Goedel's theorem is not about nonexistance of foundation in mathematics, it's ab... | Hacker News

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praptak on July 15, 2011 | parent | context | favorite | on: The Smartest Man in Europe Is Very Cautious

Goedel's theorem is not about nonexistance of foundation in mathematics, it's about existence of true but nonproveable statements in every nontrivial formal system.

One could imagine a hypothetical stronger result - maybe every nontrivial set of axioms can actually derive p^(not p)?

singular on July 15, 2011 | [–]

IANAM, but I do know it was an unassailable problem in Russell + White's efforts to found mathematics on a firm basis in Principia Mathematica [1].

[1]:http://en.wikipedia.org/wiki/Principia_mathematica#Consisten...

yequalsx on July 15, 2011 | [–]

Provided the axiom set is recursively enumerable. The second order Peano axioms for the natural numbers are complete.

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