Set theory, it has been contended, developed from its beginnings through a progression of mathematical moves, despite being intertwined with pronounced metaphysical attitudes and exaggerated foundational claims that have been held on its behalf. In this paper, the seminal results of set theory are woven together in terms of a unifying mathematical motif, one whose transmutations serve to illuminate the historical development of the subject. The motif is foreshadowed in Cantor's diagonal proof, and emerges in the interstices of the inclusion vs. membership distinction, a distinction only clarified at the turn of this century, remarkable though this may seem. Russell runs with this distinction, but is quickly caught on the horns of his well-known paradox, an early expression of our motif. The motif becomes fully manifest through the study of functions of the power set of a set into the set in the fundamental work of Zermelo on set theory. His first proof in 1904 of his Well-Ordering Theoremis a central articulation containing much of what would become familiar in the subsequent development of set theory. Afterwards, the motif is cast by Kuratowski as a fixed point theorem, one subsequently abstracted to partial orders by Bourbaki in connection with Zorn's Lemma. Migrating beyond set theory, that generalization becomes cited as the strongest of fixed point theorems useful in computer science.

Section 1 describes the emergence of our guiding motif as a line of development from Cantor's diagonal proof to Russell's Paradox, fueled by the clarification of the inclusion vs. membership distinction. Section 2 engages the motif as fully participating in Zermelo's work on the Well-Ordering Theorem and as newly informing on Cantor's basic result that there is no bijection . Then Section 3 describes in connection with Zorn's Lemma the transformation of the motif into an abstract fixed point theorem, one accorded significance in computer science.

Introduction. This survey is intended to introduce to logicians some notions, methods and theorems in set theory which arose—largely through the work of Saharon Shelah—out of (successful) attempts to solve problems in abelian group theory, principally the Whitehead problem and the closely related problem of the existence of almost free abelian groups. While Shelah's first independence result regarding the Whitehead problem used established set-theoretical methods (discussed below), his later work required new ideas; it is on these that we focus. We emphasize the nature of the new ideas and the historical context in which they arose, and we do not attempt to give precise technical definitions in all cases, nor to include a comprehensive survey of the algebraic results.

In fact, very little algebraic background is needed beyond the definitions of group and group homomorphism. Unless otherwise specified, we will use the word “group” to refer to an abelian group, that is, the group operation is commutative. The group operation will be denoted by +, the identity element by 0, and the inverse of a by −a. We shall use na as an abbreviation for a + a + … + a [n times] if n is positive, and na = (−n)(−a) if n is negative.

We present a survey of the recent applications of continuous domains for providing simple computational models for classical spaces in mathematics including the real line, countably based locally compact spaces, complete separable metric spaces, separable Banach spaces and spaces of probability distributions. It is shown how these models have a logical and effective presentation and how they are used to give a computational framework in several areas in mathematics and physics. These include fractal geometry, where new results on existence and uniqueness of attractors and invariant distributions have been obtained, measure and integration theory, where a generalization of the Riemann theory of integration has been developed, and real arithmetic, where a feasible setting for exact computer arithmetic has been formulated. We give a number of algorithms for computation in the theory of iterated function systems with applications in statistical physics and in period doubling route to chaos; we also show how efficient algorithms have been obtained for computing elementary functions in exact real arithmetic.

As an undergraduate I was taught to multiply two numbers with the help of log tables, using the formula

Having graduated to teach calculus to Engineers, I learned that log tables were to be replaced by slide rules. It was then that Imade the fateful decision that there was no need for me to learn how to use this tedious device, as I could always rely on the students to perform the necessary computations. In the course of time, slide rules were replaced by pocket calculators and personal computers, but I stuck to my original decision.

My computer phobia did not prevent me from taking an interest in the theoretical question: what is a computation? This question goes back to Hilbert's 10th problem (see Browder [1976]), which asks for an algorithm, or computational procedure, to decide whether any given polynomial equation is solvable in integers. It quickly leads to the related question: which numerical functions f : ℕn → ℕ are computable?

