Angus Macintyre

The Type Theoretic Interpretation of Constructive Set Theory

By adding to Martin-LSfs intuitionistic theory of types a ‘type of sets’ we give a constructive interpretation of constructive set theory. constructive version of the classical conception of the cumulative hierarchy of sets. This interpretation is a

Trends in Logic

Publisher Summary This chapter describes the dynamics of model theory and some threats of set theory. It is often risky to predict movements in logic. It is noted that something remains of forcing and reduced products in categorical model theory, which provides universe and semantics more relevant than the set-theoretic ones. A clear tendency is to focus on definability rather than decidability or structure theory. Most of the decidability results identify definable sets, and relations, specifically equivalence relations. A major contemporary theme is the topological structure of definable sets. Kreisel had stressed the gigantic difference in importance between algebraic topology and set-theoretic topology. The latter is ubiquitous in routine arguments and formulations, but the former is effective in advancing mathematical understanding. Applied model theory is often concerned with number theory or analytic function theory.

On definable subsets of p -adic fields

The brilliant work of Ax-Kochen [1], [2], [3] and Ersov [6], and later work by Kochen [7] have made clear very striking resemblances between real closed fields and p -adically closed fields, from the model-theoretical point of view. Cohen [5], from a standpoint less model-theoretic, also contributed much to this analogy. In this paper we shall point out a feature of all the above treatments which obscures one important resemblance between real and p -adic fields. We shall outline a new treatment of the p -adic case (not far removed from the classical treatments cited above), and establish an new analogy between real closed and p -adically closed fields. We want to describe the definable subsets of p -adically closed fields. Tarski [9] in his pioneering work described the first-order definable subsets of real closed fields. Namely, if K is a real-closed field and X is a subset of K first-order definable on K using parameters from K then X is a finite union of nonoverlapping intervals (open, closed, half-open, empty or all of K ). In particular, if X is infinite, X has nonempty interior. Now, there is an analogous question for p -adically closed fields. If K is p -adically closed, what are the definable subsets of K ? To the best of our knowledge, this question has not been answered until now. What is the difference between the two cases? Tarskis analysis rests on elimination of quantifiers for real closed fields. Elimination of quantifiers for p -adically closed fields has been achieved [3], but only when we take a cross-section π as part of our basic data. The problem is that in the presence of π it becomes very difficult to figure out what sort of set is definable by a quantifier free formula. We shall see later that use of the cross-section increases the class of definable sets.

Polynomial Bounds for VC Dimension of Sigmoidal and General Pfaffian Neural Networks

We introduce a new method for proving explicit upper bounds on the VC dimension of general functional basis networks and prove as an application, for the first time, that the VC dimension of analog neural networks with the sigmoidal activation function?(y)=1/1+e?yis bounded by a quadratic polynomialO((lm)2) in both the numberlof programmable parameters, and the numbermof nodes. The proof method of this paper generalizes to much wider class of Pfaffian activation functions and formulas and gives also for the first time polynomial bounds on their VC dimension. We present also some other applications of our method.

On algebraically closed groups

In this paper we use model-theoretic techniques to analyze the structure of algebraically closed groups. The notion of algebraically closed group first appeared in W. R. Scotts paper [24] in 1951. The intention must surely have been to provide for grouptheory an analogue of that central notion of field theory, the notion of algebraically closed field. A group G is said to be algebraically closed if every consistent finite system of equations, with parameters in G, is solvable in G. A system of equations is said to be consistent over G if it has a solution in a group extending G. A group G is said to be existentially closed if every consistent finite system of equations and inequations, with parameters in G, is solvable in G.

Polynomial bounds for VC dimension of sigmoidal neural networks

We introduce a new method for proving explicit upper bounds on the VC Dimension of general functional basis networks, and prove as an application, for the first time, the VC Dimension of analog neural networks with the sigmoid activation function o(y) = 1/1 + e-y to be bounded by a quadratic polynomial in the number of programmable parameters.

Definable sets over finite fields

We give here a new and (modulo the Lang-Weil estimates) selfcontained treatment of various logical aspects of finite (and pseudo-finite) fields. In particular we get new results on definable sets with (*) s an obvious consequence. Our main result extends the Lang-Weil estimates on absolutely irreducible varieties to arbitrary formulas with parameters. However, a finite number of case distinctions, depending on the formula, becomes necessary. Here is a precise formulation.

Totally categorical groups and rings

This paper is a contribution to applied categoricity theory. We try to classify isomorphism theorems in a certain area of algebra, by systematic use of the powerful model-theoretic methods of Morley, Shelah, and Ryll-Nardzewski. As far as we know, no results in applied categoricity were published before 1969. Since then the literature has grown quite rapidly. Before stating our main results, we give a survey of the previously known results. The main concepts under study have been K,,-categoricity, N,-categoricity, and w-stability. These concepts apply either to complete theories or to individual structures. Shelah’s notions of superstability and stability are receiving increasing attention, but only in the case of Abelian groups can we give them useful algebraic characterizations at present. In the case of &,-categoricity, satisfactory analyses have been given in the following cases:

Elimination of quantifiers in algebraic structures

One of the first techniques used in the study of the logical properties of algebraic objects was that of elimination of quantifiers. It was, for example, used by Tarski [ 161 to prove the decidability of the theory of any given algebraically closed field and the decidability of the theory of real closed ordered fields. The theories of various other algebraic structures (e.g., the padic numbers, divisible ordered abelian groups) may also be shown to eliminate quantifiers in appropriate languages. The choice of the language is especially important in these dealings. It must be chosen to lie close to natural algebraic phenomena. Any structure may be made to admit elimination of quantifiers by the choice of a sufficiently complex language: the structure may be Skolemized or Morleyized [ 151. Clearly the resulting languages are too far from the natural relations on the objects to be of immediate algebraic interest. In this paper we will show that for many natural languages those structures which have been shown in the past to admit elimination of quantifiers are, in fact, the only structures of their type to admit elimination of quantifiers in that language. In particular we will show that apart from the finite fields and the algebraically closed fields there are no other fields which admit elimination of quantifiers in the language of rings with identity (i.e., the language with primitives +, ., -, 0, l).’ We will show that the only theory of ordered fields that admits elimination of quantifiers in the natural language of ordered domains is the theory of real closed ordered fields. We show that if K is a nontrivially valued field admitting elimination of quantifiers in the language of valued fields then K is algebraically closed. Now the theory of the p-adic numbers does not admit elimination of quantifiers in the pure language of valued fields (cf. [ 121). However, if we include certain auxilary predicates {P,(x)} such that P,(x) holds if and only if x has an nth

Logarithmic-Exponential Power Series

We use generalized power series to construct algebraically a nonstandard model of the theory of the real field with exponentiation. This model enables us to show the undefinability of the zeta function and certain non-elementary and improper integrals. We also use this model to answer a question of Hardy by showing that the compositional inverse to the function (log x ) (log log x ) is not asymptotic as x →+∞ to a composition of semialgebraic functions, log and exp.