bluuebunny/crossref_metadata_2025_split huggingface dataset:
A system of abstract constructive ordinals
Authors: Howard W. A. 
Published: May 2006

As Gödel [6] has pointed out, there is a certain interchangeability between the intuitionistic notion of proof and the notion of constructive functional of finite type. He achieves this interchange in the direction from logic to functionals by his functional interpretation of Heyting arithmetic H in a free variable theory T of primitive recursive functionals of finite type. In the present paper we shall extend Gödel's functional interpretation to the case in which H and T are extended by adding an abstract notion of constructive ordinal. In other words, we obtain the Gödel functional interpretation of an intuitionistic theory U of numbers (i.e., nonnegative integers) and constructive ordinals in a free variable theory V of finite type over both numbers and constructive ordinals. This allows us to obtain an analysis of noniterated positive inductive definitions [8].The notion of constructive ordinal to be treated is as follows. There is given a function J which embeds the nonnegative integers in the constructive ordinals. A constructive ordinal of the form Jn is said to be minimal. There is also given a function δ which associates to each constructive ordinal Z and number n a constructive ordinal δZn which we denote by Zn. When Z is nonminimal, each Zn is called an immediate predecessor of Z. The basic principle for forming constructive ordinals says: for every function f from numbers n to constructive ordinals, there exists a constructive ordinal Z such that Zn = fn for all n. The principle of transfinite induction with respect to constructive ordinals says: if a property Q(Z) of constructive ordinals Z holds for minimal Z, and if ∀nQ(Zn) → Q(Z) holds for all Z, then Q(Z) holds for all Z.
On the possibility of a Σ21well-ordering of the Baire space
Authors: Mansfield Richard 
Published: May 2006

It is well known that the hypothesis that all real numbers are constructible in the sense of Gödel [1] implies the existence of a Σ21well-ordering of the Baire space [1, p. 67]. We are concerned with the converse to this theorem. From the assumption of the existence of a Σ21well-ordering with total domain, we derive various consequences which in the presence of a nonconstructible real seem highly pathological. However, while several of these consequences are obviously absurd, none have as yet been disproven. Indeed some of the stranger consistency proofs of Jensen seem to indicate that there is a possibility that they may be consistent. I am referring to such results as existence of a model for ZF set theory in which the degrees of constructibility form a sequence of typeω+ 1 with the (ω+ 1)st degree being the only one which contains a nonconstructible real, and that degree being the degree of a Δ31nonconstructible real.In what follows we shall use the small Greek lettersα, β, γto range over the Baire spaceNNwhereNis the set of natural numbers. We assume that the reader is familiar with the use of trees to encode closed subsets of this space. The class Σ21is the collection of all those subsets of the Baire space which can be defined by a formula which is Σ21in a constructible parameter. Correspondingly Π21will be taken to mean Π11in a constructible parameter. The proof of the first lemma is left as an exercise. It involves nothing more than coding perfect sets by trees and then counting quantifiers.
Encodability of Kleene's O
Authors: Jockusch Carl G. , Soare Robert I. 
Published: May 2006

Let ω be the nonnegative integers. G. E. Sacks once asked whether there exists an infinite X ⊆ ω such that, for all Y ⊆ X, ω1Yω1 where ω1 is the first nonrecursive ordinal. In this note we negatively answer this question by giving a simple proof that for every infinite set X ⊆ ω there exists Y ⊆ X such that the first recursively inaccessible ordinal. This is accomplished by proving that Hα is hyper-arithmetic in Y whereHα is the αth hyperjump of the empty set ∅, defined in a suitable sense for all ordinals Background information and undefined notation can be found in Rogers [11]. In particular, we write A ≤hB(A ≤T B) if A is hyperarithmetical (recursive) in B, and A ≡hB if A ≤hB and B ≤hA. We will say that a set A is hyperarithmetically (recursively) encodable if, for every infinite set X ⊆ ω, there exists Y ⊆ X such that A ≤hY (A ≤TY). For any set A (hyperdegree a) let A′ (a′) denote the hyperjump of A (a). Let 0 denote the hyperdegree of ∅. A function f majorizes a function g if f(n) ≥ g(n) for every n. E1 is the representing (type-2) functional ofintroduced by Tugué [13] (also Kleene [6]). Let be the smallest ordinal which is not the order type of any well-ordering recursive in E1. Information on can be found in Richter [9] and [10].
Hanf number of omitting type for simple first-order theories
Authors: Shelah Saharon 
Published: May 2006

