bluuebunny/crossref_metadata_2025 huggingface dataset:
Begründung einer strengen Implikation
Authors: Ackermann Wilhelm
Published: May 2006

Die Gründe, die C. I. Lewis [5], [6] bewogen haben, neben der gewöhnlichen Implikation eine strikte Implikation einzuführen, sind bekannt. In der vorliegenden Arbeit wird aus ähnlichen Gründen eine strenge Implikation eingeführt, die jedoch einen engeren Begriff darstellt als die strikte Implikation. Mit einer Arbeit von Arnold Schmidt [7] hat meine nur geringe Berührungspunkte, da der Verfasser sich mit der strikten Implikation beschäftigt. Für diese wird ein relativ einfaches Axiomensystem angegeben und gezeigt, wie man durch geeignete Definitionen von Notwendigkeit und Möglichkeit bei eventueller Hinzufügung von Zusatzaxiomen die bekannten Systeme des Modalitätenkalküls erhält. Den Hauptteil meiner Arbeit bildet die Einführung meines Implikationsbegriffs; im zweiten Teil der Arbeit komme ich dann auf dieser Basis auch zu einer Einführung der Modalitäten.

Um die Art der Einführung des Kalküls zu motivieren, wollen wir kurz auf das eingehen, das uns mehr oder weniger deutlich vorschwebt, wenn wir von der strengen Implikation sprechen. Die strenge Implikation, die wir durchA → Bwiedergeben, soll ausdrücken, daß zwischenAundBein logischer Zusammenhang besteht, daß der Inhalt vonBein Teil des Inhaltes vonAist, oder wie wir es sonst ausdrücken wollen. Das hat mit der Richtigkeit oder Falschheit vonAundBnichts zu tun. So würde man die Allgemeingültigkeit einer FormelA→ (B → A) ablehnen, da sie den Schluß vonAaufB → Aeinschließt und da die Richtigkeit vonAnichts damit zu tun hat, ob zwischenBundAein logischer Zusammenhang besteht.
Inner models for set theory – Part III
Authors: Shepherdson J. C.
Published: May 2006

In this third and last paper on inner models we consider some of the inherent limitations of the method of using inner models of the type defined in 1.2 for the proof of consistency results for the particular system of set theory under consideration. Roughly speaking this limitation may be described by saying that practically no further consistency results can be obtained by the construction of models satisfying the conditions of theorem 1.5, i.e., conditions 1.31, 1.32, 1.33, 1.51, viz.:

This applies in particular to the ‘complete models’ defined in 1.4. Before going on to a precise statement of these limitations we shall consider now the theorem on which they depend. This is concerned with a particular type of complete model examples of which we call “proper complete models”; they are those complete models which are essentially interior to the universe, those whose classes are sets of the universe constituting a class thereof, i.e., those for which the following proposition is true:

The main theorem of this paper is that the statement that there are no models of this kind can be expressed formally in the same way as the axioms A, B, C and furthermore it can be proved that if the axiom system A, B, C is consistent then so is the system consisting of axioms A, B, C, plus this new hypothesis that there exist no proper complete models. When combined with the axiom ‘V = L’ introduced by Gödel in (1) this new hypothesis yields a system in which any normal complete model which exists has for its universal class V, the universal class of the original system.
An informal exposition of proofs of Gödel's theorems and Church's theorem
Authors: Rosser Barkley
Published: May 2006

This paper is an attempt to explain as non-technically as possible the principles and devices used in the various proofs of Gödel's Theorems and Church's Theorem.

Roman numerals in references shall refer to the papers in the bibliography. In the statements of Gödel's Theorems and Church's Theorem, we will employ the phrase “for suitable L.” The hidden assumptions which we denote by this phrase have never been put down explicitly in a form intelligible to the average reader. The necessity for thus formulating them has commonly been avoided by proving the theorems for special logics and then remarking that the proofs can be extended to other logics. Hence the conditions necessary for the proofs of Gödel's Theorems and Church's Theorem are at present very indefinite as far as the average reader is concerned. To partly clarify this situation, we will now mention the more prominent of these assumptions.

I. In any proof of Gödel's Theorems or Church's Theorem, two logics are concerned. One serves as the “logic of ordinary discourse” in which the proof is carried out, and the other is a formal logic, L, about which the theorem is proved. The first logic may or may not be formal. However L must be formal. Among other things, this implies that the propositions of L are formulas built according to certain rules of structure.
A system of strict implication
Authors: Vredenduin P. G. J.
Published: May 2006

Lewis has made an attempt to construct a logistic system containing an implication-relation ⊰ in such a way that p⊰q shall have the meaning: q is deducible from p. Lewis and Langford admit that “the one serious doubt which can arise concerning the equivalence of p⊰q to the relation of deducibility … arises from the fact that strict implication has its corresponding paradoxes:

19.74 ˜⋄ p ⊰ · p ⊰ q,

‘If p is impossible, then p strictly implies any proposition q’; and

19.75 ˜⋄˜p ⊰ · q ⊰ p,

‘If p is necessary, then any proposition q strictly implies p.’”

Indeed, it is not obvious that 19.74 and 19.75 should hold. It is true that in many cases 19.75 (as well as 19.74) is valid; for in the system of Lewis and Langford

p · ˜p · ⊰ q

is provable, and the conformity of this law with real deduction is shown by them. In all cases in which a proposition of the form p·∼p can be derived from an impossible proposition, 19.75 holds. But it is not obvious that there are no other kinds of impossible propositions.

