R.G.
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Registered User
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Apr 8 |
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Rosserized proof-predicates and the derivability conditions. My question relates to the answer below - why do you claim that $\mathsf{Prov}_R$ is not $\Sigma_1$? |
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Mar 10 |
awarded | ● Yearling |
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Jan 28 |
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Nelson’s program to show inconsistency of ZF But you mean a proof like that by methods available in a theory not stronger that ZF(C)? |
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Jan 26 |
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Nelson’s program to show inconsistency of ZF But this is exactly my point - if you want to prove inconsistency of some axiomatizable theory $T$ extending, say Peano Artihmetic, show $T\vdash Con(T)$ and apply Goedel's 2nd. And my point is that you cannot hope to prove such a thing ONLY IF you believe in consistency of $T$. Otherwise, why not to look for a proof of $Con(T)$ within $T$ itself to demonstrate $T$ is flawed. Actually, I do not believe that this would be the easiest way to prove $T$'s inconsistency, but at least a possible way. |
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Jan 25 |
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Nelson’s program to show inconsistency of ZF I do not think I properly understand this comment. By the 2nd IT you cannot expect to prove Con(ZFC), providing ZFC is consistent. But in case you do not believe in consistency of ZFC you are not forbidden (at least by 2nd IT) to hope to show that $ZFC\vdash Con(ZFC)$. (This is not that I do not believe in consistency of ZFC, I just do not understand the argument.) |
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Dec 15 |
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Positive results coming from paradoxes It is not very correct to say that first incompleteness theorem is formalization of the Liar. As positive results which found their inspiration in paradoxes I'd rather mention development of type theory or various axiomatizations of intuitive theory of sets. But I am not sure whether OP would consider this as an answer to his question. |
