[FOM] Godel Sentence

[Vladimir Sazonov]

Torkel Franzen wrote:
> 
> Vladimir Sazonov says:
> 
>  >It could be said
>  >that they are intuitively (informally) true.
> 
>   Exactly as intuitively or informally true as the fundamental theorem
> of arithmetic, or any other mathematical theorem.

Yes, but "intuitively true" (according to some kind of intuition) 
is not the same as "mathematically true in an absolute sense" 
which (the latter) is meaningless. Say, according to one 
(minimalist) intuition, the constructibility axiom of Goedel 
may be considered as plausible (intuitively true), but according 
to other (maximalist) intuition, it is plausible to postulate 
its negation. 

> 
>   >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.
> 
>   Exactly as imaginary as large numbers n which the fundamental theorem
> of arithmetic states have a unique prime decomposition. There's nothing
> special about GЖdel's theorem.

Yes, but I stressed on a "real" meaning of consistency of PA 
which involves only feasible proofs unlike formal arithmetical 
sentence to argue that the latter is (intuitively) much more 
strong, and "believing" in it requires some additional extrapolation. 
I can believe in "real" consistency of PA, but doubt in 
consis(PA). Moreover, I also can doubt in "real" consistency 
of PA. Who knows?


> 
>   >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.
> 
>   "Correspondence to our intuition and external reality" is a very
> doubtful philosophical notion, as opposed to the mathematically
> defined "true sentence".

Of course, the definition of an interpretation of first-order 
formulas in first order structures which is usually done in the 
framework of ZFC is quite rigorous one. (And it is not a 
philosophical concept, although philosophers may be interested 
in it.) I have no doubt on that (as I wrote many times; I hope, 
you know this). It seems it should be clear from the context of 
this discussion that the issue is not this technical concept. 
The issue is whether consis(PA) is absolutely true. It is this 
concept of absolute truth (rather - a misconcept) which is doubtful 
philosophically. "Correspondence to our intuition and external 
reality" is just a "honest" concept related to SCIENCE in general. 
If a philosophy intends to ignore this concept and takes 
instead some scholastic fictions having no real meaning, 
the worse for that philosophy. 

If to use the term truth in mathematics at all, then only in 
a technical manner as above, or as a "figure of speech" or in 
a relative manner: a theorem is true if it is proved in such 
and such formal theory. Saying that consis(PA) is unprovable 
in PA, but true without mentioning (at least implicitly) where 
it is true, or according to which kind of intuition, is a 
wrong interpretation of Goedel Theorem. 

Mathematics does not consist of truths. It consists of intuitively 
reasonable formalisms and derivations in them of some theorems. 

Of course I and, I am sure, Vladimir Kanovei, have no doubts 
in the proof in ZF of the standard translation of consis(PA) 
into the language of ZF. By Goedel Theorem it follows  
also in ZF that `consis(PA) is not provable in PA'. 
But do you realize that the question of Kanovei was not 
about this at all?  It was a philosophical, not a technical 
mathematical question. 

The question was essentially whether there is some miraculous 
mathematical way not based on a derivation in a formalism  
of demonstrating a truth of mathematical sentences. The answer 
seems trivial and negative. But, nevertheless, there is some 
widespread opinion that Goedel Theorem gives a way of obtaining 
such sentences and even of demonstrating their truth. 

One more note. Establishing some physical law having a 
mathematical form (say, that the sum of angles of a triangle 
is 180 degrees) does not mean that it is mathematically true. 
Recall non-Euclidean Geometry. When we postulate some axiom, 
we just decide ourselves that it should be postulated by such 
and such reason. By some other reason, we may postulate its 
negation. The only issues are reasonableness, intuition, 
relation to some external (may be other mathematical) reality, 
i.e., applicability, and consistency (nontriviality). 

Thus, we can decide that, for any (reasonable) T, consis(T) may 
be reasonably added to T as a new axiom. We even can give some 
argumentation for that and iterate and investigate this process 
mathematically. But this does not mean that we established the 
general mathematical law of creating new truths. This is only 
a mechanism of creating new, plausible, IN A SENSE, axioms. 
And it is misleading to interpret all of this in terms of truth, 
if it is not in the technical sense mentioned above. All of 
this looks quite trivial, but, unfortunately, there are opposite, 
I believe, inconsistent opinions. 


Vladimir Sazonov

P.S. It seems I explained my opinion in quite enough detail. 
Unfortunately, because of some other business, I cannot promise 
to continue this discussion. May be with a delay. My sincere 
apologies for that to those whom I probably will not (or did 
not) reply. 


> 
> ---
> Torkel Franzen