[FOM] Godel Sentence

[Vladimir Sazonov]

Kanovei wrote:
> 

> In particular, the common belief that
> *there exist true, but unprovable sentences of PA*
> (allegedly by Goedel's theorems)
> is plain wrong in its straightforward sense, because the
> 25-centuries of development of mathematics under more or less
> current standards of rigor fail to produce anything near
> a mathematical sentence accepted to be TRUE
> (as a mathematical sentence)
> but NOT because its mathematical proof IS GIVEN

Agree completely!

> 
> Those who believe  in the paradigm
> *there exist true, but unprovable sentences of PA*
> in its straightforward sense
> are welcome to kindly present such a remarkable sentence
> along with a demonstration of its desired properties.

Of course, it is usually presented consis(PA) as such a sentence 
or just a Goedel sentence "`I' am not provable". It could be said 
that they are intuitively (informally) true. But even so, these 
sentences are expressing not exactly they are assumed (intuitively) 
to do. Thus, consis(PA) (as formulated in the metatheory PA for PA 
itself) says about *imaginary* proofs in PA of such a length which 
no mathematician can ever write even with the help of powerful 
computers whereas, intuitively, the consistency of PA may be 
considered  a very informal statement on feasible proofs, that 
*nobody* can *really* deduce a contradiction in PA. Thus, the 
formal statement consis(PA) does not correspond sufficiently well 
to its proper informal counterpart, both being usually considered 
as intuitively or believably true. But the intuition and
(quasireligious) 
beliefs alone are not enough for doing mathematics. Moreover, they 
can be sometimes misleading. Mathematics is rather an intuition and 
imagination *supported* by "mechanical" formalist devices. 
One does not exist without another. (This is a sober - not a usual 
caricature - formalist position.)

The fact that the formal statements consis(PA) and "`I' am not provable 
in PA" are provable in a stronger theory like ZF, of course, does not 
matter. Otherwise, let us consider consis(ZF), etc. But this is an 
infinite regress. 

I see no reason in considering introduction of some intuitively 
plausible axioms, like consis(PA) or the choice axiom, etc. as 
invention of mathematical truth. Some of such axioms may 
contradict to others (CH and ~CH) and even have counterintuitive
consequences. And, as Vladimir Kanovei writes quite reasonably, 
only rigorous proofs are the source of mathematical truth, if we 
can tell about truth at all. I would rather prefer to tell about 
a correspondence of provable sentences to our intuition and 
external reality, instead of "truth", and this is the real practice 
of mathematics, especially of applied one. Recall, e.g., Newton. 
"Truth" in mathematical proofs is mostly a figure of speech, 
a helper for the intuition. 


Vladimir Sazonov

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