Kurt Gödel’s name is attached to the incompleteness theorems published in 1931.
Gödel’s first incompleteness theorem showed that any consistent formal system capable of expressing basic arithmetic contains true statements that cannot be proved within that system. His second theorem showed that such a system cannot establish its own consistency using only its internal methods.
The results transformed the foundations of mathematics. They challenged hopes that all mathematical truths could be captured by a single complete and mechanically checkable formal system. Gödel’s work did not show that mathematics is useless or that every statement is undecidable; it placed precise limits on particular formal systems.
Gödel published the theorems in his 1931 paper “On Formally Undecidable Propositions of Principia Mathematica and Related Systems.” He later made major contributions to logic, set theory, and general relativity, including a solution describing rotating universes.