The End of an Era - That of The Natural Numbers
in [Math]
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From: Nam Nguyen on 20 May 2010 09:48 Jim Burns wrote: > > Look, Nam, it should be a piece of cake for > someone who can obfuscate as well as you do. > So, just do your homework, okay? G(PA) kind of formulas must be the most obfuscated kind in the history of mathematical reasoning, then.
From: William Hughes on 20 May 2010 10:46 On May 20, 1:19 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > There's a model in which the universe and all n-ary relations > are empty, and this is the model for all inconsistent formal > systems. Are you claiming that there is a model for an inconsistent formal system? - William Hughes
From: Aatu Koskensilta on 20 May 2010 17:50 Nam Nguyen <namducnguyen(a)shaw.ca> writes: > In the current FOL, > > "F is valid in every model of T" > > is equivalent to > > "T is consistent _and_ F is provable". No it isn't. -- 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 20 May 2010 17:56 Nam Nguyen <namducnguyen(a)shaw.ca> writes: > In other words, given the statement: > > (*) If the Moon is made of cheese then the Moon is blue > > (*) as a whole statement is a true statement by the inference > nature of definition of (*). And you could say, as a manner of > speaking, "the Moon is blue" is vacuously true, per this inferential > statement. Only in the uninteresting sense that anyone can say anything. On the usual meaning of the phrase it makes no sense whatever to say that "the Moon is blue" is vacuously true. Rather, it is the implication (*) itself that's vacuously true, owing to the falsity of its antecedent. -- 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 20 May 2010 17:59
Nam Nguyen <namducnguyen(a)shaw.ca> writes: > But for now would you agree there exist false models in which all > n-ary relations are empty? What is a false model? This is not a term used in logic. There are of course models in which all relations are empty, but the import of this triviality in the present context is completely obscure. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus |