![]() ![]() Gödel’s main maneuver was to map statements about a system of axioms onto statements within the system - that is, onto statements about numbers. Here’s a simplified, informal rundown of how Gödel proved his theorems. Undecidable questions have even arisen in physics, suggesting that Gödelian incompleteness afflicts not just math, but - in some ill-understood way - reality. For example, Gödel himself helped establish that the continuum hypothesis, which concerns the sizes of infinity, is undecidable, as is the halting problem, which asks whether a computer program fed with a random input will run forever or eventually halt. In the 89 years since Gödel’s discovery, mathematicians have stumbled upon just the kinds of unanswerable questions his theorems foretold. What mathematicians can prove depends on their starting assumptions, not on any fundamental ground truth from which all answers spring. His incompleteness theorems meant there can be no mathematical theory of everything, no unification of what’s provable and what’s true. ![]() He also showed that no candidate set of axioms can ever prove its own consistency. 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. 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.īut Gödel’s shocking incompleteness theorems, published when he was just 25, crushed that dream. In 1931, the Austrian logician Kurt Gödel pulled off arguably one of the most stunning intellectual achievements in history. ![]()
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