One theorem proved that mathematics can never fully prove itself

In 1931, Kurt Gödel showed that any mathematical system powerful enough to describe basic arithmetic contains true statements that system can never prove, and can never prove its own consistency from within itself. It ended a decades-long project to place all of mathematics on airtight logical foundations, and remains one of the most startling results in the history of logic.

— Kurt Gödel, On Formally Undecidable Propositions of Principia Mathematica and Related Systems I — Monatshefte für Mathematik, 1931