Asprey Lecture in Mathematics:Hugh Woodin
Monday, April 3, 2017Time
Monday, April 3, 2017
Asprey Lecture in Mathematics
Speaker: Hugh Woodin, Harvard University
Title: A short story of large infinities and small sets
The modern mathematical story of infinity began in the period 1879-84 with a series of papers by Cantor that defined the fundamental framework of the subject. Within 40 years the key ZFC axioms for Set Theory were in place and the stage was set for the detailed development of transfinite mathematics, or so it seemed. However, in a completely unexpected development, Cohen showed in 1963 that even the most basic problem of Set Theory, that of Cantor's Continuum Hypothesis, was not solvable on the basis of the ZFC axioms.
The 50 years since Cohen's announcement has seen a vast development of Cohen's method and the realization that the occurrence of unsolvable problems is ubiquitous in Set Theory. This arguably challenges the very conception of Cantor on which Set Theory is based.
However, during this same period, the detailed study of special cases of the Continuum Hypothesis led to a remarkable success. This was the discovery and validation of the axioms for Second Order Number Theory. The resulting theory is largely immune to Cohen’s method.
The prospect that this could somehow be extended to produce an analogous axiom for Set Theory itself has always seemed completely hopeless. But that belief was itself based on a misconception and recent discoveries suggest there is a resolution.