AbstractLet T be a complete countable first-order theory such that every ultrapower of a model of T is saturated. If T has a model omitting a type p in every cardinality < ℶ′, then T has a model omitting p in every cardinality. There is also a related theorem, and an example showing the ℶ′ cannot be improved.
Constructive assertions in an extension of classical mathematics
Authors: Lifschitz Vladimir 
Published: May 2006

We distinguish between two kinds of mathematical assertions: objective and constructive. An objective assertion describes the universe of mathematical objects; a constructive one describes the (idealized) mathematician's ability to find mathematical objects with various properties. The familiar formalizations of classical mathematics are based on formal languages designed for expressing objective assertions only. The constructivist program stresses, on the contrary, the importance of constructive assertions; moreover, intuitionism claims that constructive activities of the mind constitute the very subject matter of mathematics, and thus questions the semantic status of objective assertions.The purpose of this paper is to show that classical mathematics can be extended to include constructive sentences, so that both objective and constructive properties can be discussed in the framework of the same theory. To achieve this goal, we introduce a new property of mathematical objects, calculability.The word “calculable” may be applied to objects of various types: natural numbers, integers, rational or real numbers, polynomials with rational or real coefficients, etc. In each case it has a different meaning, so that actually we define not one, but many new properties.
Decidability and finite axiomatizability of theories of ℵ0-categorical partially ordered sets
Authors: Schmerl James H. 
Published: May 2006

AbstractEvery ℵ0-categorical partially ordered set of finite width has a finitely axiomatizable theory. Every ℵ0-categorical partially ordered set of finite weak width has a decidable theory. This last statement constitutes a major portion of the complete (with three exceptions) characterization of those finite partially ordered sets for which any ℵ0-categorical partially ordered set not embedding one of them has a decidable theory.
Recursive constructions in topological spaces
Authors: Kalantari Iraj , Retzlaff Allen 
Published: May 2006

AbstractWe study topological constructions in the recursion theoretic framework of the lattice of recursively enumerable open subsets of a topological spaceX. Various constructions produce complemented recursively enumerable open sets with additional recursion theoretic properties, as well as noncomplemented open sets. In contrast to techniques in classical topology, we construct a disjoint recursively enumerable collection of basic open sets which cannot be extended to a recursively enumerable disjoint collection of basic open sets whose union is dense inX.
Choice principles, the bar rule and autonomously iterated comprehension schemes in analysis
Authors: Feferman S. , Jäger G. 
Published: May 2006

In [10] Friedman showed that (-AC) is a conservative extension of (-CA)<ε0for-sentences wherei= min(n+ 2, 4), i.e.,i= 2, 3, 4 forn= 0, 1, 2 +m. Feferman [5], [7] and Tait [11], [12] reobtained this result forn= 0, 1 and even with (-DC) instead of (-AC). Feferman and Sieg established in [9] the conservativeness of (-DC) over (-CA)<ε0for-sentences (ias above) for alln. In each paper, different methods of proof have been used. In particular, Feferman and Sieg showed how to apply familiar proof-theoretical techniques by passing through languages with Skolem functionals.In this paper we study the same choice principles in the presence of theBar Rule(BR), which permits one to infer the scheme of transfinite induction on a primitive recursive relation ≺ when it has been proved that ≺ is wellfounded. The main result (Theorem 1 below) characterizes (-DC) + (BR) as a conservative extension of a system of the autonomously iterated-comprehension axiom for-sentences (idepending onnas above). Forn= 0 this has been proved by Feferman in the form that (-DC) + (BR) is a conservative extension of (-CA)<Γ0; this was first done in [8] by use of the Gödel functional interpretation for the stronger systemZω+μ+ (QF-AC) + (BR) and then more recently by the simpler methods of [9]. Jäger showed how the latter methods could also be used to obtain the general result of Theorem 1 below.
Real functions on the family of all well-ordered subsets of a partially ordered set
Authors: Todorčević Stevo 
Published: May 2006

