this paper we show how to combine the unrestricted countable choice, induction on infinite well-founded trees and restricted classical logic in a constructively given model. For readers faniliar with intuitionistic systems [14], we may succinctly describe the theory we interpret as follows. Expand the extensional version of HA

Elementary arithmetic (also known as “elementary function arithmetic”) is a fragment of first-order arithmetic so weak that it cannot prove the totality of an iterated exponential function. Surprisingly, however, the theory turns out to be remarkably robust. I will discuss formal results that show that many theorems of number theory and combinatorics are derivable in elementary arithmetic, and try to place these results in a broader philosophical context. 1

This short report presents the main topics of methods of cut-elimination which will be presented in the course at the ESSLLI'99. It gives a short introduction addressing the problem of cut-elimination in general. Furthermore we give a brief description of several methods and refer to other papers added to the course material.

Labelled sequent calculi are provided for a wide class of normal modal systems using truth values as labels. The rules for formula constructors are common to all modal systems. For each modal system, specific rules for truth values are provided that reflect the envisaged properties of the accessibility relation. Both local and global reasoning are supported. Strong completeness is proved for a natural two-sorted algebraic semantics. As a corollary, strong completeness is also obtained over general Kripke semantics. A duality result

Abstract. Paul Cohen's method of forcing, together with Saul Kripke's related semantics for modal and intuitionistic logic, has had profound effects on a number of branches of mathematical logic, from set theory and model theory to constructive and categorical logic. Here, I argue that forcing also has a place in traditional Hilbert-style proof theory, where the goal is to formalize portions of ordinary mathematics in restricted axiomatic theories, and study those theories in constructive or syntactic terms. I will discuss the aspects of forcing that are useful in this respect, and some sample applications. The latter include ways of obtaining conservation results for classical and intuitionistic theories, interpreting classical theories in constructive ones, and constructivizing model-theoretic arguments.?1. Introduction. In 1963, Paul Cohen introduced the method of forcing to prove the independence of both the axiom of choice and the continuum hypothesis from Zermelo-Fraenkel set theory. It was not long before Saul Kripke noted a connection between forcing and his semantics for modal and

It is shown that the concepts of AW*-algebras and their types are the same both in the ordinary universe and in Scott's and Solovay's Boolean valued universe of ZFC set theory. Using this transfer principle, it is proved that a finite AW*-algebra has a centre-valued trace if and only if its centre is the range of a faithful norm one projection. 1.

Both the constructive and predicative approaches to mathemat-ics arose during the period of what was felt to be a foundational crisis in the early part of this century. Each critiqued an essential logical aspect of classical mathematics, namely concerning the unre-stricted use of the law of excluded middle on the one hand, and of apparently circular \impredicative " de nitions on the other. But the positive redevelopment of mathematics along constructive, resp. pred-icative grounds did not emerge as really viable alternatives to classical, set-theoretically based mathematics until the 1960s. Now wehave a massive amount of information, to which this lecture will constitute an introduction, about what can be done by what means, and about the theoretical interrelationships between various formal systems for constructive, predicative and classical analysis. In this nal lecture I will be sketching some redevelopments of classical analysis on both constructive and predicative grounds, with an emphasis on modern approaches. In the case of constructivity, Ihave very little to say about Brouwerian intuitionism, which has been discussed extensively in other lectures at this conference, and concentrate instead on the approach since 1967 of Errett Bishop and his school. In the case of predicativity, I concentrate on developments|also since the 1960s|which take up where Weyl's work left o, as described in my second lecture. In both cases, I rst look at these redevelopments from a more informal, mathematical, point This is the last of my three lectures for the conference, Proof Theory: History and

In 1981, Takeuti introduced quantum set theory as the quantum counterpart of Boolean valued models of set theory by constructing a model of set theory based on quantum logic represented by the lattice of closed subspaces in a Hilbert space and showed that appropriate quantum counterparts of ZFC axioms hold in the model. Here, Takeuti’s formulation is extended to construct a model of set theory based on the logic represented by the lattice of projections in an arbitrary von Neumann algebra. A transfer principle is established that enables us to transfer theorems of ZFC to their quantum counterparts holding in the model. The set of real numbers in the model is shown to be in one-to-one correspondence with the set of self-adjoint operators affiliated with the von Neumann algebra generated by the logic. Despite the difficulty pointed out by Takeuti that equality axioms do not generally hold in quantum set theory, it is shown that equality axioms hold for any real numbers in the model. It is also shown that any observational proposition in quantum mechanics can be represented by a corresponding statement for real numbers in the model with the truth value consistent with the standard formulation of quantum mechanics, and that the equality relation between two real numbers in the model is equivalent with the notion of perfect correlation between corresponding observables (self-adjoint operators) in quantum mechanics. The paper is concluded with some remarks on the relevance to quantum set theory of the choice of the implication connective in quantum logic. 1

The concepts of continuity and convergence pervade the study of functional analysis, and are based on the precise formulation of the (vague) concept of closeness. Traditionally,

Abstract. This is an overview of the recent results of interaction of Boolean valued analysis and vector lattice theory.