bluuebunny/crossref_metadata_2025 huggingface dataset:
Construction of models for algebraically generalized recursive function theory
Authors: Strong H. R.
Published: May 2006

The Uniformly Reflexive Structure was introduced by E. G. Wagner who showed that the theory of such structures generalized much of recursive function theory. In this paper Uniformly Reflexive Structures are constructed as factor algebras of free nonassociative algebras. Wagner's question about the existence of a model with no computable splinter (“successor set”) is answered in the affirmative by the construction of a model whose only computable sets are the finite sets and their complements. Finally, for each countable Boolean algebra R of subsets of a countable set which contains the finite subsets, a model is constructed with R as its family of computable sets.
Strong partition properties for infinite cardinals
Authors: Kleinberg E. M.
Published: May 2006

The notion of a “partition relation”, as it has been studied in the context of set theory for the past several years, was inspired by the following theorem of F. P. Ramsey [14]:

Theorem 0.1. Let n be a positive integer and let {A, B} be a partition of those subsets of the nonnegative integers containing exactly n elements. Then there exists an infinite subset x of the nonnegative integers all of whose n-element subsets are contained in only one of A or B. (Any such set x is said to be “homogeneous” for the partition.)
Remarks on an infinitary language with constructive formulas
Authors: Lopez-Escobar E. G. K.
Published: May 2006

It is well known that L, the first-order predicate calculus, is unsuitable for characterizing certain mathematical structures. Since some of those structures are quite often considered by mathematicians, extensions of L have been considered which (partially) solve the above problem. Roughly speaking, the extensions of L fall into two main categories:

(I) extensions obtained by allowing higher order variables, and

(II) extensions obtained by allowing infinitely long formulas.
A proof of zermelo's theorem
Authors: Dévidé Vladimir
Published: May 2006

We give a short proof of the theorem that, assuming the axiom of choice, every set can be well-ordered. The proof makes direct use of neither induction nor ordinals.

Let S be any given set and its power-set. Let be a choice function, i.e. we denote Ζ — {γΖ} by Ζ'.
Effectively retractable theories and degrees of undecidability
Authors: Jones J. P.
Published: May 2006

In this paper a new property of theories, called effective retractability is introduced and used to obtain a characterization for the degrees of subtheories of arithmetic and set theory. By theory we understand theory in standard formalization as defined by Tarski [10]. The word degree refers to the Kleene-Post notion of degree of recursive unsolvability [2]. By the degree of a theory we mean, of course, the degree associated with its decision problem via Gödel numbering.
The degrees of hyperhyperimmune sets
Authors: Jockusch Carl G.
Published: May 2006

In [5, Corollary 3.1] D. A. Martin characterized the degrees of the hyperhypersimple sets as those r.e. degrees a satisfying a′ = 0″. In the present paper we investigate the degrees of the hyperhyperimmune (h.h.i.) sets. We do not achieve a characterization of these degrees, but we do show that the condition a′ ≥ 0″ is sufficient, and the condition a′ ≥ 0′ is necessary, for a degree a to contain an h.h.i. set.
Partially ordered sets representable by recursively enumerable classes
Authors: Florence J. B.
Published: May 2006

A partially ordered (p.o.) set (P, ≼) is represented by the recursively enumerable (r.e.) class C if (P, ≼) is order isomorphic to (C, ⊆), that is to the p.o. set consisting of C ordered by the inclusion relation. (P, ≼.) is representable if it is represented by some r.e. class. N will denote the set of natural numbers.
Tensor products of polyadic algebras
Authors: Daigneault Aubert
Published: May 2006

A basic concept of the theory of models is that of elementary equivalence of similar relational systems: two such systems are said to be elementarily equivalent if they satisfy the same first-order statements or, in other words, if they have the same (first-order) complete theory. It is possible to reformulate this notion of elementary equivalence of systems within the framework of algebraic logic by replacing theories by algebraic structures derived from them or more directly from the systems which are models of these theories. To any such theory T (or model of it), is indeed associated a locally finite polyadic algebra with equality, the underlying Boolean algebra of which is simply the well-known Tarski-Lindenbaum algebra of the theory. It is not hard to prove (see Section 6.1) that two systems are elementarily equivalent iff (i.e. if and only if) they have isomorphic polyadic. algebras. The possibility of replacing theories by algebraic structures and of reducing the purely logical concept of elementary equivalence to the algebraic one of isomorphism can be exploited to give a purely algebraic treatment of model-theoretic problems and suggests natural questions concerning these structures. The present paper illustrates that possibility.
Independent Axioms for Infinite-Valued Logic
Authors: Turquette Atwell R.
Published: May 2006

Recent interest in Łukasiewicz' Lℵ0, raises the question whether this axiomatic system can be simplified [2]. It is known that Łukasiewicz' fourth axiom CCCPQCQPCQP is dependent [4] p. 51. The axiomatic system resulting from deleting the fourth axiom from Lℵ0 will be shown to be “minimal” in the sense that the axioms and rules of inference are mutually independent; consequently, no further simplification of Lℵ0, is possible. By switching basic operators, however, simplified modifications of Lℵ0, can be constructed.
Finite inseparability of some theories of cylindrification
algebras
Authors: Comer Stephen D.
Published: May 2006

