The End of an Era - That of The Natural Numbers
in [Math]
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From: herbzet on 14 May 2010 03:19 herbzet wrote: > 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"? Oh, wait, I just noticed the set brackets. You're defining theory T as the singleton {(x=x) /\ ~(x=x)}. You had previously invoked the "context of FOL reasoning", so I was assuming the context of FOL reasoning. C U. -- hz
From: herbzet on 14 May 2010 03:21 herbzet wrote: > herbzet wrote: > > > 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 > > 3) ((x=x) /\ ~(x=x)) -> (x=x) by substitution in (2) > > 4) (x=x) (1),(3), 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"? > > Oh, wait, I just noticed the set brackets. You're defining > theory T as the singleton {(x=x) /\ ~(x=x)}. > > You had previously invoked the "context of FOL reasoning", so > I was assuming the context of FOL reasoning. > > C U. Typos corrected in proof. -- hz
From: Alan Smaill on 14 May 2010 05:22 Nam Nguyen <namducnguyen(a)shaw.ca> writes: > 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)}? What does "true in a theory" mean? Please be precise. -- Alan Smaill
From: Marshall on 14 May 2010 21:17 On May 13, 9:38 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > 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? This question is merely a diversion from the discussion of whether "1+1=2" in an intuition or not, and furthermore is a question we both know the answer to. It is a waste of your time and mine. Marshall
From: Marshall on 14 May 2010 21:22
On May 13, 9:30 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > 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)}? The formula is true in every model of T. In fact, the formula is true in every model. The formula is provable in T. The formula is provable in every theory. Marshall |