In treatises or advanced textbooks on theoretical physics, it is apparent that the way mathematics is used is very different from what is to be found in books of mathematics. There is, for example, no close connection between books on analysis, on the one hand, and any classical textbook in quantum mechanics, for example, Schiff, [11], or quite recent books, for example Ryder, [10], on quantum field theory. The differences run a good deal deeper than the fact that the books on theoretical physics are not written in the definition-theorem-proof style characteristic of pure mathematics. Although a good many propositions are proved in the books on physics, there are almost with exception no existential proofs, and consequently there is no really serious systematic use of quantifiers. Another important characteristic is the free use of infinitesimals. In fact, most results would not lose anything, from a physicist's point of view, by leaving them in approximate form, i.e., instead of strict equalities or inequalities, using equalities or inequalities only up to an infinitesimal.

The discrepancy between the way mathematics is ordinarily done in theoretical physics and the way it is built up from a foundational standpoint in any of the standard modern views raises the question of whether it might be possible to construct quite directly a rigorous foundation that reflects a significant part of this standard practice in theoretical physics. Other parts of standard practice in physics, for example, the use of physically intuitive but nonrigorous arguments, are not present in our system.

Can a supercompact cardinal κ be Laver indestructible when there is a level-by-level agreement between strong compactness and supercompactness? In this article, we show that if there is a sufficiently large cardinal above κ, then no, it cannot. Conversely, if one weakens the requirement either by demanding less indestructibility, such as requiring only indestructibility by stratified posets. or less level-by-level agreement, such as requiring it only on measure one sets, then yes. it can.

A typical approach to semantics for relevance (and other) logics: specify a class of algebraic structures and take a model to be one of these structures, α, together with some function or relation which associates with every formula A a subset of α. (This is the approach of, among others, Urquhart, Routley and Meyer and Fine.) In some cases there are restrictions on the class of subsets of α with which a formula can be associated: for example, in the semantics of Routley and Meyer [1973], a formula can only be associated with subsets which are closed upwards. It is natural to take a proposition of α to be such a subset of α, and, further, to take the propositional quantifiers to range over these propositions. (Routley and Meyer [1973] explicitly consider this interpretation.) Given such an algebraic semantics, we call (following Routley and Meyer [1973], who follow Henkin [1950]) the above-described interpretation of the quantifiers the primary interpretation associated with the semantics.

A logical calculus Κ was defined in a previous paper and then shown in a subsequent paper to contain within itself a representation of every constructively definable subclass of expressions of a certain infinite class U of expressions, where Κ itself is one such subclass. (The former paper will be referred to as BL and the latter paper as RC.) The calculus Κ was called a “basic calculus” and its theorems were thought of as expressing the asserted propositions of a “basic logic,” that is, of a logic within which is definable every constructively definable system of logic and indeed every constructively definable class or relation. The notion of constructive definability was essentially equated with the notion of recursive enumerability.

Since Gilmore showed that some theory with a positive comprehension scheme is consistent when the axiom of extensionality is dropped and inconsistent with it (see [1] and [2]), the problem of the consistency of various positive comprehension schemes has been investigated. We give here a short classification, which shows clearly the importance of the axiom of extensionality and of the abstraction operator in these consistency problems. The most difficult problem was to show the consistency of the comprehension scheme for positive formulas, with extensionality but without abstraction operator. In his unpublished thesis, Set theory in which the axiom of foundation fails [3], Malitz solved partially this problem but he needed to assume the existence of some unusual kind of large cardinal; as his original construction is very interesting and his thesis is unpublished, we give a short summary of it. M. Forti solved the problem completely by working in ZF with a free-construction principle (sometimes called an anti-foundation axiom), instead of ZF with the axiom of foundation, as Malitz did.

This permits one to obtain the consistency of this positive theory, relative to ZF. In his general investigations about “topological set theories” (to be published), E. Weydert has independently proved the same result. The authors are grateful to the Mathematisches Forshungsinstitut Oberwolfach for giving them the opportunity of discussing these subjects and meeting E. Weydert during the meeting “New Foundations”, March 1–7, 1987.

In this paper we introduce the notion of a first order topological structure, and consider various possible conditions on the complexity of the definable sets in such a structure, drawing several consequences thereof.

Our aim is to develop, for a restricted class of unstable theories, results analogous to those for stable theories. The “material basis” for such an endeavor is the analogy between the field of real numbers and the field of complex numbers, the former being a “nicely behaved” unstable structure and the latter the archetypal stable structure. In this sense we try here to situate our work on o-minimal structures [PS] in a general topological context. Note, however, that the p-adic numbers, and structures definable therein, will also fit into our analysis.

In the remainder of this section we discuss several ways of studying topological structures model-theoretically. Eventually we fix on the notion of a structure in which the topology is “explicitly definable” in the sense of Flum and Ziegler [FZ]. In §2 we introduce the hypothesis that every definable set is a Boolean combination of definable open sets. In §3 we introduce a “dimension rank” on (closed) definable sets. In §4 we consider structures on which this rank is defined, and for which also every definable set has a finite number of definably connected definable components. We show that prime models over sets exist under such conditions.