An elementary theory T in a language L is
(strongly) finitely inseparable if the set of logically
valid sentences of L and the set of
T-finitely refutable sentences are recursively inseparable.
In §1 we establish a sufficient condition for the elementary theory of a
class of BA's with operators to be finitely inseparable. This is done using
the methods developed independently by M. Rabin and D. Scott (see [6]) on
the one hand and by Ershov on the other (see [2]).
On the structure of quantum logic
Authors: Finch P. D.
Published: May 2006

In the axiomatic development of the logic of nonrelativistic quantum
mechanics it is not difficult to set down certain plausible axioms which
ensure that the quantum logic of propositions has the structure of an
orthomodular poset. This can be done in a number of ways, for example, as in
Gunson [2], Mackey [4], Piron [5], Varadarajan [7] and Zierler [8], and we
summarise one of these ways in §2 below.
Algebraic completeness results for R-mingle and its extensions
Authors: Michael Dunn J.
Published: May 2006

Schiller Joe Scroggs in [9] established remarkable facts concerning “normal” extensions of the modal sentential calculus S5, the most notable of these facts being that all such proper extensions have finite characteristic matrices. The major import of the present paper is that like facts hold for the relevant sentential calculus R-Mingle (RM). Robert K. Meyer in [6] has obtained an important completeness result for RM, which will play a central role in our results. However, in §2 we shall obtain a new proof of Meyer's result as a by-product of the algebraic logic that we develop in §1. Also in §2 we shall obtain the promised results for extensions of RM. In §3 we shall provide a strong completeness theorem for RM by generalizing the semantics of Meyer.
Turing degrees and many-one degrees of maximal sets
Authors: Lerman Manuel
Published: May 2006

Martin [4, Theorems 1 and 2] proved that a Turing degree a is the degree of a maximal set if, and only if, a′ = 0″. Lachlan has shown that maximal sets have minimal many-one degrees [2, §1] and that every nonrecursive r.e. Turing degree contains a minimal many-one degree [2, Theorem 4]. Our aim here is to show that any r.e. Turing degree a of a maximal set contains an infinite number of maximal sets whose many-one degrees are pairwise incomparable; hence such Turing degrees contain an infinite number of distinct minimal many-one degrees. This theorem has been proved by Yates [6, Theorem 5] in the case when a = 0′.

The need for this theorem first came to our attention as a result of work done by the author [3, Theorem 2.3]. There we looked at the structure / obtained from the recursive functions of one variable under the equivalence relation f ~ g if, and only if, f(x) = g(x) a.e., that is, for all but finitely many x ∈ , where M is a maximal set, and M is its complement. We showed that /1 ≡ /2 if, and only if, 1 =m2, i.e., 1. and 2 have the same many-one degree. However, it might be possible that some Turing degree of a maximal set contains exactly one many-one degree of a maximal set. Theorem 1 was proved to show that this was not the case, and hence that the theory of / is not an invariant of Turing degree.
An algebraic characterization of indistinguishable cardinals
Authors: Slomson A. B.
Published: May 2006

Two cardinals are said to be indistinguishable if there is no sentence of second order logic which discriminates between them. This notion, which is defined precisely below, is closely related to that of characterizable cardinals, introduced and studied by Garland in [3]. In this paper we give an algebraic criterion for two cardinals to be indistinguishable. As a consequence we obtain a straightforward proof of an interesting theorem about characterizable cardinals due to Zykov [6].
Ultraproducts which are not saturated
Authors: Jerome Keisler H.
Published: May 2006

In this paper we continue our study, begun in [5], of the connection between ultraproducts and saturated structures. IfDis an ultrafilter over a setI, andis a structure (i.e., a model for a first order predicate logicℒ), the ultrapower ofmoduloDis denoted byD-prod. The ultrapower is important because it is a method of constructing structures which are elementarily equivalent to a given structure(see Frayne-Morel-Scott [3]). Our ultimate aim is to find out what kinds of structure are ultrapowers of. We made a beginning in [5] by proving that, assuming the generalized continuum hypothesis (GCH), for each cardinalαthere is an ultrafilterDover a set of powerαsuch that for all structures,D-prodisα+-saturated.
What is strict implication?
Authors: Hacking Ian
Published: May 2006

C. I. Lewis intended his systems S1–S5 as contributions to the study of “strict implication”, but in his formulation, strict implication is so thoroughly intertwined with other notions, such as possibility and negation, that it remains a problem, to separate out the properties of strict implication itself. I shall solve this problem for S2–5 and von Wright's M. The results for S3–5 are given below, while the implicative parts of S2 and M, which are rather more complicated, are given in §5.