Fortunately it is easy to see what is the origin of the paradoxes. They are introduced into the system by the definition

11.02 p ⊰ q · = ˜ ⋄(p · ˜q).

Assume this definition. Then if q is necessary, p·∼q is impossible, and so, in accordance with 11.02, p⊰q is valid.
On simplifying truth-functional formulas
Authors: Bing Kurt
Published: May 2006

The following are some consequences of a recent result concerning the prime implicants of disjunctive normal formulas.

I. Introduce the void fundamental formula λ, and define the consensus of αϕ and as in [5], but allowing ϕ or ψ or both to be λ. Then a normal formula Φ is valid if and only if the operations of dropping subsuming clauses and adjoining consensus formulas end by transforming Φ into the single formula λ. (For under the notions of fundamental formula and consensus as now extended, λ is a (unique) prime implicant of all valid formulas and of no others; and the argument of [5] now shows that the operations mentioned transform every normal formula, whether valid or not, into the disjunction of its prime implicants.)

II. A normal formula ψ implies a normal formula Φ if and only if every clause of ψ subsumes one of the prime implicants of Φ, which are found as in I above. (The condition is clearly sufficient. It is necessary, for if ψ implies Φ, then every clause of ψ implies Φ and subsumes a shortest fundamental formula implying Φ.) In this condition, the prime implicants of Φ may be replaced by the clauses of the result of merely adjoining, as long as possible, consensus formulas to Φ.
A rudimentary definition of addition
Authors: Ritchie R. W.
Published: May 2006

In [S, pp. 77–88], Smullyan introduced the class of rudimentary relations, and showed that they form a basis for the recursively enumerable sets. He also asked [S, p. 81] if the addition and multiplication relations were rudimentary. In this note we answer one of these questions by showing that the addition relation is rudimentary. This result was communicated to Smullyan orally in 1960 and is announced in [S, p. 81, footnote 1]. However, the proof has not yet appeared in print. (Shortly after the publication of [S], James H. Bennett, using much more subtle methods than those of this note, showed that the multiplication relation is also rudimentary. That result appears in his doctoral dissertation [B], and is being prepared for publication.)

Let us begin by reviewing Smullyan's definition [S, p. 10] of dyadic notation for the positive integers. Each positive integerais identified with the unique stringanan−1…a1a0of 1's and 2's such thata= Σin=0ai2iBecause of this identification, we are able to speak of typographical properties of numbers.
Une Généralisation de l'ultraproduit
Authors: Fraïssé Roland
Published: May 2006

Plusieurs auteurs ont déjà considéré l'ultraproduit et l'ultrapuissance en prenant certaines, et non forcément toutes les fonctions qui, à chaque indice i, associent un élément de la base de la relation de rang i (voir par exemple [6]). La généralisation ici considérée, appelée ultraproduit complet (voir §3) est définie à l'aide des isomorphismes locaux. On y retrouve l'essentiel de la théorie de l'ultraproduit classique avec évidemment des exemples plus variés.
On Meyer and Lambert's quantificational calculus FQ
Authors: Leblanc Hugues
Published: May 2006

The semantical account that Meyer and Lambert give in [7] of their quantificational calculus FQ can be considerably simplified, and—supposing as the authors do at the close of their paper that ‘ = ’ counts as a primitive sign—so can their axiom system for FQ.

(1) As the authors remark, axiom schema 102 can be simplified to read : A ⊃ (∀Χ)A, where Χ does not occur free in A. Following Tarski's [9], it can be further simplified: A ⊃ (∀Χ)A, where Χ does not occur in A.
On measures on complete Boolean algebras
Authors: Prikry Karel
Published: May 2006

In this paper we prove some theorems concerning measures on complete Boolean algebras. Among other things, in §I of this paper, we construct a counterexample to the following conjecture of W. Luxemburg: Every measure on a nonatomic hyperstonian Boolean algebra is normal. (See [3, p. 57].) This result is expressed by Theorem 1, §I. In order to construct this example we have to suppose that a real-valued measurable cardinal exists. This hypothesis is independent of the usual axioms of set theory. Luxemburg proved that our assumption is necessary. Our second result is stated in Theorem 2 near the end of the paper.
Some calculi with strong negation primitive
Authors: Jay Zeman J.
Published: May 2006

We shall take “strong negation” to be a unary operator with some of the properties usually associated with negation, and such that the strong negation of a statement implies the “ordinary” negation ofthat statement, but not vice-versa. This paper will consider two varieties of strong negation: first, strong negation set in the context of the propositional calculus, specifically, the intuitionist PC, and secondly, strong negation in modal systems, that is, strong negation as impossibility. In the latter part of the paper, we shall present axiomatizations of several classical Lewis-modal systems having only material implication and impossibility primitive.
Functionals defined by transfinite recursion
Authors: Tait W. W.
Published: May 2006

This paper deals mainly with quantifier-free second order systems (i.e., with free variables for numbers and functions, and constants for numbers, functions, and functionals) whose basic rules are those of primitive recursive arithmetic together with definition of functionals by primitive recursion and explicit definition. Precise descriptions are given in §2. The additional rules have the form of definition by transfinite recursion up to some ordinal ξ (where ξ is represented by a primitive recursive (p.r.) ordering). In §3 we discuss some elementary closure properties (under rules of inference and definition) of systems with recursion up to ξ. Let Rξ denote (temporarily) the system with recursion up to ξ. The main results of this paper are of two sorts:

