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.