Re: THIS STATEMENT HAS NO PROOF IN ANY SYSTEM = true or false?

From: Barb Knox (see_at_sig.below)
Date: 01/13/05 

Date: Thu, 13 Jan 2005 12:44:01 +1300

In article <gxiFd.399412$b5.19413518@news3.tin.it>,
 "LordBeotian" <pokipsy76@CANCELLAMIyahoo.it> wrote:

>"Barb Knox" <see@sig.below> ha scritto
>
>> >> Would you consider statements of the form "proposition P has a proof in
>> >> theory T" to be candidate mathematical truths? If so, then every
>formal
>> >> proof of anything also represents a computationally-verifiable
>> >mathematical
>> >> truth, without any ambiguity or additional philosphical committment.
>> >
>> >Even if the existence of the proof is proved by some nonconstructive
>> >methods?
>>
>> If it's a *formal* non-constructive proof (in some system TT) of the
>> existence of a proof in T of P, then yes. The importance of it being a
>> formal proof is that it can be algorithmically verified, not needing any
>> additional philosophical assumptions. Note in this case that the "truth"
>is
>> TT |- (T |- P), not T |- P itself.
>
>Uhm... and what if I prove T |- P (in TT) using things like axiom of choice,
>transfinite induction or existence of some large cardinal?
>If we accept this proof as valid why don't we consider "mathematical truth"
>the whole set theory without restricting the notion of "mathematical truth"
>to statements of the form "proposition P has a proof in theory T"?

I'm not claiming that statements of the form "T |- P" exhaust all possible
sorts of mathematical truth, but that they are a subset. I am not
addressing whether or not (e.g.) AC is "true" in some other sense. 

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