Gödel's Incompleteness Theorem

Fri, 31 Jul 2026 19:12:40 +1000

In 1931 a twenty-five-year-old in Vienna published a paper proving that mathematics cannot fully formalise itself: any consistent formal system rich enough to do arithmetic contains true statements it cannot prove, and — worse — cannot even prove its own consistency (Gödel, Kurt, 1931). 𐃏 This page walks the actual argument: what the theorem says precisely, the two technical inventions that power it (Gödel numbering and diagonalisation), and — just as important — the long list of things it does not say.

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Gödel's Incompleteness Theorem