Kurt Gödel

Austrian-American logician who proved the incompleteness theorems — any consistent formal system strong enough to express arithmetic contains true statements it cannot prove — transforming logic, philosophy of mathematics, and epistemology.

First incompleteness theorem: any consistent, sufficiently powerful formal system contains unprovable true statements. Second incompleteness theorem: no such system can prove its own consistency. Completeness theorem (1929): every valid first-order sentence is provable. Developed Gödelian platonism: mathematical intuition gives access to abstract objects.