Definition 1 (Kurepa [3, p. 99]). Let E be a partially ordered set. Then σE denotes the set of all bounded well-ordered subsets of E. We consider σE as a partially ordered set with ordering defined as follows: st if and only if s is an initial segment of t.Then σE is a tree, i.e., {s ∈ σ E∣ st} is well-ordered for every t ∈ σE. The trees of the form αE were extensively studied by Kurepa in [3]–[10]. For example, in [4], he used σQ and σR to construct various sorts of Aronszajn trees. (Here Q and R denote the rationals and reals, respectively.) While considering monotone mapping between some kind of ordered sets, he came to the following two questions several times:P.1. Does there exist a strictly increasing rational function on σQ? (See [4, Problème 2], [5, p. 1033], [6, p. 841], [7, Problem 23.3.3].)P.2. Let T be a tree in which every chain is countable and every level has cardinality <2ℵ0. Does there exist a strictly increasing real function on T? (See [6, p. 246] and [7].)It is known today that Problem 2 is independent of the usual axioms of set theory (see [1]). Concerning Problem 1 we have the following.
QE rings in characteristic pn
Authors: Berline Chantal , Cherlin Gregory 
Published: May 2006

AbstractWe show that all QE rings of prime power characteristic are constructed in a straightforward way out of three components: a filtered Boolean power of a finite field, a nilpotent Jacobson radical, and the ring Zp. or the Witt ring W2(F4) (which is the characteristic four analogue of the Galois field with four elements).
Game sentences, recursive saturation and definability
Authors: Harnik Victor 
Published: May 2006

In the past 10 years much progress occurred in the direction of unifying the treatment of finitary Lωω and infinitary logic. We have especially in mind the study of countable admissible fragments (of which Lωω is a particular case) initiated by Barwise in [1] (cf. also [2]), the introduction of -recursively saturated structures by Ressayre in [22] (for these were independently introduced in [3]) and game sentences as studied by Vaught in [26] (see also Svenonius [25] and Moschovakis [19], [20]). A good reference for these topics is [17].In this paper we illustrate the use of these tools for deducing definability theorems. We go beyond the known results as seen from the discussion that follows.Let L be a language and P a (say, unary) additional predicate. Consider a sentence σ(P) in the language L(P). There are two types of definability theorems: the “Beth type” and the “Svenonius type”. Beth's theorem [4] characterizes the condition:(i)′ for any ;while Svenonius' result [24] characterizes:(i)″ for any , every L-automorphism of is an L(P)-automorphism.
Two-dimensional partial orderings: Undecidability
Authors: Manaster Alfred B. , Rosenstein Joseph G. 
Published: May 2006

In this paper we examine the class of two-dimensional partial orderings from the perspective of undecidability. We shall see that from this perspective the class of 2dpo's is more similar to the class of all partial orderings than to its one-dimensional subclass, the class of all linear orderings. More specifically, we shall describe an argument which lends itself to proofs of the following four results:(A) the theory of 2dpo's is undecidable:(B) the theory of 2dpo's is recursively inseparable from the set of sentences refutable in some finite 2dpo;(C) there is a sentence which is true in some 2dpo but which has no recursive model;(D) the theory of planar lattices is undecidable.It is known that the theory of linear orderings is decidable (Lauchli and Leonard [4]). On the other hand, the theories of partial orderings and lattices were shown to be undecidable by Tarski [14], and that each of these theories is recursively inseparable from its finitely refutable statements was shown by Taitslin [13]. Thus, the complexity of the theories of partial orderings and lattices is, by (A), (B) and (D), already reflected in the 2dpo's and planar lattices.As pointed out by J. Schmerl, bipartite graphs can be coded into 2dpo's, so that (A) and (B) could also be obtained by applying a Rabin-Scott style argument [9] to Rogers' result [11] that the theory of bipartite graphs is undecidable and to Lavrov's result [5] that the theory of bipartite graphs is recursively inseparable from the set of sentences refutable in some finite bipartite graph. (However, (C) and (D) do not seem to follow from this type of argument.)
Minimal forms inλ-cakulus computations
Authors: Böhm Corrado , Micali Silvio 
Published: May 2006

AbstractThe notion of a minimal form is defined as an extension of the notion of a normal form inλ-β-calculus and its meaning is discussed in a computational environment. The features of the Knuth-Gross reduction strategy are used to prove that to possess a minimal form, for a generic term, is a semidecidable predicate.
An invariance notion in recursion theory
Authors: Byerly Robert E. 
Published: May 2006