The purpose of this paper is to show that some of the difficulties confronting logicians are caused by too narrow an interpretation of the function of mathematical logic. I hope to show that very fruitful results can be obtained by considering a much wider range of problems than usually found in the literature.

I believe that the function of mathematical logic is to study all logical systems. The problems now considered usually concern some particular system or set of systems. Of course we would have to agree first on what a logical system is. I shall suggest a tentative definition. The rest of my paper will proceed on the assumption that we have agreed on this question. However, it is only necessary for this paper that some definition should be accepted, it is not necessary that it should be the one I suasest.

Assume ZFC + “There is a weakly compact cardinal” is consistent. Then:

(i) For every S ⊆ ω, ZFC + “S and the monadic theory of ω2 are recursive each in the other” is consistent; and

(ii) ZFC + “The full second-order theory of ω2 is interpretable in the monadic theory of ω2” is consistent.

Our main goal in this paper is to establish a Glimm-Effros type dichotomy for arbitrary analytic equivalence relations.

The original Glimm-Effros dichotomy, established by Effros [Ef], [Ef1], who generalized work of Glimm [G1], asserts that if an Fσ equivalence relation on a Polish space X is induced by the continuous action of a Polish group G on X, then exactly one of the following alternatives holds:

(I) Elements of X can be classified up to E-equivalence by “concrete invariants” computable in a reasonably definable way, i.e., there is a Borel function f: X → Y, Y a Polish space, such that xEy ⇔ f(x) = f(y), or else

(II) E contains a copy of a canonical equivalence relation which fails to have such a classification, namely the relation xE0y ⇔ ∃n∀m ≥ n(x(n) = y(n)) on the Cantor space 2ω (ω = {0,1,2, …}), i.e., there is a continuous embedding g: 2ω → X such that xE0y ⇔ g(x)Eg(y).

Moreover, alternative (II) is equivalent to:

(II)′ There exists an E-ergodic, nonatomic probability Borel measure on X, where E-ergodic means that every E-invariant Borel set has measure 0 or 1 and E-nonatomic means that every E-equivalence class has measure 0.

We confirm a conjecture, about neat embeddings of cylindric algebras, made in 1969 by J. D. Monk, and a later conjecture by Maddux about relation algebras obtained from cylindric algebras. These results in algebraic logic have the following consequence for predicate logic: for every finite cardinal α ≥ 3 there is a logically valid sentence X, in a first-order language ℒ with equality and exactly one nonlogical binary relation symbol E, such that X contains only 3 variables (each of which may occur arbitrarily many times), X has a proof containing exactly α + 1 variables, but X has no proof containing only α variables. This solves a problem posed by Tarski and Givant in 1987.

This paper deals with Hilbert's substitution method for eliminating bound variables from first order proofs. With a first order system S framed in the ε-calculus [2] the problem is to associate a system S' without bound variables and an effective procedure for transforming derivations in S into derivations in S′. The transform of a formula A derived in S is to be an “ε-substitution instance” of A, i.e. it is obtained by replacing terms εxB(x) in A by terms of S′. In general the choice of these terms will depend on the particular derivation of A, and not on A alone. Cf. [4]. The present formulation sharpens Hilbert's original statement of the problem, i.e. that the transform of A should be finitistically verifiable, by making explicit the methods of verification used, namely those formalized in S′; on the other hand, it generalizes Hilbert's formulation since S′ need not be restricted to finitist systems.

The bound variable elimination procedure can always be taken to be primitive recursive in (the Gödel number of) the derivation of A. Constructions which transcend primitive recursion can simply be built into S′.

In this paper we show that if S′ is taken to be a second order system with constants for functionals, then the existence of suitable ε-substitution instances can be expressed by the solvability of certain functional equations in S′. We deal with two cases here. If S is number theory without induction, i.e. essentially predicate calculus with identity, then we can solve the equations in question by taking for S′ the free variable part S* of S with an added rule of definition of functionals by cases (recursive definition on finite ordinals), which is a conservative extension of S*.

The purpose of the present article is to formulate in an intuitionistically meaningful manner the completeness problems for the intuitionistic predicate calculus, and to establish the completeness of certain fragments of it. In these proofs certain translations of classical logic into intuitionistic logic are used, in particular those discovered by Kolmogorov [11] and Gödel [3], and a new one in which negations of prenex formulae are central. Since the last one is of interest also independently of the completeness problems, the details are presented separately at the end of this paper.

All arguments are intended to be intuitionistically valid unless the opposite is stated; in particular, ‘proof’ means intuitionistic proof, and ‘formal proof’ means proof in the relevant system of Heyting [4], [5].

Open problems are mentioned in Remarks 3.1 and 8.1.

German capitals denote formulae of predicate logic, Latin (italic) capitals denote predicate symbols or propositional letters.

A Turing degree a is said to be almost everywhere dominating if, for almost all X ∈ 2ω with respect to the “fair coin” probability measure on 2ω, and for all g: ω → ω Turing reducible to X, there exists f: ω → ω of Turing degree a which dominates g. We study the problem of characterizing the almost everywhere dominating Turing degrees and other, similarly defined classes of Turing degrees. We relate this problem to some questions in the reverse mathematics of measure theory.