In this presentation, ‘⊃’ stands for strict implication, and missing brackets for ‘⊃’ are restored by associationtothe right. Astrictformula is one of the form A ⊃ B. Axiom schemes are used throughout.
Axiomatization of the infinite-valued predicate calculus
Authors: Hay Louise Schmir
Published: May 2006

The infinite-valued statement calculus to which this paper refers is that of Łukasiewicz [10], whose axiomatization was proved complete in [5]. In [9], Rutledge extended this system to include predicates and quantifiers2 and presented a deductively complete set of axioms for the monadic predicate calculus. This paper represents an attempt to axiomatize the full predicate calculus; for the proposed axiomatization, a property akin to but weaker than completeness is proved. An attempt to prove full completeness along similar lines failed; it has since been shown [11] that the set of valid formulas of the infinite-valued predicate calculus is not recursively enumerable. The method of this paper was suggested by Professor J. Barkley Rosser.
Logical connectives for intuitionistic propositional logic
Authors: McCullough Dean P.
Published: May 2006

In classical propositional logic it is well known that {7, ⊃ } is a functionally complete set with respect to a two-valued truth function modeling. I.e. all definable logical connectives are definable from 7 and ⊃. Other modelings of classical type propositional logics may have different functionally complete sets; for example, multivalued truth function modelings.

This paper examines the question of a functionally complete set of logical connectives for intuitionistic propositional logic with respect to S. Kripke's modeling for intuitionistic logic.
Relative necessity
Authors: Smiley Timothy
Published: May 2006

This paper was suggested by some work of Alan Ross Anderson on deontic logic. Anderson proposed in [1] that the various deontic modalities (‘it is obligatory that’, ‘it is forbidden that’, ‘it is permitted that’, etc.) should be defined in terms of the ordinary alethic modalities (‘it is necessary that’, ‘it is possible that’, etc.). The definition I shall consider is not in fact Anderson's own, though it is closely related to it. His definition, as set out in [1], involves the idea of an (unspecified) penalty or sanction. A forbidden action is defined as one whose commission entails the application of the sanction; an obligatory action is one whose omission entails the application of the sanction.
Recursion, metarecursion, and inclusion
Authors: Owings James C.
Published: May 2006

In this paper it will be shown that the ordering of the recursively enumerable (r.e.) sets under inclusion modulo finite differences (m.f.d.), the ordering of the II111 sets under inclusion m.f.d., and the ordering of the metarecursively enumerable (meta-r.e.) sets under inclusion m.f.d. are all distinct. In fact, it will be shown that the three orderings are pairwise elementarily inequivalent when interpreted in the obvious way in a first order language with one binary relation “≥.” Our result answers a question of Hartley Rogers, Jr. [3, p. 203]. All necessary background material may be found in [2] and [4].
On the derivability of instantiation properties
Authors: Friedman Harvey
Published: May 2006

Every recursively enumerable extension of arithmetic which obeys the disjunction property obeys the numerical existence property [Fr, 1]. The requirement of recursive enumerability is essential. For extensions of intuitionistic second order arithmetic by means of sentences (in its language) with no existential set quantifiers, the numerical existence property implies the set existence property. The restriction on existential set quantifiers is essential. The numerical existence property cannot be eliminated, but in the case of finite extensions of HAS, can be replaced by a weaker form of it. As a consequence, the set existence property for intuitionistic second order arithmetic can be proved within itself.
Experimental logics and Π30 theories
Authors: Hájek Petr
Published: May 2006

In this paper we are going to consider experimental logics introduced by Jeroslow [4] as models of human reasoning proceeding by trial and error, i.e. admitting changes of axioms in time (some axioms are deleted, some new ones accepted). Jeroslow's notion is based on the idea that events which may cause changes in axioms and rules of reasoning are mechanical. Suppose a finite alphabet Γ to be fixed and let Γ* be the set of words in the alphabet Γ. N denotes the set of natural numbers.

0.1. Definition. An experimental logic is a recursive relation H ⊆ N × Γ*; H(t, φ) is read “the expression is accepted at the point of time t”. φ is recurring w.r.t. H (notation: RecH(φ)) if H(t, φ) holds for infinitely many t; is stable w.r.t. H (notation: StblH(φ)) if H(t,φ) holds for all but finitely many t. In symbols:

H is convergent if every recurring expression is stable.

0.2. We have the following facts: Let X ∈ Γ*. (1) X ∈ iff there is an experimental logic H such that X = {φ; RecH(φ)}- (2) X ∈ iff there is an experimental logic H such that X = {φ; StblH(φ)}. (3) X ∈ iff there is a convergent experimental logic H such that X is the set of all expressions recurring ( = stable) w.r.t. H. (See [4], [3], [7]; cf. also [5].)
One hundred and two problems in mathematical logic
Authors: Friedman Harvey
Published: May 2006

This expository paper contains a list of 102 problems which, at the time of publication, are unsolved. These problems are distributed in four subdivisions of logic: model theory, proof theory and intuitionism, recursion theory, and set theory. They are written in the form of statements which we believe to be at least as likely as their negations. These should not be viewed as conjectures since, in some cases, we had no opinion as to which way the problem would go.

In each case where we believe a problem did not originate with us, we made an effort to pinpoint a source. Often this was a difficult matter, based on subjective judgments. When we were unable to pinpoint a source, we left a question mark. No inference should be drawn concerning the beliefs of the originator of a problem as to which way it will go (lest the originator be us).

