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(math) Hilbert's second problem
Posted on: May 29, 2008 - 6:12am
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(math) Hilbert's second problem
Does anybody know what must be demonstrated to show that the axioms of arithmetic are consistent, or that Gentzen's consistency proof is correct? It certainly seems to me that the definition of ε0 covers it; am I missing something? I like to know what my limits are.
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Hilbert was looking for finitary methods, because omega is generally considered to just be infinite epsilon zero really doesn't satisfy his program. (that being said, the likelihood of satisfying Hilberts program as intended is pretty low as of Godel Incompleteness 2). Gentzen's consistency proof lacks a finite standpoint in terms of accessibility (= a constructive argument) for ordinals up to epsilon zero.
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Ah, alright. Thanks!