Can any system of math rules ever prove every true statement about numbers?
Mathematicians once hoped to find one complete, consistent set of axioms strong enough to prove every true statement about arithmetic, with no contradictions. Is that even possible, for any system powerful enough to describe whole numbers and their addition and multiplication?
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No. Kurt Godel proved in 1931 that any consistent formal system powerful enough to express basic arithmetic must contain true statements it cannot prove, and it can never prove its own consistency from within itself. He built a single self-referential sentence that effectively says 'this statement cannot be proved,' showing mathematics can never fully complete itself. Explained for general readers in Ernest Nagel and James Newman's 'Godel's Proof' (1958).
— Ernest Nagel and James R. Newman, Godel's Proof — New York University Press, 1958 (theorem originally published by Kurt Godel, 1931)
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