Incompleteness refers to mathematical statements that cannot be proved or refuted. Precise formulations under mild hypotheses are credited to Kurt Goedel (1930s). Incompleteness has its roots in ancient algebra and geometry.

After discussing the First and Second Incompleteness Theorems, we present the generally accepted foundation for mathematics, ZFC. This is presented as an immediate extension of provable facts about finite sets.

We discuss work of Goedel and Cohen concerning Incompleteness of ZFC, surrounding the axiom of choice and the continuum hypothesis.

This lecture is part of Invitation to Mathematics.
Pre-candidacy students can sign up for this lecture series by registering for one credit hour of Math 693, Call # 16732 (with Prof H. Moscovici).