Truth, Proof, and Paradox

(3 Credit Hours) · Undergraduate

The mathematician Andre Weil famously declared that "God exists because mathematics is consistent, but the Devil exists because we cannot prove it." This course will examine the discovery that prompted Weil's declaration: Gödel's incompleteness theorems. The incompleteness theorems are a pair of puzzling results on the limits of what can be mathematically proven. They raise fundamental questions about the limits of mathematical knowledge and the relation between truth and proof. They are also intimately connected to two topics at the heart of modern computer science and logic: Alan Turing's theory of computation and Alfred Tarski's work on the indefinability of truth. This course will discuss the foundational results of Gödel, Tarski, and Turing, and explore some of their consequences for philosophy, mathematics, computer science, and artificial intelligence.