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title
Azimuth
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image
site name
Azimuth
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updated
2025-12-19 22:46:13
raw text
Azimuth Azimuth Home About The Probability of Undecidability 15 March, 2024 There’s a lot we don’t know. There’s a lot we can’t know. But can we at least know how much we can’t know? What fraction of mathematical statements are undecidable—that is, can be neither proved nor disproved? There are many ways to make this question precise… but it remains a bit mysterious. The best results I know appear, not in a published paper, but on MathOverflow! In 1998, the Fields-medal winning topologist Michael Freedman conjectured that the fraction of statements that are provable in Peano Arithmetic approaches zero quite rapidly as you go to longer and longer statements: He must also have been conjecturing that Peano Arithmetic is consistent, since if it’s inconsistent then all its statements are provable. From now on let’s assume that PA is consistent. In 2005, Cristian Calude and Konrad Jürgensen published a paper arguing that Freedman was on the ri...
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