The choice of these problems was based on five criteria. Firstly, we are only including problems which call for the truth value of a particular mathematical statement. A second criterion is the extent to which the concepts involved in the statements are concepts that are well known, well denned, and well understood, as well as having been extensively considered in the literature. A third criterion is the extent to which these problems have natural, simple and attractive formulations. A fourth criterion is the extent to which there is evidence that a real difficulty exists in finding a solution. Lastly and unavoidably, the extent to which these problems are connected with the author's research interests in mathematical logic.
Types of simple α-recursively enumerable sets
Authors: Leggett Anne , Shore Richard A.
Published: May 2006

One general program of α-recursion theory is to determine as much as possible of the lattice structure of (α), the lattice of α-r.e. sets under inclusion. It is hoped that structure results will shed some light on whether or not the theory of (α) is decidable with respect to a suitable language for lattice theory. Fix such a language ℒ.

Many of the basic results about the lattice structure involve various sorts of simple α-r.e. sets (we use definitions which are definable in ℒ over (α)). It is easy to see that simple sets exist for all admissible α. Chong and Lerman [1] have found some necessary and some sufficient conditions for the existence of hhsimple α-r.e. sets, although a complete determination of these conditions has not yet been made. Lerman and Simpson [9] have obtained some partial results concerning r-maximal α-r.e. sets. Lerman [6] has shown that maximal α-r.e. sets exist iff a is a certain sort of constructibly countable ordinal. Lerman [5] has also investigated the congruence relations, filters, and ideals of (α). Here various sorts of simple sets have also proved to be vital tools. The importance of simple α-r.e. sets to the study of the lattice structure of (α) is hence obvious.

Lerman [6, Q22] has posed the following problem: Find an admissible α for which all simple α-r.e. sets have the same 1-type with respect to the language ℒ. The structure of (α) for such an α would be much less complicated than that of (ω). Lerman [7] showed that such an α could not be a regular cardinal of L. We show that there is no such admissible α.
Discernible elements in models for Peano arithmetic
Authors: Ehrenfeucht Andrzej
Published: May 2006

Combinations of capital letters (e.g., G, EVEN) will be used as names of formulas of Peano arithmetic, x, y, z will be variables. Let be a model for Peano arithmetic. The names of formulas with a bar (e.g., G−, EVEN−(x)) will denote the relations defined by the formulas in . The elements of are called numbers.

By a finite sequence of numbers 〈a0, …, an〉 we mean a number . Note that n Pi's and ai's can be nonstandard elements and therefore “finite,” and all concepts which will use it are not absolute but relative to .
Infinite exponent partition relations and well-ordered choice
Authors: Kleinberg E. M. , Seiferas J. I.
Published: May 2006

The study of partition relations for cardinal numbers introduced by Erdös and his school in the 1950's has, for the past several years, had a profound impact in logic. Unfortunately, quite early in their development, it was noticed by Rado [1] that the potentially most fruitful class of such relations, infinite exponent partition relations, were always in contradiction with the axiom of choice (AC). As a result, such relations were overlooked. This turned out to be a mistake; for, as has been noticed recently, a close study of infinite exponent partition relations is both interesting and rewarding. For example, there are weakened versions of such relations which are provable in ZF and which have valuable applications in recursion theory and set theory. In addition, the pure theory of these relations, like that of the axiom of determinateness, is fruitful as well as elegant. For more background here one should refer to [3].

At any rate, with a more detailed look at infinite exponent partition relations came a more refined version of Rado's original theorem. Specifically, Rado used the full axiom of choice to carefully construct partitions to violate any desired relation—a more sophisticated look at the actual theory of such relations indicated how one could put together some desired partitions using only well-ordered choice [3].

The distinction between well-ordered choice and full choice is by no means vacuous in this context. For Mathias has shown [4] that the simplest infinite exponent partition relation, ω → (ω)ω, is consistent with countable choice (well-ordered choice of length ℵ0) and, in fact, is consistent with dependent choice.
Bounds for the closure ordinals of replete monotonic increasing functions
Authors: Schmidt Diana
Published: May 2006

Whenever particular ordinals are used as tools in a proof or a definition, it is necessary to find a way of representing them. If the ordinals are sufficiently small, there is a standard way (e.g. Cantor normal forms for ordinals less than ε0); in general, representations are often found by using functions on initial segments of the ordinals: Each term which can be obtained from by applications of a function symbol is regarded as a notation for the ordinal obtained by the same applications of the function f to the ordinal 0. In this way, f provides representations for all the ordinals in Clf(0), the closure set of f (se e §1). (For an introduction to and development of this principle, see Feferman [F1]; and for a discussion of the significance of such representations in proof theory, see Kreisel [K1, pp. 22–34].) Thus it is natural to ask whether there are connections between frequently encountered properties of ordinal functions and the size of the ordinals for which they can provide representations.