While Hilbert's 10th problem was only resolved in 1970, this related question had some earlier answers, of which I shall single out the following three:

(1) f is recursive (Gödel, Kleene),

(2) f is computable on an abstract machine (Turing, Post),

(3) f is definable in the untyped λ-calculus (Church, Kleene).

These tentative answers were shown to be equivalent by Church [1936] and Turing [1936–7].

I shall discuss here some of the more recent developments of these notions of computability and their relevance to linguistics and logic. I hope to be forgiven for dwelling on some of the work I have been involved with personally, with greater emphasis than is justified by its historical significance.

We describe a model-theoretic approach to ordinal analysis via the finite combinatorial notion of an α-large set of natural numbers. In contrast to syntactic approaches that use cut elimination, this approach involves constructing finite sets of numbers with combinatorial properties that, in nonstandard instances, give rise to models of the theory being analyzed. This method is applied to obtain ordinal analyses of a number of interesting subsystems of first- and second-order arithmetic.

We present here an approach to the fine structure of L based solely on elementary model theoretic ideas, and illustrate its use in a proof of Global Square in L. We thereby avoid the Lévy hierarchy of formulas and the subtleties of master codes and projecta, introduced by Jensen [3] in the original form of the theory. Our theory could appropriately be called ”Hyperfine Structure Theory”, as we make use of a hierarchy of structures and hull operations which refines the traditional Lα -or Jα-sequences with their Σn-hull operations.

§1. Introduction. In 1938, K. Gödel defined the model L of set theory to show the relative consistency of Cantor's Continuum Hypothesis. L is defined as a union

of initial segments which satisfy: L0 = ∅, Lλ = ∪α<λLα for limit ordinals λ, and, crucially, Lα + 1 = the collection of 1st order definable subsets of Lα. Since every transitive model of set theory must be closed under 1st order definability, L turns out to be the smallest inner model of set theory. Thus it occupies the central place in the set theoretic spectrum of models.

The proof of the continuum hypothesis in L is based on the very uniform hierarchical definition of the L-hierarchy. The Condensation Lemma states that if π : M → Lα is an elementary embedding, M transitive, then some ; the lemma can be proved by induction on α. If a real, i.e., a subset of ω, is definable over some Lα,then by a Löwenheim-Skolem argument it is definable over some countable M as above, and hence over some , < ω1. This allows one to list the reals in L in length ω1 and therefore proves the Continuum Hypothesis in L.

Alonzo Church's mathematical work on computability and undecidability is well-known indeed, and we seem to have an excellent understanding of the context in which it arose. The approach Church took to the underlying conceptual issues, by contrast, is less well understood. Why, for example, was “Church's Thesis” put forward publicly only in April 1935, when it had been formulated already in February/March 1934? Why did Church choose to formulate it then in terms of Gödel's general recursiveness, not his own λ-definability as he had done in 1934? A number of letters were exchanged between Church and Paul Bernays during the period from December 1934 to August 1937; they throw light on critical developments in Princeton during that period and reveal novel aspects of Church's distinctive contribution to the analysis of the informal notion of effective calculability. In particular, they allow me to give informed, though still tentative answers to the questions I raised; the character of my answers is reflected by an alternative title for this paper, Why Church needed Gödel's recursiveness for his Thesis. In Section 5, I contrast Church's analysis with that of Alan Turing and explore, in the very last section, an analogy with Dedekind's investigation of continuity.

One of the most important contributions of A. Church to logic is his invention of the lambda calculus. We present the genesis of this theory and its two major areas of application: the representation of computations and the resulting functional programming languages on the one hand and the representation of reasoning and the resulting systems of computer mathematics on the other hand.

We use the category of presheaves over PTIME-functions in order to show that Cook and Urquhart's higher-order function algebra PVω defines exactly the PTIME-iunctions. As a byproduct we obtain a syntax-free generalisation of PTIME-computability to higher types.

