Math Club Colloquium: Big Sets, Bigger Sets, and Their Discontents

In the closing decades of the 19th century. mathematicians and logicians made a number of foundational breakthroughs. Georg Cantor’s pioneering work in set theory met with resistance at first in many quarters, but it was not long before David Hilbert (another giant in both mathematics and foundations) could be motivated by the hope that we would never be expelled from the paradise that Cantor had created for us. And set theory has indeed proved to be an essential tool in the mathematician’s kit.

In this talk with Professor Matthew E. Moore, we will review some of the apparently paradoxical discoveries (such as multiple grades of infinity) that occasioned the early resistance to Cantor’s work, and some of the genuine paradoxes (including Russell’s Paradox and Cantor’s own) that led to refinements in the theory, issuing in such standard axiomatizations as Zermelo-Fraenkel set theory with Choice (ZFC). We will also talk about the radical incompleteness of ZFC, and about the philosophical implications of the search for new axioms that can settle the open question of the size of the continuum.