The purpose of this paper is to show that, for any integer n, the ordinal (see §2) is a bound for the closure ordinals of replete monotonic increasing n-place functions. This result is optimal for n > 2 (the bound is attained by where θα = 1 + α) but not for n < 2. D. H. de Jongh has recently proved, using a completely different method, that the (least possible) bound for n = 2 is ε0. (Trivially, that for n = 1 is φ.)
The nonderivability in intuitionistic formal systems of theorems on the continuity of effective operations
Authors: Beeson Michael J.
Published: May 2006

The primary purpose of this work is to analyze the constructive interpretations of classical theorems concerning the continuity of effective operations. These theorems are the “recursivizations” of the statements that all functions (on various spaces) are continuous. To discuss the question whether these statements are constructively valid, it would be necessary to analyze the notion of “constructively valid,” which is beyond the scope of this paper. We now proceed to describe our results. We deal with two kinds of effective operations: (i) partial operations on indices of partial functions, and (ii) total operations on indices of total functions. The two principal theorems of the subject assert that each such operation is continuous; for (i) this is due to Myhill-Shepherdson [7] and called MS here; for (ii) it is due to Kreisel, Lacombe, and Shoenfield [5] and called KLS. (Explicit formulations of these and other principles mentioned in the introduction will be given in §1 below.)

Several interpretations (of, e.g., Heyting's arithmetic HA) in the literature satisfy Markov's principle MP, and for these MS and KLS are valid, since a close inspection of the classical proofs of MS and KLS shows that they are derivable in HA + MP. But we also find an interpretation in the literature under which MS and KLS are valid while MP is not (hence MS and KLS do not imply MP in systems for which this interpretation is sound). This work is in §2.
Modal reduction principles
Authors: van Benthem J.F.A.K.
Published: May 2006

Modal reduction principles (MRPs) are modal formulas of the following form: Mp → Np, where M, N are (possibly empty) sequences of modal operators (i.e. □ or ◊). The notation M, N will be used to abbreviate such an MRP. We study a certain semantic correspondence between modal formulas and relational properties and obtain two main results. (1) On transitive semantic structures every MRP corresponds to a first-order relational property. (2) For the general case a syntactic criterion exists for distinguishing modal formulas with corresponding first-order properties from the others.
Modal formulas are either elementary or not ΣΔ-elementary
Authors: van Benthem J. F. A. K.
Published: May 2006

In this paper we prove that if L is a set of modal propositional formulas then FR(L) (the class of all frames in which every formula of L holds) is elementary, Δ-elementary or not ΣΔ-elementary. For single modal formulas the second of these cases does not occur.

The model theoretic terminology and results used here are from [1]. (The underlying first order language contains only one, binary, predicate letter in addition to the identity symbol.) We presuppose familiarity with the usual notions and notations of propositional modal logic. A structure for our first order language is called a frame. (So a frame is an ordered couple 〈W, R〉 with domain W and R a binary predicate on W, i.e. a subset of W × W.) A valuation V on F is a function from the set of proposition letters to the power set of W. Using the well-known Kripke truth definition V can be extended to a function from the set of all modal propositional formulas to the power set of W. A modal propositional formula φ holds in a frame F (= 〈W, R〉) if, for all V on F, V(φ) = W. Notation: FR(φ) for the class of all frames in which φ holds. For a set L of modal propositional formulas we define FR(L) as ⋂φ∈LFR(φ). Obviously both FR(L) and cFR(L) (the complement of FR(L)) are closed under isomorphisms.
Order types of ordinals in models of set theory
Authors: Hutchinson John E.
Published: May 2006

An ordinal in a model of set theory is truly countable if its set of predecessors is countable in the real world. We classify the order types of the sets of truly countable ordinals. Models with indiscernibles and other related results are discussed.
A counterexample in the theory of model companions
Authors: Saracino D.
Published: May 2006

In [7] we proved that (I) if T is a countable ℵ0-categorical theory without finite models then T has a model companion; and several people have observed that (II) if T is a countable theory without finite models which is ℵ1-categorical and forcingcomplete for infinite forcing (i.e., T= TF) then T is model-complete. It is natural to ask (1) whether in (I) we can replace ℵ0 by ℵ1; (2) whether in (II) we can replace TF by Tf; and (3) in connection with (II), whether the categoricity of the class of infinitely generic structures for a theory K in some or all infinite powers implies the existence of a model companion for K. The purpose of this note is to provide negative answers to (1), (2), and (3). Specifically, we will prove:

Theorem. There exists a countable theory T such that

(i) T has no finite models and is ℵ-categorical;

(ii) T is forcing-complete for finite forcing, i.e., T = Tf;

(iii) T has no model companion (i.e., in light of (ii), T is not model-complete);

(iv) the class of infinitely generic structures for T is categorical in every infinite power;

(v) every uncountable existentially complete structure for T is infinitely generic;

(vi) there is, up to isomorphism, precisely one countable existentially complete model of Tf, and there are no uncountable e.c. models of Tf (in particular, there is just one countable finitely generic structure and there are no uncountable ones);

(vii) there are precisely ℵ0isomorphism types of countable existentially complete structures for T.
First-order definability in modal logic
Authors: Goldblatt R. I.
Published: May 2006

In the early days of the development of Kripke-style semantics for modal logic a great deal of effort was devoted to showing that particular axiom systems were characterised by a class of models describable by a first-order condition on a binary relation. For a time the approach seemed all encompassing, but recent work by Thomason [6] and Fine [2] has shown it to be somewhat limited—there are logics not determined by any class of Kripke models at all. In fact it now seems that modal logic is basically second-order in nature, in that any system may be analysed in terms of structures having a nominated class of second-order individuals (subsets) that serve as interpretations of propositional variables (cf. [7]). The question has thus arisen as to how much of modal logic can be handled in a first-order way, and precisely which modal sentences are determined by first-order conditions on their models. In this paper we present a model-theoretic characterisation of this class of sentences, and show that it does not include the much discussed LMp → MLp.

