# The Gödel-Deligne theorem

# The Gödel-Deligne theorem

Gödel’s celebrated completeness theorem says that if a mathematical proposition is true then it is also provable. This is a a founding result of mathematical logic and well-known among logicians. In algebraic geometry, Deligne’s theorem says that certain well-behaved toposes behave just like normal topological spaces. This is a founding result of Grothendieck-style algebraic geometry and well-known among geometers. In a very curious twist, these two theorems turn out to be equivalent. In this talk I will introduce the basics of first-order logic and of topos theory needed to understand both theorems, sketch a proof that they are equivalent and explain in what sense geometry and logic become one and the same thing in the setting of toposes. If time permits, I will riff on how these methods connect to recent developments in homotopy theory.