The End of an Era - That of The Natural Numbers
in [Math]
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From: Nam Nguyen on 14 May 2010 00:30 Marshall wrote: > On May 13, 7:13 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >> ... the fact that nobody could have a single example of a _FOL_ >> absolute (formula) truth. > > x=x > Is that formula true in the theory T = {(x=x) /\ ~(x=x)}?
From: Nam Nguyen on 14 May 2010 00:38 Marshall wrote: > On May 13, 8:01 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >> The thing escapes my understanding is why my opponents and the >> "standard theorists" never want to admit we only have an intuitive >> knowledge of the natural numbers. Why is that? > > Because it's wrong. > > 1+1=2 is not an intuition. S0 + S0 = SS0 is also true in arithmetic modulo 2. So do the naturals form the arithmetic modulo 2?
From: herbzet on 14 May 2010 01:23 Nam Nguyen wrote: > James Burns wrote: > > Nam Nguyen wrote: > >> Nam Nguyen wrote: > >>> herbzet wrote: > >>>> Nam Nguyen wrote: > >>>> > >>>>> In other word there's no absolute truth. > >>>> > >>>> That's absolutely true. > >>> > >>> Relatively speaking. > >> > >> All truths are relative, including this truth. > >> (Kind of rings the bell, doesn't it?) > > > > Please put this statement of yours in context. > > Otherwise, I may not understand it. > > Sure. > > > > > And then, please put the context in context. > > Sure, 1 more time. > > > > > And then, continue contextualizing context > > until you're done, of course. > > If I'm sure at any given context, I'd be sure 1 more time. > All of which means I'd be done - by induction reasoning. > > > > > Thanks in advance. > > You're welcomed. > > *** > > On a more serious note, my > > >>>>> In other word there's no absolute truth. > > only meant _within the context of FOL reasoning_ there's > no such thing as an absolute truth of a formula! > > That's all I ever meant and herbzet should have realized that > and not jumped on the bandwagon Oh-Nam-is-making-philosophical- > comment-again, which the "standard theorists" tend to jump, to > hide the fact that nobody could have a single example of a _FOL_ > absolute (formula) truth. > > I hope you understand the context and the situation now. I don't know if I agree, but thanks for the clarification. -- hz
From: Nam Nguyen on 14 May 2010 02:06 herbzet wrote: > > Nam Nguyen wrote: >> James Burns wrote: >>> Nam Nguyen wrote: >>>> Nam Nguyen wrote: >>>>> herbzet wrote: >>>>>> Nam Nguyen wrote: >>>>>> >>>>>>> In other word there's no absolute truth. >>>>>> That's absolutely true. >>>>> Relatively speaking. >>>> All truths are relative, including this truth. >>>> (Kind of rings the bell, doesn't it?) >>> Please put this statement of yours in context. >>> Otherwise, I may not understand it. >> Sure. >> >>> And then, please put the context in context. >> Sure, 1 more time. >> >>> And then, continue contextualizing context >>> until you're done, of course. >> If I'm sure at any given context, I'd be sure 1 more time. >> All of which means I'd be done - by induction reasoning. >> >>> Thanks in advance. >> You're welcomed. >> >> *** >> >> On a more serious note, my >> >> >>>>> In other word there's no absolute truth. >> >> only meant _within the context of FOL reasoning_ there's >> no such thing as an absolute truth of a formula! >> >> That's all I ever meant and herbzet should have realized that >> and not jumped on the bandwagon Oh-Nam-is-making-philosophical- >> comment-again, which the "standard theorists" tend to jump, to >> hide the fact that nobody could have a single example of a _FOL_ >> absolute (formula) truth. >> >> I hope you understand the context and the situation now. > > I don't know if I agree, but thanks for the clarification. For what it's worth, I actually didn't believe you intended to jump on the bandwagon. They didn't want to call me as a "crank" so they labeled me "philosophical" and somehow that might have stayed in people's minds. As for agreement, I'm not looking for sophisticated examples. Just one wff which they believe is *intrinsically true in _all_ contexts* and in which it's impossible for me to have a context to say its truth value is otherwise, hence is relative. Just one single example.
From: herbzet on 14 May 2010 03:05
Nam Nguyen wrote: > herbzet wrote: > > Nam Nguyen wrote: > >> James Burns wrote: > >>> Nam Nguyen wrote: > >>>> Nam Nguyen wrote: > >>>>> herbzet wrote: > >>>>>> Nam Nguyen wrote: > >>>>>> > >>>>>>> In other word there's no absolute truth. > >>>>>> That's absolutely true. > >>>>> Relatively speaking. > >>>> All truths are relative, including this truth. > >>>> (Kind of rings the bell, doesn't it?) > >>> Please put this statement of yours in context. > >>> Otherwise, I may not understand it. > >> Sure. > >> > >>> And then, please put the context in context. > >> Sure, 1 more time. > >> > >>> And then, continue contextualizing context > >>> until you're done, of course. > >> If I'm sure at any given context, I'd be sure 1 more time. > >> All of which means I'd be done - by induction reasoning. > >> > >>> Thanks in advance. > >> You're welcomed. > >> > >> *** > >> > >> On a more serious note, my > >> > >> >>>>> In other word there's no absolute truth. > >> > >> only meant _within the context of FOL reasoning_ there's > >> no such thing as an absolute truth of a formula! > >> > >> That's all I ever meant and herbzet should have realized that > >> and not jumped on the bandwagon Oh-Nam-is-making-philosophical- > >> comment-again, which the "standard theorists" tend to jump, to > >> hide the fact that nobody could have a single example of a _FOL_ > >> absolute (formula) truth. > >> > >> I hope you understand the context and the situation now. > > > > I don't know if I agree, but thanks for the clarification. > > For what it's worth, I actually didn't believe you intended to jump > on the bandwagon. They didn't want to call me as a "crank" so they > labeled me "philosophical" and somehow that might have stayed in > people's minds. I was just cruising thru, and took one of my favorite cheap shots. But it provoked a fun comment from James Burns, and a clarification from you, so I guess things work out. > As for agreement, I'm not looking for sophisticated examples. Just > one wff which they believe is *intrinsically true in _all_ contexts* > and in which it's impossible for me to have a context to say its truth > value is otherwise, hence is relative. I see. > Just one single example. In another post where you reply to Marshall, you ask "Is that formula [x=x] true in the theory T = {(x=x) /\ ~(x=x)}?" Well, it has a proof in that theory: 1) ((x=x) /\ ~(x=x)) axiom of T 2) (A /\ B) -> A theorem of FOL 2) ((x=x) /\ ~(x=x)) -> (x=x) by substitution in (2) 3) (x=x) (1),(2), detachment. Does the fact that (x=x) has a proof in the FOL theory T mean that it is true in T? What do you mean by "true in T"? What do you mean by "true in a context"? If we knew that, we might have a shot at finding a formula "true in all contexts". C U. -- hz |