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by strangestchild
4777 days ago
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What I mean is that since Goodstein's theorem is provably true for the naturals, but is not a consequence of the Peano axioms, then the definition of the naturals used to demonstrate Goodstein's must be strictly stronger than the Peano axioms themselves. I was wondering what this definition might be. I'm familiar with the distinction between formalism and Platonism, although I still haven't made my mind up yet :) |
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Gödel's first incompleteness theorem [1] states this fact, that no theory above a certain expressiveness (read as can express natural numbers with addition and multiplication) can be consistent and complete. Assuming Peano arithmetic is consistent, it can not be complete and complete means you can prove all true facts expressible in the system within the system itself.
The (standard) proof of Goodstein's theorem uses ordinal numbers [2] which are outside of Peano arithmetic and the Kirby–Paris theorem proves that there is no proof inside Peano arithmetic [3].
[1] http://en.wikipedia.org/wiki/G%C3%B6dels_incompleteness_theo...
[2] http://en.wikipedia.org/wiki/Ordinal_number
[3] http://en.wikipedia.org/wiki/Goodsteins_theorem#Proof_of_Goo...