Sections 5–7 are concerned with less elementary closure properties of the systems Rξ. Namely, we show that certain classes of functional equations in Rη can be solved in Rη for some explicitly determined η < ε(η) (the least ε-number > ξ). The classes of functional equations considered all have roughly the form of definition by recursion on the partial ordering of unsecured sequences of a given functional F, or on some ordering which is obtained from this by simple ordinal operations. The key lemma (Theorem 1) needed for the reduction of these equations to transfinite recursion is simply a sharpening of the Brouwer-Kleene idea.
Some theorems on R-maximal sets and major subsets of recursively enumerable sets
Authors: Lerman Manuel
Published: May 2006

In [5], we studied the relational systems /Ā obtained from the recursive functions of one variable by identifying two such functions if they are equal for all but finitely many х ∈ Ā, where Ā is an r-cohesive set. The relational systems /Ā with addition and multiplication defined pointwise on them, were once thought to be potential candidates for nonstandard models of arithmetic. This, however, turned out not to be the case, as was shown by Feferman, Scott, and Tennenbaum [1]. We showed, letting A and B be r-maximal sets, and letting denote the complement of X, that /Ā and are elementarily equivalent (/Ā ≡ ) if there are r-maximal supersets C and D of A and B respectively such that C and D have the same many-one degree (C =mD). In fact, if A and B are maximal sets, /Ā ≡ if, and only if, A =mB. We wish to study the relationship between the elementary equivalence of /Ā and , and the Turing degrees of A and B.
The Craig-Lyndon interpolation theorem in 3-valued logic
Authors: Rockingham Gill R. R.
Published: May 2006

The purpose of this paper is to provide a formal system which is (a) adequate (functionally complete), (b) consistent, and (c) complete, relative to 3-valued logic with one designated value, and for which, furthermore, (d) a simple normal form theorem, and (e) the Craig-Lyndon Interpolation Theorem [1], [2] holds.
Admissible ordinals and intrinsic consistency
Authors: Machtey Michael
Published: May 2006

This paper deals with a fundamental problem in the theory of recursion on initial segments of the ordinal numbers which was originated by Kripke [4] and Platek [8]. Following Kreisel-Sacks [3], we define a notion of relative recursiveness for this abstract recursion theory which allows us to speak of a function ƒ being weakly recursive in a set A via a partial recursive function g; this makes it reasonable to speak of partial recursive functions as ‘reduction procedures’. We say that a reduction procedure g is intrinsically consistent if for any set A there is a partial function ƒ which is weakly recursive in A via g. It is an elementary result of ordinary recursion theory that an arbitrary reduction procedure can be replaced in a uniform manner by an intrinsically consistent reduction procedure without the loss of any function f being reducible to any set A. In §3 we show that for recursion theory on the ordinals less than any infinite cardinal the analogous result is consistent with the axioms of set theory. In §5 we show that for all other cases of abstract recursion theory the analogous result fails in a very strong way to be true.
Consistency of lesniewski's mereology
Authors: Lejewski Czesław
Published: May 2006

According to Sobociński's recollection, the consistency of Mereology was proved by Leśniewski by means of an appropriate interpretation within the framework of the theory of real numbers. His proof was never published, but in a recent paper R. E. Clay has succeeded in reconstructing a version of it.1 Clay's result amounts to showing that if Lesniewski's Ontology expanded by the addition of the axioms for the real numbers is consistent then Mereology is consistent. Without casting any doubts on the validity of the proof one can hardly fail to note that here we have a case where the consistency of a conceptually simple theory is made to depend on the consistency of a theory which from the point of view of intuition is far from being obvious.
Identity, equivalence and isomorphism of problems
Authors: Materna Pavel
Published: May 2006

Zdzislaw Cackowski [1], commenting on ideas of Rubinštejn [3], points out that it is possible for two questions with the same answer to have different “content”.1 In the present paper we shall adopt a more general semantical approach to the question of identity of problems and will distinguish between equivalence and isomorphism.
Note on a paper in tense logic
Authors: Bull R. A.
Published: May 2006

In [1, §4], my ‘proof’ that GH1 has the finite model property is incorrect;
there are considerable obscurities towards the end of §1, particularly on p.
33; and I should have exhibited the finite models for GH1. In §1 of this
paper I expand the analysis of the sub-directly irreducible models for GH1
which I give in §1 of [1]. In §2 I give a correct proof that GH1 has the
finite model property. In §3 I exhibit these finite models as models on
certain ordered sets.
A complete and consistent modal set theory
Authors: Fitch Frederic B.
Published: May 2006

1.1. The aim of this paper is the construction of a demonstrably consistent system of set theory that (1) contains roughly the same amount of mathematics as the writer's system K′ [3], including a theory of continuous functions of real numbers, and (2) provides a way for expressing in the object language various propositions which, in the case of K′, could be expressed only in the metalanguage, for example, general propositions about all real numbers. It was not originally intended that the desired system should be a modal logic, but the modal character of the system appears to be a natural outgrowth of the way it is constructed. A detailed treatment of the natural, rational, and real numbers is left for a subsequent paper.
Interpretations of set theory and ordinal number theory
Authors: Yasugi Mariko
Published: May 2006

In [3], Takeuti developed the theory of ordinal numbers (ON) and constructed a model of Zermelo-Fraenkel set theory (ZF), using the primitive recursive relation ∈ of ordinal numbers. He proved:

(1) If A is a ZF-provable formula, then its interpretation A0 in ON is ON-provable;

(2) Let B be a sentence of ordinal number theory. Then B is a theorem of ON if and only if the natural translation B* of B in set theory is a theorem of ZF;

(3) (V = L)° holds in ON.
Intensional interpretations of functionals of finite type I
Authors: Tait W. W.
Published: May 2006

T0 will denote Gödel's theory T[3] of functionals of finite type (f.t.) with intuitionistic quantification over each f.t. added. T1 will denote T0 together with definition by bar recursion of type o, the axiom schema of bar induction, and the schema

of choice. Precise descriptions of these systems are given below in §4. The main results of this paper are interpretations of T0 in intuitionistic arithmetic U0 and of T1 in intuitionistic analysis is U1. U1 is U0 with quantification over functionals of type (0,0) and the axiom schemata AC00 and of bar induction.
Strong reduction and normal form in combinatory logic
Authors: Lercher Bruce
Published: May 2006

The notion of strong reduction is introduced in Curry and Feys' book Combinatory logic [1] as an analogue, in the theory of combinatore, to reduction (more exactly, βη-reduction) in the theory of λ-conversion. The existence of an analogue and its possible importance are suggested by an equivalence between the theory of combinatore and λ-conversion, and the Church-Rosser theorem in λ-conversion. This theorem implies that if a formula X is convertible to a formula X* which cannot be further reduced—is irreducible, or in normal form—then X is convertible to X* by a reduction alone. Moreover, the reduction may be performed in a certain prescribed order.
The decidability of hindley's axioms for strong reduction
Authors: Lercher Bruce
Published: May 2006

In his paper [3] Hindley shows that strong reduction in combinatory logic (see [1] for the basic discussion of this notion) cannot be axiomatized with a finite number of axiom schemes, and then he presents an infinite system of axiom schemes for strong reduction. Hindley lists one axiom and six axiom schemes, together with a method (clause (viii) below) for generating further axiom schemes from these. The results of this note are that different applications of clause (viii) yield different axiom schemes, and that the property of being an axiom is decidable.
A simplification of the Bachmann method for generating large countable ordinals
Authors: Bridge Jane
Published: May 2006

In [2] Bachmann showed how a hierarchy of normal functions on the countable ordinals Ω can be constructed using certain uncountable ordinals together with appropriate fundamental sequences for limit ordinals, as indexing ordinals for the hierarchy. This method, explained in more detail in §1 below, has been generalized by Pfeiffer [6] and Isles [4] to produce larger hierarchies. Using the hierarchies constructed in this way it is possible, as an immediate consequence of the definitions, to assign ω-sequences 〈αn〉n∈ω to limit ordinals α in an initial segment I of Ω such that . Indeed this was the motivation for Bachmann's original construction. However the initial segments I constructed in this way are of interest even without the fundamental sequences because they occur naturally as the proof-theoretic ordinals associated with particular formal theories. The fundamental sequences necessary for the Bachmann method are not intrinsically of interest proof-theoretically and only serve to obscure the definitions of the associated initial segments. Feferman and Weyhrauch [9] suggested an alternative definition for a sequence of functions on the countable ordinals. This definition was generalized by Aczel [1] who showed how the new functions corresponded to those in Bachmann's hierarchy. (Independently, Weyhrauch established the same results for an initial segment of the sequence of Bachmann functions, i.e. those functions indexed by α < εΩ+1.)
Controlling the dependence degree of a recursively enumerable vector space
Authors: Shore Richard A.
Published: May 2006

Early work combining recursion theory and algebra had (at least) two different sets of motivations. First the precise setting of recursion theory offered a chance to make formal classical concerns as to the effective or algorithmic nature of algebraic constructions. As an added benefit the formalization gives one the opportunity of proving that certain constructions cannot be done effectively even when the original data is presented in a recursive way. One important example of this sort of approach is the work of Frohlich and Shepardson [1955] in field theory. Another motivation for the introduction of recursion theory to algebra is given by Rabin [1960]. One hopes to mathematically enrich algebra by the additional structure provided by the notion of computability much as topological structure enriches group theory. Another example of this sort is provided in Dekker [1969] and [1971] where the added structure is that of recursive equivalence types. (This particular structural view culminates in the monograph of Crossley and Nerode [1974].)

More recently there is the work of Metakides and Nerode [1975], [1977] which combines both approaches. Thus, for example, working with vector spaces they show in a very strong way that one cannot always effectively extend a given (even recursive) independent set to a basis for a (recursive) vector space.
Model existence theorems for modal and intuitionistic logics
Authors: Fitting Melvin
Published: May 2006

In classical logic a collection of sets of statements (or equivalently, a property of sets of statements) is called a consistency property if it meets certain simple closure conditions (a definition is given in §2). The simplest example of a consistency property is the collection of all consistent sets in some formal system for classical logic. The Model Existence Theorem then says that any member of a consistency property is satisfiable in a countable domain. From this theorem many basic results of classical logic follow rather simply: completeness theorems, the compactness theorem, the Lowenheim-Skolem theorem, and the Craig interpolation lemma among others. The central position of the theorem in classical logic is obvious. For the infinitary logic the Model Existence Theorem is even more basic as the compactness theorem is not available; [8] is largely based on it.