Definition 1. A modal frame ℱ = 〈W, R〉 consists of a set W on which a binary relation R is defined. A valuation V on ℱ is a function that associates with each propositional variable p a subset V(p) of W (the set of points at which p is “true”).
The word problem for free fields: a correction and an addendum
Authors: Cohn P. M.
Published: May 2006

In [1] it was claimed that the word problem for free fields with infinite centre can be solved. In fact it was asserted that if K is a skew field with infinite central subfield C, then the word problem in the free field on a set X over K can be solved, relative to the word problem in K.

As G. M. Bergman has pointed out (in a letter to the author), it is necessary to specify rather more precisely what type of problem we assume to be soluble for K: We must be able to decide whether or not a given finite set in K is linearly dependent over its centre. This makes it desirable to prove that the free field has a corresponding property (and not merely a soluble word problem). This is done in §2; interestingly enough it depends only on the solubility of the word problem in the free field (cf. Lemma 2 and Theorem 1′ below).

Bergman also notes that the proof given in [1] does not apply when K is finite-dimensional over its centre; this oversight is rectified in §4, while §3 lifts the restriction on C (to be infinite). However, we have to assume C to be the precise centre of K, and not merely a central subfield, as claimed in [1].

I am grateful to G. M. Bergman for pointing out the various inaccuracies as well as suggesting remedies.
A new proof that analytic sets are Ramsey
Authors: Ellentuck Erik
Published: May 2006

We give a direct mathematical proof of the Mathias-Silver theorem that every analytic set is Ramsey.
Embedding first order predicate logic in fragments of intuitionistic logic
Authors: Löb M. H.
Published: May 2006

Some syntactically simple fragments of intuitionistic logic possess considerable expressive power compared with their classical counterparts.

In particular, we consider in this paper intuitionistic second order propositional logic (ISPL) a formalisation of which may be obtained by adding to the intuitionistic propositional calculus quantifiers binding propositional variables together with the usual quantifier rules and the axiom scheme (Ex), where is a formula not containing x.

The main purpose of this paper is to show that the classical first order predicate calculus with identity can be (isomorphically) embedded in ISPL.

It turns out an immediate consequence of this that the classical first order predicate calculus with identity can also be embedded in the fragment (PLA) of the intuitionistic first order predicate calculus whose only logical symbols are → and (.) (universal quantifier) and the only nonlogical symbol (apart from individual variables and parentheses) a single monadic predicate letter.

Another consequence is that the classical first order predicate calculus can be embedded in the theory of Heyting algebras.

The undecidability of the formal systems under consideration evidently follows immediately from the present results.

We shall indicate how the methods employed may be extended to show also that the intuitionistic first order predicate calculus with identity can be embedded in both ISPL and PLA.

For the purpose of the present paper it will be convenient to use the following formalisation (S) of ISPL based on [3], rather than the one given above.
Weak compactness and square bracket partition relations
Authors: Kleinberg E. M. , Shore R. A.
Published: May 2006

Although there are many characterizations of weakly compact cardinals (e.g. in terms of indescnbability and tree properties as well as compactness) the most interesting set-theoretic (combinatorial) one is in terms of partition relations. To be more precise we define for κ and α cardinals and n an integer the partition relation of Erdös, Hajnal and Rado [2] as follows:

For every function F: [κ]n→ α (called a partition of [κ]n, the n-element subsets of κ, into α pieces), there exists a set C⊆ κ (called homogeneous for F) such that card C = κ and F″[C]n≠ α, i.e. some element of the range is omitted when F is restricted to the n-element subsets of C. It is the simplest (nontrivial) of these relations, i.e. , that is the well-known equivalent of weak compactness.1

Two directions of inquiry immediately suggest themselves when weak compactness is described in terms of these partition relations: (a) Trying to strengthen the relation by increasing the superscript—e.g., —and (b) trying to weaken the relation by increasing the subscript—e.g., . As it turns out, the strengthening to is only illusory for using the equivalence of to the tree property one quickly sees that implies (and so is equivalent to) for every n. Thus is the strongest of these partition relations. The second question seems much more difficult.
α-degrees of α-theories
Authors: Metakides George
Published: May 2006

Let α be a limit ordinal with the property that any “recursive” function whose domain is a proper initial segment of α has its range bounded by α. α is then called admissible (in a sense to be made precise later) and a recursion theory can be developed on it (α-recursion theory) by providing the generalized notions of α-recursively enumerable, α-recursive and α-finite. Takeuti [12] was the first to study recursive functions of ordinals, the subject owing its further development to Kripke [7], Platek [8], Kreisel [6], and Sacks [9].

