The End of an Era - That of The Natural Numbers
in [Math]
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From: Aatu Koskensilta on 8 Jun 2010 22:36 Nam Nguyen <namducnguyen(a)shaw.ca> writes: > Well then he has yet to demonstrate the formula is true in a false > model (where U is empty). To your satisfaction? I doubt that's possible. -- 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 8 Jun 2010 22:42 Nam Nguyen <namducnguyen(a)shaw.ca> writes: > Aatu Koskensilta wrote: > >> It is patently obvious mathematical logic is a branch of mathematics. > > So what's your definition of "a branch of mathematics" and how would > you demonstrate "mathematical logic" would fit to your definition? I have no private definition of "branch of mathematics" or "mathematical logic". Are you saying you don't know what these terms mean? >> It indicates they have no idea what they're talking about. > > Neither would you, in their views. If that's all there is to it. Naturally everyone will make their own mind about such matters. > But do you believe so? A simple consistency proof for PA consists essentially in the observation that all of its axioms are true in the naturals. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Daryl McCullough on 8 Jun 2010 22:56 Nam Nguyen says... > >Aatu Koskensilta wrote: >> Nam Nguyen <namducnguyen(a)shaw.ca> writes: >> >>> Daryl McCullough wrote: >>> >>>> Because (Ax x=x) is true in every model. >>> You meant every model where U is non-empty obviously. >> >> No he didn't. >> > >Well then he has yet to demonstrate the formula is true >in a false model (where U is empty). I've explained it to you several times already. First, there is no such thing as a "false model". But the way that truth in a model works for classical logic is this: If S is a structure for a language L, and Phi(x) is a formula of L, then: If there is any way to assign a value to variable x so as to make Phi(x) true, then the formula Ex Phi(x) is true in the structure. Otherwise, ~Ex Phi(x) is true in the structure. If U happens to be empty, then it immediately follows that ~Ex Phi(x) is true for every formula Phi(x). And also, ~Ex ~Phi(x) is true for every formula Phi(x). Classically, ~Ex ~Phi(x) is interpreted to mean the same thing as Ax Phi(x). So it follows that: if S is a structure for L with an empty domain, and Phi(x) is a formula of L, then Ax Phi(x) is true in that structure, and Ex Phi(x) is false in that structure. A special case is the formula x=x. So we have: If S is a structure with empty domain, then Ax x=x is true in that structure. -- Daryl McCullough Ithaca, NY
From: Nam Nguyen on 8 Jun 2010 23:00 Aatu Koskensilta wrote: > Nam Nguyen <namducnguyen(a)shaw.ca> writes: > >> Aatu Koskensilta wrote: >> >>> It is patently obvious mathematical logic is a branch of mathematics. >> So what's your definition of "a branch of mathematics" and how would >> you demonstrate "mathematical logic" would fit to your definition? > > I have no private definition of "branch of mathematics" or "mathematical > logic". Are you saying you don't know what these terms mean? I know what the term means "loosely", which is in no way the reason to classify "mathematical logic" as such. Technically a branch of mathematics is just a formal system or a collection of related formal systems, as far I understand the term how the is used. In any rate, can you share with us what you think the "public" definition of "branch of mathematics" be? > >>> It indicates they have no idea what they're talking about. >> Neither would you, in their views. If that's all there is to it. > > Naturally everyone will make their own mind about such matters. > >> But do you believe so? > > A simple consistency proof for PA consists essentially in the > observation that all of its axioms are true in the naturals. But what are the naturals? A model of PA? I'm sure you know what circularity means!
From: Daryl McCullough on 8 Jun 2010 23:02
Nam Nguyen says... > >Daryl McCullough wrote: >> Nam Nguyen says... >> >>> Of course Shoenfield did treat the 2 formulas differently: he mentioned >>> 1 kind on pg. 19 but not the other kind. But how does that have anything >>> to bear on the arguments here? Specifically, how does such observation >>> help you to prove there's no context in FOL in which x=x (or even Ax[x=x]) >>> is false? >> >> Because (Ax x=x) is true in every model. > >You meant every model where U is non-empty obviously. But note my >"prove" and "no context in FOL". Ax x=x is true for every model, empty domain or not. -- Daryl McCullough Ithaca, NY |