In this paper we define appropriate notions of consistency properties for the first-order modal logics S4, T and K (without the Barcan formula) and for intuitionistic logic. Indeed we define two versions for intuitionistic logic, one deriving from the work of Gentzen, one from Beth; both have their uses. Model Existence Theorems are proved, from which the usual known basic results follow. We remark that Craig interpolation lemmas have been proved model theoretically for these logics by Gabbay ([5], [6]) using ultraproducts. The existence of both ultra-product and consistency property proofs of the same result is a common phenomena in classical and infinitary logic. We also present extremely simple tableau proof systems for S4, T, K and intuitionistic logics, systems whose completeness is an easy consequence of the Model Existence Theorems.
Bases of countable Boolean algebras
Authors: Pierce R. S.
Published: May 2006

The purpose of this note is to give a short proof of a conjecture of Feiner that every countable Boolean algebra has an ordered basis that is a lexicographic sum of well-ordered sets over the ordered set η of all rational numbers. Actually, we prove a slightly more precise fact, which is formulated below as Theorem 3. An earlier proof of Feiner's conjecture was obtained by David Cossack (unpublished), using a different method.

Our proof will use the following property of Cantor's dyadic discontinuum D.
A finite arithmetic
Authors: Shepard Philip T.
Published: May 2006

In this paper I shall argue that the presumption of infinitude may be excised from the area of mathematics known as natural number theory with no substantial loss. Except for a few concluding remarks, I shall restrict my concern in here arguing the thesis to the business of constructing and developing a first-order axiomatic system for arithmetic (called ‘FA’ for finite arithmetic) that contains no theorem to the effect that there are infinitely many numbers.

The paper will consist of five parts. Part I characterizes the underlying logic of FA. In part II ordering of natural numbers is developed from a restricted set of axioms, induction schemata are proved and a way of expressing finitude presented. A full set of axioms are used in part III to prove working theorems on comparison of size. In part IV an ordinal expression is defined and characteristic theorems proved. Theorems for addition and multiplication are derived in part V from definitions in terms of the ordinal expression of part IV. The crucial final constructions of part V present a new method of replacing recursive characterizations by strict definitions.

In view of our resolution not to assume that there are infinitely many numbers, we shall have to deal with the situation where singular arithmetic terms of FA may fail to refer. For I know of no acceptable and systematic way of avoiding such situations. As a further result, singular-term instances of universal generalizations of FA are not to be inferred directly from the generalizations themselves. Nevertheless, (i) (x)(y)(x + y = y + x), for example, and all its instances are provable in FA.
Some models for intuitionistic finite type arithmetic with fan functional
Authors: Troelstra A. S.
Published: May 2006

In this note we shall assume acquaintance with [T4] and the parts of [T1] which deal with intuitionistic arithmetic in all finite types. The bibliography just continues the bibliography of [T4].

The principal purpose of this note is the discussion of two models for intuitionistic finite type arithmetic with fan functional (HAω+ MUC). The first model is needed to correct an oversight in the proof of Theorem 6 [T4, §5]: the model ECF+as defined there cannot be shown to have the required properties inEL+ QF-AC, the reason being that a change in the definition ofW12alone does not suffice—if one wishes to establish closure under the operations of HAωthe definitions ofW1σfor other σ have to be adopted as well. It is difficult to see how to do this directly in a uniform way — but we succeed via a detour, which is described in §2.

For a proper understanding, we should perhaps note already here thaton the assumption of the fan theorem, ECF+as defined in [T4] and the new model of this note coincide (since then the definition ofW12[T4, p. 594] is equivalent to the definition forW12in the case of ECF); but inELit is impossible to prove this (and under assumption of Church's thesis the two models differ).
Sheaves and normal submodels
Authors: Mansfield Richard
Published: May 2006

Ellerman, Comer, and Macintyre have all observed that sheaves are an interesting generalization of models and are deserving of model theoretic attention. Scott has pointed out that sheaves are Heyting algebra valued models. The reverse does not hold however since almost no genuine Boolean valued model is a sheaf.

In §1 we shall review the definition of a sheaf and prove a theorem about Boolean valued models using the sheaf construction. In §2 we shall be concerned with the set of sentences preserved by global sections. Our principal result is that global section sentences are also normal submodel sentences. (We define as a normal submodel of if is a submodel of and every point of B − A can be moved by an automorphism of which fixes each point of A.) In §3 we prove that every normal submodel sentence is the negation of a disjunction of Horn sentences and that the set of normal submodel sentences is r.e. but not recursive. §3 involves only traditional model theory and can be read independently of the first two sections.
A model theoretic approach to Malcev conditions
Authors: Baldwin John T. , Berman Joel
Published: May 2006

A varietyV(equational class of algebras) satisfies a strong Malcev condition ∃f1,…, ∃fnθ(f1, …,fn,x1, …,xm) where θ is a conjunction of equations in the function variablesf1, …,fnand the individual variablesx1, …,xm, if there are polynomial symbolsp1, …,pnin the language ofVsuch that ∀x1, …,xmθ(p1…,pn,x1, …,xm) is a law ofV. Thus a strong Malcev condition involves restricted second order quantification of a strange sort. The quantification is restricted to functions which are “polynomially definable”. This notion was introduced by Malcev [6] who used it to describe those varieties all of whose members have permutable congruence relations. The general formal definition of Malcev conditions is due to Grätzer [1]. Since then and especially since Jónsson's [3] characterization of varieties with distributive congruences there has been extensive study of strong Malcev conditions and the related concepts: Malcev conditions and weak Malcev conditions.

In [9], Taylor gives necessary and sufficient semantic conditions for a class of varieties to be defined by a (strong) Malcev condition. A key to the proof is the translation of the restricted second order concepts into first order concepts in a certain many sorted language. In this paper we show that, given this translation, Taylor's theorem is an easy consequence of a result of Tarski [8] and the standard preservation theorems of first order logic.
L(Q)-preservation theorems
Authors: Flum Jörg
Published: May 2006

Much effort has been spent to prove that the reduced product operation preserves, and sometimes even strengthens the L-equivalence of structures, where L is some infinitary language. A similar result is suggested by the following well-known fact:

Assume D is a nonprincipal ultrafilter on ω and, for n Є ω, Cn is a set. If the ultra-product ΠωCn/D is infinite, it has a cardinality ≥ Hence, by Łos' theorem, (i) if and φ(x) is a first-order formula, then iff where Q is the unary quantifier “there are many.”

We shall prove some generalizations of (i). In particular, we show

(ii) if D is a nonprincipal ultrafilter over I = ω, and then where L(Q) is the language obtained from the first-order language by adding the quantifier Q.

(ii) remains true, if D is an ω-regular or an atomless filter over a set I.

Lipner [7] proved that if is regular, then the L(Q)-equivalence is preserved under direct products. We show that the assumption “ is regular” is necessary.
Types of simple α-recursively enumerable sets
Authors: Lerman Manuel
Published: May 2006

Let α be an admissible ordinal, and let (α) denote the lattice of α-r.e. sets, ordered by set inclusion. An α-r.e. set A is α*-finite if it is α-finite and has ordertype less than α* (the Σ1 projectum of α). An a-r.e. set S is simple if (the complement of S) is not α*-finite, but all the α-r.e. subsets of are α*-finite. Fixing a first-order language ℒ suitable for lattice theory (see [2, §1] for such a language), and noting that the α*-finite sets are exactly those elements of (α), all of whose α-r.e. subsets have complements in (α) (see [4, p. 356]), we see that the class of simple α-r.e. sets is definable in ℒ over (α). In [4, §6, (Q22)], we asked whether an admissible ordinal α exists for which all simple α-r.e. sets have the same 1-type. We were particularly interested in this question for α = ℵ1L (L is Gödel's universe of constructible sets). We will show that for all α which are regular cardinals of L (ℵ1L is, of course, such an α), there are simple α-r.e. sets with different 1-types.

The sentence exhibited which differentiates between simple α-r.e. sets is not the first one which comes to mind. Using α = ω for intuition, one would expect any of the sentences “S is a maximal α-r.e. set”, “S is an r-maximal α-r.e. set”, or “S is a hyperhypersimple α-r.e. set” to differentiate between simple α-r.e. sets. However, if α > ω is a regular cardinal of L, there are no maximal, r-maximal, or hyperhypersimple α-r.e. sets (see [4, Theorem 4.11], [5, Theorem 5.1] and [1,Theorem 5.21] respectively). But another theorem of (ω) points the way.
Weak liberated versions of T and S4
Authors: Morgan Charles G.
Published: May 2006

The usual semantics for the modal systems T, S4, and S5 assumes that the set of possible worlds contains at least one member. Recently versions of these modal systems have been developed in which this assumption is dropped. The systems discussed here are obtained by slightly weakening the liberated versions of T and S4. The semantics does not assume the existence of possible worlds, and the accessibility relation between worlds is only required to be quasi-reflexive instead of reflexive. Completeness and independence results are established.
Embedding theorems for boolean algebras and consistency results on ordinal definable sets
Authors: Štěpánek Petr , Balcar Bohuslav
Published: May 2006

The existence of complete rigid Boolean algebras was first proved by McAloon [8] who also showed that every Boolean algebra can be completely embedded in a rigid complete Boolean algebra. McAloon was interested in consistency results on ordinal definable sets. His approach was based on forcing. Recently, Shelah [10] proved that for every uncountable cardinal κ there exists a Boolean algebra of power κ with rigid completion. Extending his method, we get the following theorems.

Theorem 1. Any Boolean algebra B can be completely embedded in a complete Boolean algebra C with no nontrivial σ-complete one-one endomor-phism. If B satisfies the κ-chain condition for an uncountable cardinal κ, the same holds true for C.

Since every automorphism is a complete endomorphism, it follows from Theorem 1 that C is rigid. The other extreme case of Boolean algebras are homogeneous algebras. It was proved by Kripke [7] that every Boolean algebra can be completely embedded in a homogeneous complete Boolean algebra. In his proof, the homogeneous algebra contains antichains of cardinality equal to the power of the embedded Boolean algebra. The following result shows that this is essential: the analogue of Theorem 1 is not provable in set theory even for Boolean algebras with a very weak homogeneity property. We use a Suslin tree with particular properties constructed by Jensen [6] in conjunction with a forcing argument.
An abstract Church-Rosser theorem. II: Applications
Authors: Hindley R.
Published: May 2006

This paper is a continuation of An abstract form of the Church-Rosser theorem. I (this Journal, vol. 35 (1969), pp. 545–560). In Part I, the Church-Rosser property was deduced from abstract premises (A1)–(A8). The original draft of Part II contained some applications of this result, and a fairly simple abstract result by which the Church-Rosser property could be extended from λβ-reduction to λβη-reduction (Curry's notation [3, Chapter 3]). But since this draft was written, these results have been obtained independently and improved by other workers, and a simple and natural new proof for λβ-reduction has been discovered by W. W. Tait and P. Martin-Löf (see §11 later, and [17, §2.4.3]).

