Weak set theories in foundational debates.

Abstract

Perception of the relationship of the discipline of logic to other exact sciences changes with the years. No twentieth-century proposal for a single logical system that would support the whole of mathematics satisfied everyone, so weaker formal systems with applications in many different contexts are now sought, in mathematics, philosophy, computer science or elsewhere. As always, logicians help by asking such questions as 'Is this theory correct for the given context?' and 'Can it be used in other contexts?' The Introduction of this paper surveys the history of logic and its links to other scientific disciplines such as biology, economics, engineering and physics, and some details of current and future research projects are given. Then the very weak set theory PROVI is introduced and its support for the techniques of constructibility (Gödel 1935) and forcing (Cohen PJ 1963 The independence of the continuum hypothesis, I. Proc. Natl Acad. Sci. USA 50, 1143-1148. (doi:10.1073/pnas.50.6.1143)) is given in outline, full details being already in print. Relations with other known techniques are explored and developed. New results coming from the study of illfounded [Formula: see text]-models of PROVI and other systems are given; and new formal systems in the style of Quine (1937 Quine WV. 1936 Set-theoretic foundations for logic. J. Symb. Log. 1, 45-57. (doi:10.2307/2268548)) are described. This article is part of the theme issue 'Modern perspectives in Proof Theory'.