Infinitary logic on the other hand (i.e., the study of languages which allow expressions of infinite length) was quite extensively studied by Scott [11], Tarski, Kreisel, Karp [5] and others. Kreisel suggested in the late '50's that these languages (even which allows countable expressions but only finite quantification) were too large and that one should only allow expressions which are, in some generalized sense, finite. This made the application of generalized recursion theory to the logic of infinitary languages appear natural. In 1967 Barwise [1] was the first to present a complete formalization of the restriction of to an admissible fragment (A a countable admissible set) and to prove that completeness and compactness hold for it. [2] is an excellent reference for a detailed exposition of admissible languages.
Completeness of an ancient logic
Authors: Corcoran John
Published: May 2006

In previous articles ([4], [5]) it has been shown that the deductive system developed by Aristotle in his “second logic” (cf. Bochenski [2, p. 43]) is a natural deduction system and not an axiomatic system as previously had been thought [6]. It was also pointed out that Aristotle's logic is self-sufficient in two senses: First, that it presupposed no other logical concepts, not even those of propositional logic; second, that it is (strongly) complete in the sense that every valid argument formable in the language of the system is demonstrable by means of a formal deduction in the system. Review of the system makes the first point obvious. The purpose of the present article is to prove the second. Strong completeness is demonstrated for the Aristotélian system.
Deducibility and many-valuedness
Authors: Shoesmith D. J. , Smiley T. J.
Published: May 2006

Lindenbaum's construction of a matrix for a propositional calculus, in which the wffs themselves are taken as elements and the theorems as the designated elements, immediately establishes two general results: that every prepositional calculus is many-valued, and that every many-valued propositional calculus is also ℵ0-valued. These results are however concerned exclusively with theoremhood, the inferential structure of the calculus being relevant only incidentally, in that it may serve to determine the set of theorems. We therefore ask what happens when deducibility is taken into consideration on a par with theoremhood. The answer is that in general the Lindenbaum construction is no longer adequate and both results fail.
Simplifications of the recursion scheme
Authors: Gladstone M. D.
Published: May 2006

This paper resolves 3 problems left open by R. M. Robinson in [3].

We recall that the set of primitive recursive functions is the closure under (i) substitution (or “composition”), and (ii) recursion, of the set P consisting of the zero, successor and projection functions (see any textbook, for instance p. 120 of [2]).
Semantics for relevant logics
Authors: Urquhart Alasdair
Published: May 2006

In what follows there is presented a unified semantic treatment of certain “paradox-free” systems of entailment, including Church's weak theory of implication (Church [7]) and logics akin to the systems E and R of Anderson and Belnap (Anderson [3], Belnap [6]). We shall refer to these systems generally as relevant logics.

The leading idea of the semantics is that just as in modal logic validity may be defined in terms of certain valuations on a binary relational structure so in relevant logics validity may be defined in terms of certain valuations on a semilattice—interpreted informally as the semilattice of possible pieces of information. Completeness theorems can be given relative to these semantics for the implicational fragments of relevant logics. The semantical viewpoint affords some insights into the structure of the systems—in particular light is thrown upon admissible modes of negation and on the assumptions underlying rejection of the “paradoxes of material implication”.

The systems discussed are formulated in fragments of a first-order language with → (entailment), &, ⋁, ¬,(x) and (∃x) primitive, omitting identity but including a denumerable list of propositional variables (p, q, r, p1,…etc.), and (for each n > 0), a denumerable list of n-ary predicate letters. The schematic letters A, B, C, D, A1,… are used on the meta-level as variables ranging over formulas. The conventions of Church [9] are followed in abbreviating formulas. The semantics of the systems are given in informal terms; it is an easy matter to turn the informal descriptions into formal set-theoretical definitions.
Nonrecursive tilings of the plane. II
Authors: Myers Dale
Published: May 2006

We show that there is a finite set of tiles which can tile the plane but not in any recursive way. This answers a natural sequel to Hao Wang's problem of the existence of a finite set of tiles which can tile the plane but not in any periodic way. In the proof, an elaboration of Robinson's method of transforming origin-constrained problems into unconstrained problems is applied to Hanf's origin-constrained tiling of Part I. We will assume familiarity with §§2, 3, and 7 of [3].

Following Robinson, we will mark the edges of our tiles with symbols and configurations of arrow heads and tails as well as with colors. The matching condition for abutting tiles will be that the symbols on adjacent edges must be identical, every arrow head must match with a tail, every tail with a head, and the colors must be the same. This is, of course, mathematically equivalent to the original condition which involved only colors. A tiling of the plane by a set of tiles is a covering of the plane with translated copies of the tiles such that adjacent edges of abutting tiles satisfy the above matching condition. A set of tiles is consistent if the plane can be tiled by the set. A set of tiles with a designated origin tile is origin-consistent if there is a tiling of the plane with the origin tile at the center.

A square block of tiles is a tiling if every pair of abutting tiles satisfies the above matching condition. If an origin tile has been designated, a block of tiles is an origin-constrained tiling if it is a tiling with the origin tile at the center. Two blocks have the “same” center row if the blocks are of the same size and have identical center rows or if the smaller block's center row is a centered segment of the larger block's center row.
The λ-calculus is ω-incomplete
Authors: Plotkin G. D.
Published: May 2006

The ω-rule in the λ-calculus (or, more exactly, the λK-β, η calculus) is

In [1] it was shown that this rule is consistent with the other rules of the λ-calculus. We will show the rule cannot be derived from the other rules; that is, we will give closed terms M and N such that MZ = NZ can be proved without using the ω-rule, for each closed term Z, but M = N cannot be so proved. This strengthens a result in [4] and answers a question of Barendregt.

The language of the λ-calculus has an alphabet containing denumerably many variables a, b, c, … (which have a standard listing e1, e2, …), improper symbolsλ, ( , ) and a single predicate symbol = for equality.

Terms are defined inductively by the following:

(1) A variable is a term.

(2) If M and N are terms, so is (MN); it is called a combination.

(3) If M is a term and x is a variable, (λx M) is a term; it is called an abstraction.

We use ≡ for syntactic identity of terms.

If M and N are terms, M = N is a formula.

BV(M), the set of bound variables in M, and FV(M), its free variables, are defined inductively by

A term M is closed iff FV(M) = ∅.
The Hanf number of second order logic
Authors: Barwise K. Jon
Published: May 2006

We prove, among other things, that the number mentioned above cannot be shown to exist without using some instance of the axiom of replacement.
Toward model theory through recursive saturation
Authors: Schlipf John Stewart
Published: May 2006

One of the most significant by-products of the study of admissible sets with urelements is the emphasis it has given to recursively saturated models. As suggested in [Schlipf, 1977], countable recursively saturated models (for finite languages) possess many of the desirable properties of saturated and special models. The notion of resplendency was introduced to isolate some of these desirable properties. In §§1 and 2 of this paper we study these parallels, showing how they can be exploited to give new proofs of some traditional model theoretic theorems. This yields both pedagogical and philosophical advantages: pedagogical since countable recursively saturated models are easier to build and manipulate than saturated and special models; philosophical since it shows that uncountable models — which the downward Lowenheim–Skolem theorem tells us are in some sense not basic in the study of countable theories — are not needed in model theoretic proofs of these theorems. In §3 we apply our local results to get results about resplendent models of ZF set theory and PA (Peano arithmetic). In §4 we shall examine certain analogous results for admissible languages, most similar to, and seemingly generally slightly weaker than, already known results. (The Chang–Makkai sort of result, however, is new.)

Although this paper is an outgrowth of work with admissible sets with urelements, I have tried to keep it as accessible as possible to those with a background only in finitary model theory. Thus §§1,2, and 3 should not involve any work with admissible sets. §4, however, is concerned with some admissible analogues to results in §2 and necessarily uses certain technical results of §11 5 of [Schlipf, 1977].
A type-free Gödel interpretation
Authors: Beeson Michael
Published: May 2006

In 1930, A. Heyting first specified a formal system for part of intuitionistic mathematics. Although his rules were presumably motivated by the “intended interpretation” or meaning of the logical symbols, over the years a number of other possible interpretations have been discovered for which the rules are also valid. In particular, one might mention the realizablity interpretation of Kleene, the (Dialectica) interpretation of Gödel, and various semantic interpretations, such as Kripke models. (Each of these has several variants or close relatives.) Each such interpretation can be regarded as defining precisely a certain “notion of constructivity”, the study of which may illuminate the still rather vague notions which underlie the intended interpretation; or, if one doubts that there is a single interpretation “intended” by all constructivist mathematicians, the study of precisely defined interpretations may help to delineate and distinguish the possibilities.

In the last few years, Heyting's systems have been vastly extended, in order to encompass the large and growing body of constructive mathematics. Several kinds of new systems have been put forward and studied. The present author has extended the various realizability interpretations to several of these systems [B1], [B2] and drawn a number of interesting applications. The mathematical content of the present paper is an interpretation in the style of Godel's Dialectica interpretation, but applicable to the new systems put forward by Feferman [Fl]. The original motivation for this work was to obtain certain metamathematical applications: roughly speaking, Markov's rule and its variants.
Some anomalies in Fitch's system QD
Authors: Bunder M. W. , Seldin Jonathan P.
Published: May 2006

This note shows that the argument used in the proof of the inconsistency of Curry's system (see [1]) can also be applied to Fitch's system QD (see [3, Chapter 6]). As one vital rule of is not present in QD this argument does not lead to an actual contradiction, but it does lead to a theorem which is not a proposition if the system is consistent.

The method used below is that of [2], which is simpler than that of [1].

The axioms and rules of QD that we require are also present in [1] and [2] provided that Fitch's D is replaced by H. These axioms and rules are the following:

DD int: ⊦ D(Da),

m p: If ⊦ a ⊃ b2 and ⊦ a then ⊦ b, and

res imp int: If a ⊦ b then Da ⊦ a ⊃ b.

Let a be arbitrary and let G be [x](Dx ⊃ (x ⊃ a)). Then let X be BWBG(BWBG). We then have the following proof:

Thus, we have ⊦ X. Now suppose we also had ⊦ DX. Then by the method of the innermost subproof in the above proof, we would have ⊦ a, and since a is

arbitrary the system would be inconsistent. Hence, if QD is consistent, we do not have ⊦ DX, and so X is a theorem which is not a proposition.