“Consider the following sentence: “This statement is false.” Is that true? If so, that would make the statement false. But if it’s false, then the statement is true. This sentence creates an unsolvable paradox; if it’s not true and it’s not false– what is it? This question led a logician to a discovery that would change mathematics forever.”
This TED-Ed video, a lesson written by mathematician Marcus du Sautoy with animation by the team at BASA, introduces Gödel’s Incompleteness Theorem, the paradox at the heart of mathematics. The concept was introduced by 25-year-old Austrian logician Kurt Gödel in 1931.
Learn more with this 2020 summary from Wired:
“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.”
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