Workshop on Set Theory and the Philosophy of Mathematics

Hilbert, in his celebrated address to the Second International Congress of Mathematicians held at Paris in 1900, expressed the view that all mathematical problems are solvable by the application of pure reason. At that time, he could not have anticipated the fate that awaited the first two problems on his list of twenty-three, namely, Cantor's Continuum Hypothesis and the problem of the consistency of an axiom system adequate to develop real analysis. The Gödel Incompleteness Theorems and the Gödel-Cohen demonstration of the independence of CH from ZFC make clear that continued confidence in the unrestricted scope of pure reason in application to mathematics cannot be founded on trust in its power to squeeze the utmost from settled axiomatic theories which are constitutive of their respective domains. The goal of our Workshop is to consider the extent to which it may be possible to frame new axioms for set theory that both settle the Continuum Hypothesis and satisfy reasonable standards of justification. The recent success of set theorists in establishing deep connections between large cardinal hypotheses and hypotheses of definable determinacy suggests that it is possible to find rational justification for new axioms that far outstrip the evident truths about the cumulative hierarchy of sets, first codified by Zermelo and later supplemented and refined by others, in their power to settle questions about real analysis. The Workshop will focus on both the exploration of promising mathematical developments and on philosophical analysis of the nature of rational justification in the context of set theory.