His incompleteness theorems destroyed the search for a mathematical theory of everything. Nearly a century later, we’re still coming to grips with the consequences.
In 1931, the Austrian logician Kurt Gödel pulled off arguably one of the most stunning intellectual achievements in history.
Mathematicians of the era sought a solid foundation for mathematics: a set of basic mathematical facts, or axioms, that was both consistent — never leading to contradictions — and complete, serving as the building blocks of all mathematical truths.
But Gödel’s shocking incompleteness theorems, published when he was just 25, crushed that dream. He proved that any set of axioms you could posit as a possible foundation for math will inevitably be incomplete; there will always be true facts about numbers that cannot be proved by those axioms. He also showed that no candidate set of axioms can ever prove its own consistency.
His incompleteness theorems meant there can be no mathematical theory of everything, no unification of what’s provable and what’s true. What mathematicians can prove depends on their starting assumptions, not on any fundamental ground truth from which all answers spring.
Undecidable questions have even arisen in physics, suggesting that Gödelian incompleteness afflicts not just math, but — in some ill-understood way — reality.
The article next outlines a “simplified, informal rundown of how Gödel proved his theorems.” What the results imply is discussed next.
No Proof of Consistency
We’ve learned that if a set of axioms is consistent, then it is incomplete. That’s Gödel’s first incompleteness theorem. The second — that no set of axioms can prove its own consistency — easily follows.
What would it mean if a set of axioms could prove it will never yield a contradiction? It would mean that there exists a sequence of formulas built from these axioms that proves the formula that means, metamathematically, “This set of axioms is consistent.” By the first theorem, this set of axioms would then necessarily be incomplete. But “The set of axioms is incomplete” is the same as saying, “There is a true formula that cannot be proved.”
Gödel’s proof killed the search for a consistent, complete mathematical system. The meaning of incompleteness “has not been fully fathomed,” Nagel and Newman wrote in 1958. It remains true today.Full article available at Quanta Magazine.
“Truth is bigger than proof.”