AbstractA set of gödel numbers is invariant if it is closed under automorphisms of (ω, ·), where ω is the set of all gödel numbers of partial recursive functions and · is application (i.e., n · m ≃ φn(m)). The invariant arithmetic sets are investigated, and the invariant recursively enumerable sets and partial recursive functions are partially characterized.
On the expressibility hierarchy of Magidor-Malitz quantifiers
Authors: Rubin Matatyahu , Shelah Saharon 
Published: May 2006

AbstractWe prove that the logics of Magidor-Malitz and their generalization by Rubin are distinct even for PC classes.Let M ⊨ Qnx1 … xnφ(x1 … xn) mean that there is an uncountable subset A of ∣M∣ such that for every a1 …, an ∈ A, M ⊨ φ[a1, …, an].Theorem 1.1 (Shelah) (♢ℵ1). For every n ∈ ωthe classKn+1 = {‹A, R› ∣ ‹A, R› ⊨ ¬ Qn+1x1 … xn+1R(x1, …, xn+1)} is not an ℵ0-PC-class in the logic ℒn, obtained by closing first order logic underQ1, …, Qn. I.e. for no countable ℒn-theory T, isKn+1the class of reducts of the models of T.Theorem 1.2 (Rubin) (♢ℵ1). Let M ⊨ QE x yφ(x, y) mean that there is A ⊆ ∣M∣ such thatEA, φ = {‹a, b› ∣ a, b ∈ A and M ⊨ φ[a, b]) is an equivalence relation on A with uncountably many equivalence classes, and such that each equivalence class is uncountable. Let KE = {‹A, R› ∣ ‹A, R› ⊨ ¬ QExyR(x, y)}. Then KE is not an ℵ0-PC-class in the logic gotten by closing first order logic under the set of quantifiers {Qn ∣ n ∈ ω) which were defined in Theorem 1.1.
Some model theory of modules. I. On total transcendence of modules
Authors: Rothmaler Philipp 
Published: May 2006

This is the first part of a series of papers on certain topics of model theory of modules, which will further consist of Part II. On stability and categoricity of flat modules and Part III. On infiniteness of sets definable in modules. Although these parts are only weakly connected, it is convenient to have common background and notation. The first part was written in Summer 1980, when I did not know about either footnote (3) appearing in the final version of [GA 2] or about the work of the “Bedford College Group” concerning stability theory of modules. I would like to thank Wilfrid Hodges and Mike Prest for their interest in this note, which encouraged me to include it as the first part of the present series, and for acquainting me with their unpublished work, to which I refer in footnotes. I should also like to thank the referee for his suggestions, in particular for correcting the argument in Remark (5) and for drawing my attention to [Z-H].A syntactical analysis of Zimmermann's notion of “endlich matriziellen Untergruppen” of modules [ZI, p. 1087] shows that it coincides with the notion of subgroups definable by positive primitive formulae (without parameters) [GA 1, p. 80]. Then combination of [ZI, Folgerung 3.4] with [GA 1, Lemma 5] (or [GA 2, Theorem 1] for uncountable rings) yields purely algebraic criteria for total transcendence of modules. I present here a proof of this result using only the following facts.(G1) A module M is totally transcendental iff it satisfies the minimal condition on subgroups definable by positive primitive formulae (without parameters) [GA 1, Lemma 5] (and [GA 2, Theorem 1] for uncountable rings).
Arithmetical interpretations of dynamic logic
Authors: Hájek Petr 
Published: May 2006

AbstractAn arithmetical interpretation of dynamic propositional logic (DPL) is a mapping ƒ satisfying the following: (1) ƒ associates with each formula A of DPL a sentence ƒ(A) of Peano arithmetic (PA) and with each program α a formula ƒ(α) of PA with one free variable describing formally a supertheory of PA; (2) ƒ commutes with logical connectives; (3) ƒ([α]A) is the sentence saying that ƒ(A) is provable in the theory ƒ(α); (4) for each axiom A of DPL, ƒ(A) is provable in PA (and consequently, for each A provable in DPL, ƒ(A) is provable in PA). The arithmetical completeness theorem is proved saying that a formula A of DPL is provable in DPL iff for each arithmetical interpretation ƒ, ƒ(A) is provable in PA. Various modifications of this result are considered.
Rungs and trees
Authors: Masseron Marcel 
Published: May 2006