By restricting to sheaves for a suitable topology we obtain a model for intuitionistic predicate logic with -induction over PVω and use this to re-establish that the provably total functions in this system are polynomial time computable. Finally, we apply the category-theoretic approach to a new higher-order extension of Bellantoni-Cook's system BC of safe recursion.

We announce two new dichotomy theorems for Borel equivalence relations, and present the results in context by giving an overview of related recent developments.

§1. Introduction. For X a Polish (i.e., separable, completely metrizable) space and E a Borel equivalence relation on X, a (complete) classification of X up to E-equivalence consists of finding a set of invariants I and a map c : X → I such that xEy ⇔ c(x) = c(y). To be of any value we would expect I and c to be “explicit” or “definable”. The theory of Borel equivalence relations investigates the nature of possible invariants and provides a hierarchy of notions of classification.

The following partial (pre-)ordering is fundamental in organizing this study. Given equivalence relations E and F on X and Y, resp., we say that E can be Borel reduced to F, in symbols

if there is a Borel map f : X → Y with xEy ⇔ f(x)Ff(y). Then if is an embedding of X/E into Y/F, which is “Borel” (in the sense that it has a Borel lifting).

Intuitively, E ≤BF might be interpreted in any one of the following ways:

(i) The classi.cation problem for E is simpler than (or can be reduced to) that of F: any invariants for F work as well for E (after composing by an f as above).

(ii) One can classify E by using as invariants F-equivalence classes.

(iii) The quotient space X/E has “Borel cardinality” less than or equal to that of Y/F, in the sense that there is a “Borel” embedding of X/E into Y/F.

We identify the computational complexity of the satisfiability problem for FO2, the fragment of first-order logic consisting of all relational first-order sentences with at most two distinct variables. Although this fragment was shown to be decidable a long time ago, the computational complexity of its decision problem has not been pinpointed so far. In 1975 Mortimer proved that FO2 has the finite-model property, which means that if an FO2-sentence is satisiable, then it has a finite model. Moreover, Mortimer showed that every satisfiable FO2-sentence has a model whose size is at most doubly exponential in the size of the sentence. In this paper, we improve Mortimer's bound by one exponential and show that every satisfiable FO2-sentence has a model whose size is at most exponential in the size of the sentence. As a consequence, we establish that the satisfiability problem for FO2 is NEXPTIME-complete.

Tarski's 1936 paper, “On the concept of logical consequence”, is a rather philosophical, non-technical paper that leaves room for conflicting interpretations. My purpose is to review some important issues that explicitly or implicitly constitute its themes. My discussion contains four sections: (1) terminological and conceptual preliminaries, (2) Tarski's definition of the concept of logical consequence, (3) Tarski's discussion of omega-incomplete theories, and (4) concluding remarks concerning the kind of conception that Tarski's definition was intended to explicate. The third section involves subsidiary issues, such as Tarski's discussion concerning the distinction between material and formal consequence and the important question ofthe criterion for distinguishing between logical and non-logical terms.

§1. Preliminaries. In this paper an argument is a two-part system composed of a set of propositions P (the premise-set) and a single proposition c (the conclusion). The expression ‘c is a [logical] consequence of P’ is used with the same meaning as the expression ‘c is [logically] implied by P’. The expressions ‘is a logical consequence of’ and the converse ‘implies’ are relational. Often, I shall be talking in the same sense of validity of an argument. Validity is a property of arguments; an argument with premise-set P and conclusion c is valid if and only if P implies c; i.e., c is a logical consequence of P. Notice that this notion of argument is strictly ontic; it does not involve any agent that thinks, determines or establishes that a given proposition is or is not a consequence of a given set of propositions.

A new games model of the language FPC, a type theory with products, sums, function spaces and recursive types, is described. A definability result is proved, showing that every finite element of the model is the interpretation of some term of the language.