this property, then the larger models would also be nonisomorphic. The notion of extendibility introduced in §2 is such a property which allows a version of this naive proof to work. Let us preview the sections.
The classification of excellent classes
Authors: Grossberg R. , Hart B. 
Published: May 2006

In [9] and [12], Shelah defined a certain type of Scott sentence which he called excellent. He proved, among other things, that if a Scott sentence is excellent and categorical in some uncountable power then it is categorical in all uncountable powers: the analog of the Morley categoricity theorem. Proving such an analog is often the starting point in the classification of a family of classes. Before beginning this classification in the case of excellent Scott sentences, let us say a few words about what this paper is and what it is not.It is not the beginning of a classification theory for complete sentences in where is countable. Although excellence arises in the study of the model theory of Scott sentences, it is not a dividing line in a classification of them. In particular, the assumption of nonexcellence does not yield much information. In fact, in [3] there is an example of a nonexcellent Scott sentence, categorical in ℵ1 which is. not fully categorical. It seems to the second author that a classification of sentences analogous to the classification of first order theories is a long way off and may not be accomplishable in ZFC.This is not to say that the study of excellent Scott sentences (or the class of models of such which we will call excellent classes) is unproductive. Besides its extreme usefulness in [12], Mekler and Shelah have shown that excellence plays a decisive role in the study of almost free algebras (see [7]). Moreover, as the class of ω-saturated models of an ω-stable theory is an example of an excellent class, the study of excellent classes is at least as difficult as the study of first order ω-stable theories.
The consistency problem for positive comprehension principles
Authors: Forti M. , Hinnion R. 
Published: May 2006

Since Gilmore showed that some theory with a positive comprehension scheme is consistent when the axiom of extensionality is dropped and inconsistent with it (see [1] and [2]), the problem of the consistency of various positive comprehension schemes has been investigated. We give here a short classification, which shows clearly the importance of the axiom of extensionality and of the abstraction operator in these consistency problems. The most difficult problem was to show the consistency of the comprehension scheme for positive formulas, with extensionality but without abstraction operator. In his unpublished thesis, Set theory in which the axiom of foundation fails [3], Malitz solved partially this problem but he needed to assume the existence of some unusual kind of large cardinal; as his original construction is very interesting and his thesis is unpublished, we give a short summary of it. M. Forti solved the problem completely by working in ZF with a free-construction principle (sometimes called an anti-foundation axiom), instead of ZF with the axiom of foundation, as Malitz did.This permits one to obtain the consistency of this positive theory, relative to ZF. In his general investigations about “topological set theories” (to be published), E. Weydert has independently proved the same result. The authors are grateful to the Mathematisches Forshungsinstitut Oberwolfach for giving them the opportunity of discussing these subjects and meeting E. Weydert during the meeting “New Foundations”, March 1–7, 1987.
Reflecting on incompleteness
Authors: Feferman Solomon 
Published: May 2006

To what extent can mathematical thought be analyzed in formal terms? Gödel's theorems show the inadequacy of single formal systems for this purpose, except in relatively restricted parts of mathematics. However at the same time they point to the possibility of systematically generating larger and larger systems whose acceptability is implicit in acceptance of the starting theory. The engines for that purpose are what have come to be called reflection principles. These may be iterated into the constructive transfinite, leading to what are called recursive progressions of theories. A number of informative technical results have been obtained about such progressions (cf. Feferman [1962], [1964], [1968] and Kreisel [1958], [1970]). However, for some years I had hoped to give a more realistic and perspicuous finite generation procedure. This was first done in a rather special way in Feferman [1979] for the characterization of predicativity, which may be regarded as that part of mathematical thought implicit in our acceptance of elementary number theory. What is presented here is a new and simple notion of the reflective closure of a schematic theory which can be applied quite generally.Two examples of schematic theories in the sense used here are versions of Peano arithmetic and Zermelo set theory.