The concern of this paper is with recursively enumerable and co-recursively enumerable subspaces of a recursively presented vector spaceV∞ over a (finite or infinite) recursive field F which is defined in [6] to consist of a recursive subset U of the natural numbers N and operations of vector addition and scalar multiplication which are partial recursive and under which V∞ becomes a vector space. Throughout this paper, we will identify V∞ with N, say via some fixed Gödel numbering, and assume V∞ is infinite dimensional and has a dependence algorithm, i.e., there is a uniform effective procedure which determines whether or not any given n-tuple v0, …, vn−1 from V∞ is linearly dependent. Various properties of V∞ and its sub-spaces have been studied by Dekker [1], Guhl [3], Metakides and Nerode [6], Kalantari and Retzlaff [4], and the author [7].

Given a subspace W of V∞, we say W is r.e. (co-r.e.) if W(V∞ − W) is an r.e. subset of N and write dim(V) for the dimension of V. Given subspaces V, W of V∞, V + W will denote the weak sum of V and W and if V ⋂ M = {0} (where 0 is the zero vector of V∞), we write V ⊕ Winstead of V + W. If W ⊇ V, we write Wmod V for the quotient space. An independent set A ⊆ V∞ is extendible if there is an r.e. independent set I ⊇ A such that I − A is infinite and A is nonextendible if it is not the case An is extendible. A r.e. subspace M ⊇ V∞ is maximal if dim(V∞ mod M) = ∞ and for any r.e. subspace W ⊇ Meither dim(W mod M) < ∞ or dim(V∞ mod W) < ∞.

Let P0 be the subsystem of Peano arithmetic obtained by restricting induction to bounded quantifier formulas. Let M be a countable, nonstandard model of P0 whose domain we suppose to be the standard integers. Let T be a recursively enumerable extension of Peano arithmetic all of whose existential consequences are satisfied in the standard model. Then there is an initial segment M′ of M which is a model of T such that the complete diagram of M′ is Turing reducible to the atomic diagram of M. Moreover, neither the addition nor the multiplication of M is recursive.

Let E be a computably enumerable (c.e.) equivalence relation on the set ω of natural numbers. We say that the quotient set $\omega /E$ (or equivalently, the relation E) realizes a linearly ordered set ${\cal L}$ if there exists a c.e. relation ⊴ respecting E such that the induced structure ($\omega /E$; ⊴) is isomorphic to ${\cal L}$. Thus, one can consider the class of all linearly ordered sets that are realized by $\omega /E$; formally, ${\cal K}\left( E \right) = \left\{ {{\cal L}\,|\,{\rm{the}}\,{\rm{order}}\, - \,{\rm{type}}\,{\cal L}\,{\rm{is}}\,{\rm{realized}}\,{\rm{by}}\,E} \right\}$. In this paper we study the relationship between computability-theoretic properties of E and algebraic properties of linearly ordered sets realized by E. One can also define the following pre-order $ \le _{lo} $ on the class of all c.e. equivalence relations: $E_1 \le _{lo} E_2 $ if every linear order realized by E1 is also realized by E2. Following the tradition of computability theory, the lo-degrees are the classes of equivalence relations induced by the pre-order $ \le _{lo} $. We study the partially ordered set of lo-degrees. For instance, we construct various chains and anti-chains and show the existence of a maximal element among the lo-degrees.

Retraceable sets were denned in Dekker and Myhill [5]. In that paper it was pointed out that if A is a retraceable set, then A is recursive in each of its infinite subsets, i.e. A is introreducible. We would like to know to what extent theorems on retraceable sets also hold for introreducible sets; however, we have made very little progress in this direction. On the other hand, “uniformly introreducible” sets (which will be defined in §2) are more tractable, and it turns out that some results on retraceable sets extend to uniformly introreducible sets while others fail.

We give a direct, purely arithmetical and elementary proof of the strong normalization of the cut-elimination procedure for full (i.e., in presence of all the usual connectives) classical natural deduction.

Limiting identification of r.e. indexes for r.e. languages (from a presentation of elements of the language) and limiting identification of programs for computable functions (from a graph of the function) have served as models for investigating the boundaries of learnability. Recently, a new approach to the study of “intrinsic” complexity of identification in the limit has been proposed. This approach, instead of dealing with the resource requirements of the learning algorithm, uses the notion of reducibility from recursion theory to compare and to capture the intuitive difficulty of learning various classes of concepts. Freivalds, Kinber, and Smith have studied this approach for function identification and Jain and Sharma have studied it for language identification.

The present paper explores the structure of these reducibilities in the context of language identification. It is shown that there is an infinite hierarchy of language classes that represent learning problems of increasing difficulty. It is also shown that the language classes in this hierarchy are incomparable, under the reductions introduced, to the collection of pattern languages.

Richness of the structure of intrinsic complexity is demonstrated by proving that any finite, acyclic, directed graph can be embedded in the reducibility structure. However, it is also established that this structure is not dense. The question of embedding any infinite, acyclic, directed graph is open.