The End of an Era - That of The Natural Numbers
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From: Nam Nguyen on 8 Jun 2010 20:19 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".
From: Aatu Koskensilta on 8 Jun 2010 20:20 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. -- 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 21:24 Nam Nguyen <namducnguyen(a)shaw.ca> writes: > Aatu Koskensilta wrote: > >> There's nothing controversial in the observation that mathematical >> logic is a branch of mathematics. > > Just like that, huh? Sure. It is patently obvious mathematical logic is a branch of mathematics. > So what happens if others believe the idea mathematical logic is a > branch of mathematics is at the very least controversial? It indicates they have no idea what they're talking about. > Doesn't you school of thought believe that we could use the naturals > to prove PA's consistency? I have no "school of thought". -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Nam Nguyen on 8 Jun 2010 21:27 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).
From: Nam Nguyen on 8 Jun 2010 21:35
Aatu Koskensilta wrote: > Nam Nguyen <namducnguyen(a)shaw.ca> writes: > >> Aatu Koskensilta wrote: >> >>> There's nothing controversial in the observation that mathematical >>> logic is a branch of mathematics. >> Just like that, huh? > > Sure. 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? Surely anybody including the cranks could also say anything they wish to say without bothering to prove, demonstrate anything! > >> So what happens if others believe the idea mathematical logic is a >> branch of mathematics is at the very least controversial? > > 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. > >> Doesn't you school of thought believe that we could use the naturals >> to prove PA's consistency? > > I have no "school of thought". But do you believe so? |