In a Consistent System, Can a True Sentence Imply a False One?
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From: Aatu Koskensilta on 31 Jul 2010 03:39 Nam Nguyen <namducnguyen(a)shaw.ca> writes: > Really? Or is this the case just couldn't read simple English > sentences? Come now, don't you think it's a bit tacky to deride others for their poor English skills? This is not an English usage group, after all. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechen kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Aatu Koskensilta on 31 Jul 2010 03:44 Marshall <marshall.spight(a)gmail.com> writes: > Just picture him in that clown outfit, waving his coffee-stained copy > of Schoenfeld, ranting at the undergrads outside the clown college. Shoenfield, you beef-eating invasion-monkey, Shoenfield! -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechen kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Aatu Koskensilta on 31 Jul 2010 03:46 Nam Nguyen <namducnguyen(a)shaw.ca> writes: > Where in that list would it contain your explanation why the given > definition of "disprovable" would make sense in the case of an > inconsistent theory? Make sense how? There's no apparent obscurity, ambiguity, difficulty, in applying the definition in case of an inconsistent theory. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechen kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Aatu Koskensilta on 31 Jul 2010 04:05 Sergei Tropanets <trop.sergei(a)gmail.com> writes: > By Godel's second incompleteness theorem we know that PA can't prove > Con(PA). This implies Con(PA + Con(PA)) and Con(PA + not Con(PA)). No it doesn't. It only implies Con(PA + not Con(PA)). You need Sigma-1 soundness of PA to conclude that PA + Con(PA) is consistent. > So if we think of PA and its theorems as true statements than we have > to think so also of PA + not Con(PA). We have to do no such thing. PA + not Con(PA) is a consistent theory that proves arithmetical falsehoods. > Then true (and consistent) theory PA + not Con(PA) would prove its own > inconsistency which may be interpreted as something false. This was > one of the Godel's key arguments against Hilbert's program: formal > system may be syntactically consistent but semantically inconsistent! No it wasn't. > So just syntactical consistency is not enough for foundation of > mathematics! We can just observe that the consistent theory PA + not Con(PA) proves an arithmetical falsity to see that consistency is a piddling correctness condition. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechen kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Aatu Koskensilta on 31 Jul 2010 04:09
Nam Nguyen <namducnguyen(a)shaw.ca> writes: > What does it mean when we say we "prove" a theorem? Of course that's > an easy question: to prove a theorem is to find a syntactical proof > through rules of inference. So when Andrew Wiles proved Fermat's last theorem he did so by finding a "syntactical proof through rules of inference"? > And we can! Hooray! -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechen kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus |