Here are the slides from The Liar’s Paradox and the Foundations of Mathematics, presented to Talk Math With Your Friends, May 6, 2021.
The Liar’s Paradox, stated most directly as, “This sentence is false,” was around as a puzzle and curiosity for over 2000 years. Then variants of it caused crises in the modern foundations of mathematics. First, Russell’s Paradox forced mathematics to rethink the concept of sets that underlies it all. Then Kurt Gödel proved that the holy grail of mathematics—a complete axiomatization of mathematics–was impossible. In particular, no matter how sophisticated your mathematical machinery, there will always be true statements that cannot be proved. This holds even if you restrict yourself to arithmetic.
We will discuss these paradoxes and how they changed mathematics. We will outline Gödel’s ingenious proof and reflect on what it tells us about mathematics. This talk is accessible to anyone with elementary familiarity with sets and proofs.