In a recent work on -recursivity ([M] in which the principal result of [G.2] is obtained by more direct methods) I have given a very simple lemma on the structure of the so-called rungs [G.1] which enables us to describe precisely the relation they have with Jervell's tree [J] in the wellfounded case (R.O. Gandy has also worked out this relation in the wellfounded case, but with techniques very different from those introduced here).In the present work I treat the general case in §§I and II. The results are summarized in §III. These results are then exploited by means of the operator Σ*, which is the principal tool of Lemma (4.1) (i.e. the generalized version of the previous structural lemma). A systematical use of Σ* is also the basis of the study of ladders in the last section.Recall that the notions of rungs and ladders were introduced by J.-Y. Girard [G, 0] and [G.1] as a basic tool for ordinal analysis of Π½-logic (i.e. roughly speaking, logic with β-rule) and the introduction of trees and homogeneous trees by H.R. Jervell is a first attempt to give to these notions a geometrical interpretation without any explicit mention of functors and categorical limits. In fact, as it is evident in the last section of the present paper, limits play a modest role in the study of ladders, but their properties with respect to limits are of fundamental interest in their applications (a domain which is out of the scope of this basic work. Readers interested in applications to ordinal analysis are invited to consult the references). Note that my trees are not exactly those introduced by H.R. Jervell: his extra symbols λ and ρ are respectively changed into 0 and the type of the concerned tree. More important is the fact that we consider only the extremal nodes of the trees. The objects so obtained are then as close as possible to Girard's dendroids.I am indebted to J.-Y. Girard for his advice about my work.
Double jumps of minimal degrees
Authors: Jockusch Carl G. , Posner David B. 
Published: May 2006

This paper concerns certain relationships between the ordering of degrees of unsolvability and the jump operation. It is shown that every minimal degree a < 0′ satisfies a″ = 0″. To restate this result in more suggestive language and compare it with related results, we shall use notation based on the now standard terminology of “high” and “low” degrees. Let Hn be the class of degrees a < 0′ such that a(n) = 0(n+1), and let Ln be the class of degrees a ≤ 0′ such that an = 0n. (Observe that Hi ⊆ Lj and Li ⊆ Lj, whenever i ≤ j, and Hi ∩ Lj = ∅ for all i andj.) The result mentioned above may now be restated in the form that every minimal degree a ≤ 0′ is in L2. This extends an earlier result of S. B. Cooper ([1], see also [4]) that no minimal degree a < 0′ is in H1. In the other direction, Sasso, Epstein, and Cooper ([10], [15]) have shown that there is a minimal degree a < 0′ which is not in L1. Also, C.E.M. Yates [14, Corollary 11.14], showed the existence of a minimal degree a < 0′ in L1. Thus each minimal degree a < 0′ lies in exactly one of the two classes L1L2 – L1 and each of the classes contains minimal degrees.Our results are not restricted to the degrees below 0′. We show in fact that every minimal degree a satisfies a″ = (a ∪ 0′)′. To restate this result and discuss extensions of it, we extend the “high-low” classification of degrees from the degrees below 0′ to degrees in general. There are a number of fairly plausible ways of doing this, but we choose the one way we know of doing so which leads to interesting results. Let GHn be the class of degrees a such that a(n) = (a ∪ 0′)(n), and let GLn be the class of degrees a such that a(n) = (a ∪ 0′)(n−1).
Further consistency and independence results in NF obtained by the permutation method
Authors: Forster T. E. 
Published: May 2006

The permutation method was first applied to NF by Scott; other workers have published results ([2], [3]; or see [4] for a survey). Some of the results proved here have a more metalogical character than most previously yielded by this method. Hinnion and Petry (unpublished, but see [4]) proved that the existence of objects x such that x = {y: x∈y} is consistent with NF. (The significance of this is that the operation that sends x to {y: x∈y} respects ∈ and is thus an embedding.) It is demonstrated below that the existence of such objects is independent of the axioms of NF.The existence of nontrivial automorphisms of the universe is not an interesting possibility in ZF, since it contradicts wellfoundedness. Similar auguments are not available in NF, however, and it is shown below that if NF + AC for pairs is consistent, then we can consistently add an axiom stating that there is a ∈-automorphism of the universe that is a set of the model. (In the proof given below, the automorphism is in fact of order 2, but natural enrichments of the construction enable one to find automorphisms of other orders with suitable versions of choice as additional hypotheses.)