The End of an Era - That of The Natural Numbers
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From: Frederick Williams on 23 May 2010 10:46 "Jesse F. Hughes" wrote: > Hence, a model must have non-empty support, Free logic may have empty models. -- I can't go on, I'll go on.
From: Aatu Koskensilta on 23 May 2010 11:10 "Jesse F. Hughes" <jesse(a)phiwumbda.org> writes: > Strictly speaking, does "empty model" mean anything at all? Empty models are usually disregarded, for reasons of technical convenience. > To be sure, this doesn't negate Daryl's primary point. If we try to > interpret a language in the trivial "model", we will find that every > universal statement is true. My point is only that such > interpretations don't have the property that, if P is provable, then P > is true under the interpretation. If we allow empty models obviously we must modify the deductive calculi accordingly. It's a bit messy but not at all difficult. This sort of stuff goes under the heading "free logic". -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Aatu Koskensilta on 23 May 2010 11:17 Nam Nguyen <namducnguyen(a)shaw.ca> writes: > For example, if we re-intuit the concept of the naturals as that in > which G(T) (per GIT) be false, then we'd arrive at the meta theorem > GIT': > > For any consistent T as strong as arithmetic of the naturals [i.e. > the new "naturals"], there's a formula G(T) which is false and not > provable! Now there's an exciting theorem. But I think I'll give up at this point and thank you for the helpful reminder that it's useless to attempt to discuss logic with some people. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Jesse F. Hughes on 23 May 2010 11:20 Frederick Williams <frederick.williams2(a)tesco.net> writes: > "Jesse F. Hughes" wrote: > >> Hence, a model must have non-empty support, > > Free logic may have empty models. Yes, but I believe the context here is classical FOL. -- Jesse F. Hughes "Being wrong is easy, knowing when you're right can be hard, but actually being right and knowing it, is the hardest thing of all." -- James S. Harris
From: Nam Nguyen on 23 May 2010 12:43
Aatu Koskensilta wrote: > Nam Nguyen <namducnguyen(a)shaw.ca> writes: > >> For example, if we re-intuit the concept of the naturals as that in >> which G(T) (per GIT) be false, then we'd arrive at the meta theorem >> GIT': >> >> For any consistent T as strong as arithmetic of the naturals [i.e. >> the new "naturals"], there's a formula G(T) which is false and not >> provable! > > Now there's an exciting theorem. Yes, as exciting as GIT, which is as exciting as the contradiction that we do know the naturals are but that's not intuition! > But I think I'll give up at this point > and thank you for the helpful reminder that it's useless to attempt to > discuss logic with some people. The feeling is mutual. By now I think I'm well reminded that even in the modern day, long after the Inquisition, there's a class of people who seem to "own" mathematical reasoning and what they don't want to hear or discuss will automatically become crackpot. |