So the main purpose of the present Part II is merely to justify the claim in Part I that the abstract theorem does cover the case of λβ-reduction (and various modifications). I shall also include a summary of the main kinds of Church-Rosser proofs. The paragraph and theorem numbers in Part II will continue from those of Part I.

In §§5 and 6 below, Theorem 1 will be specialised to reductions defined by replacements of parts of expressions by others (Theorems 2 and 2A). At the end of §6 an important subclass of such reductions will be treated (Theorem 3).

In §7, Theorem 3 will be applied to prove the Church-Rosser property for combinatory weak reduction [10, §11B], with or without type-restrictions and extra “arithmetical” reduction-rules (Theorems 4 and 5). (In the original draft Theorem 5 was deduced directly from Theorem 2A; the present intervening Theorem 3 is an independent result of B. Rosen [7].)
Some remarks on changing cofinalities
Authors: Devlin Keith J.
Published: May 2006

In [2], Prikry showed that if κ is a weakly inaccessible cardinal which carries a Rowbottom filter, then there is a Boolean extension of V (the universe), having the same cardinals as V, in which cf(κ) = ω. In this note, we obtain necessary and sufficient conditions which a filter D on κ must possess in order that this may be done.
A proof theoretic proof of Scott's general interpolation theorem
Authors: Africk Henry
Published: May 2006

In [5] Scott asked if there was a proof theoretic proof of his interpolation theorem. The purpose of this paper is to provide such a proof, working with the first order system LK of Gentzen [2]. Our method is an extension of the one in Maehara [3] for Craig's interpolation theorem. We will also sketch the original model theoretic proof and show how Scott used his result to obtain a definability theorem of Svenonius [7].

A language for LK contains the usual logical symbols: , ∧, ∨, ⊃, ∀, ∃; countably many free variables a0, a1, … and bound variables x0, x1, …; and some or all of the following nonlogical symbols: n-ary predicates ; n-ary functions ; and individual constants c0, c1, …. Semiterms are defined as follows: (1) Free variables, bound variables and individual constants are semiterms. (2) If f is an n-ary function and s1 …, sn are semiterms, then f(s1 …, sn), is a semiterm. A term is a semiterm that does not contain a bound variable. Formulas are defined as follows: (1) If R is an n-ary predicate and t1 …, tn are terms, then R(t1 …, tn) is a formula. (2) If A and B are formulas, then A, A ∧ B, A ∨ B and A ⊃ B are formulas. (3) If A(t) is a formula which has zero or more occurrences of the term t, and if x is a bound variable not contained in A(t), then ∀xA(x) and ∃xA(x) are formulas where A(x) is obtained from A(t) by substituting x for t at all indicated places.
Forcing for the impredicative theory of classes
Authors: Chuaqui Rolando
Published: May 2006

The purpose of this work is to formulate a general theory of forcing with classes and to solve some of the consistency and independence problems for the impredicative theory of classes, that is, the set theory that uses the full schema of class construction, including formulas with quantification over proper classes. This theory is in principle due to A. Morse [9]. The version I am using is based on axioms by A. Tarski and is essentially the same as that presented in [6, pp. 250–281] and [10, pp. 2–11]. For a detailed exposition the reader is referred there. This theory will be referred to as .

The reflection principle (see [8]), valid for other forms of set theory, is not provable in . Some form of the reflection principle is essential for the proofs in the original version of forcing introduced by Cohen [2] and the version introduced by Mostowski [10]. The same seems to be true for the Boolean valued models methods due to Scott and Solovay [12]. The only suitable form of forcing for found in the literature is the version that appears in Shoenfield [14]. I believe Vopěnka's methods [15] would also be applicable. The definition of forcing given in the present paper is basically derived from Shoenfield's definition. Shoenfield, however, worked in Zermelo-Fraenkel set theory.

I do not know of any proof of the consistency of the continuum hypothesis with assuming only that is consistent. However, if one assumes the existence of an inaccessible cardinal, it is easy to extend Gödel's consistency proof [4] of the axiom of constructibility to .
A discrete chain of degrees of index sets
Authors: Hay Louise
Published: May 2006

Let {Wi} be a standard enumeration of all recursively enumerable (r.e.) sets, and for any class A of r.e. sets, let θA denote the index set of A = {n ∣ Wn ∈ A}. (Clearly, .) In [1], the index sets of nonempty finite classes of finite sets were classified under one-one reducibility into an increasing sequence {Ym}, 0 ≤ m < ∞. In this paper we examine further properties of this sequence within the partial ordering of one-one degrees of index sets. The main results are as follows: (1) For each m, Ym < Ym + 1 and < Ym + 1; (2) Ym is incomparable to ; (3) Ym + 1 and ; are immediate successors (among index sets) of Ym and m; (4) the pair (Ym + 1, ) is a “least upper bound” for the pair (Ym, ) in the sense that any successor of both Ym and is ≥ Ym + 1or; (5) the pair (Ym, ) is a “greatest lower bound” for the pair (Ym + 1, ) in the sense that any predecessor of both Ym + 1 and is ≤ Ym or . Since and all Ym are in the bounded truth-table degree of K, this yields some local information about the one-one degrees of index sets which are “at the bottom” in the one-one ordering of index sets.
A note on frame extensions
Authors: Hay Louise
Published: May 2006

In [2] “recursive frames” were introduced as a means of extending relations R on the nonnegative integers to relations RΛ on the isols. In [1], this extension procedure was generalized by the introduction of “partial recursive frames”; the resulting extended relation on the isols was called RΛ. It was shown in [1] that the two extension procedures agree for recursive relations R, while RΛ ⊇ RΛ if R is . The case when R is , nonrecursive was left open. We show in this note that the extension procedures in fact agree for all relations R.

In the following, the notation and terminology is that of [1] and [2].

Theorem. If R ⊆ XκQ is a recursively enumerable (r.e.) relation, then RΛ = RΛ.

Proof. Clearly RΛ ⊆ RΛ, since every recursive frame is partial recursive. To prove RΛ ⊆ RΛ, we give a uniform effective method for expanding any partial recursiveR-frame F to a recursiveR-frame G such that F ⊆ G, so that So let F be a (nonempty) partial recursive R-frame, with CF(α) generated by . Let Rn denote the result of performing n steps in a fixed recursive enumeration of R. If g(α) is a partial recursive function, “g(α) is defined in n steps” means that in whichever coding of recursive computations is being used, a terminating computation for g with argument α has length ≤ n.
The model-companion of a class of structures
Authors: Cherlin G. L.
Published: May 2006

If Σ is the class of all fields and Σ* is the class of all algebraically closed fields, then it is well known that Σ* is characterized by the following properties:

(i) Σ* is the class of models of some first order theory K*.

(ii) If m1m2 are in Σ* and m1 ⊆ m2 then m1 ≺ m2 (m1 is an elementary substructure of m2, i.e. any first order sentence true in m1 is true in m2).

(iii) If m1 is in Σ then there is a structure m2 in Σ* such that m1 ⊆ m2.

If Σ is some other class of models of a first order theory K and a subclass Σ* of Σ exists satisfying (i)–(iii) then Σ* is uniquely determined and K* (which is unique up to logical equivalence) is called the model-companion of K. This notion is a generalization of the fundamental notion of model-completion introduced and extensively studied by A. Robinson [6], When the model-companion exists it provides the basis for a satisfactory treatment of the notion of an algebraically closed model of K.

Recently A. Robinson has developed a more general formulation of the notion of “algebraically closed” structures in Σ, which is applicable to any inductive elementary class Σ of structures (by elementary we always mean ECΔ). Condition (i) must be weakened to

(i′) Σ* is closed under elementary substructure (i.e. if m1 is in Σ* and m2 ≺ m1 then m2 is in Σ*).
Some representations of Diophantine sets
Authors: Robinson Raphael M.
Published: May 2006

A set D of natural numbers is called Diophantine if it can be defined in the form

where P is a polynomial with integer coefficients. Recently, Ju. V. Matijasevič [2], [3] has shown that all recursively enumerable sets are Diophantine. From this, it follows that a bound for n may be given.

We use throughout the logical symbols ∧ (and), ∨ (or), → (if … then …), ↔ (if and only if), ⋀ (for every), and ⋁ (there exists); negation does not occur explicitly. The variables range over the natural numbers 0,1,2,3, …, except as otherwise noted.

It is the purpose of this paper to show that if we do not insist on prenex form, then every Diophantine set can be defined existentially by a formula in which not more than five existential quantifiers are nested. Besides existential quantifiers, only conjunctions are needed. By Matijasevič [2], [3], the representation extends to all recursively enumerable sets. Using this, we can find a bound for the number of conjuncts needed.

Davis [1] proved that every recursively enumerable set of natural numbers can be represented in the form

where P is a polynomial with integer coefficients. I showed in [5] that we can take λ = 4. (A minor error is corrected in an Appendix to this paper.) By the methods of the present paper, we can again obtain this result, and indeed in a stronger form, with the universal quantifier replaced by a conjunction.
Jump equivalence of the Δ20 hyperimmune sets
Authors: Cooper S. B.
Published: May 2006

An infinite set A is said to be hyperhyperimmune (h.h.i.) if, for any collection of disjoint simultaneously recursively enumerable (r.e.) finite sets, A must fail to intersect with one of those sets. Thus the elements of an h.h.i. set are, in a sense, very elusive. D. A. Martin [3] showed that the degrees of h.h.i. sets with r.e. complements are exactly the r.e. degrees with jump 0″. More generally, C. G. Jockusch [2] found a′ ≥ 0″ to be a sufficient condition for a to be the degree of an h.h.i. set and found a′ ≥ 0′ to be necessary. However, it was also shown that in the degrees as a whole neither condition gave a characterization of the h.h.i. degrees. The purpose of this note is to prove that a′ = 0″does characterize the h.h.i. degrees below 0′.

Theorem. The degrees below 0′ containing h.h.i. sets are exactly those degrees below 0′ with jump 0″.

Proof. From [2], if a′ ≥ 0″, then a contains an h.h.i. set.

Conversely, let A ∈ a where a′ < 0″ and a < 0′. Let {As ∣ s ≥ 0} be a recursive sequence of finite sets such that for each x, lims, Ax(x) exists and equals A(x).

For a set B, let B[m] denote B ∩ [0, m], and (if B is finite) let ∣B∣ denote the cardinality of B.