[FOM] Is Godel's Theorem surprising?

[Antonino Drago]

11/12/06 I wrote:
> According to Poincaré's criticism to formalists methods, we cannot prove
> the consistency of the totality of theorems of arithmetics unless we apply
> the induction principle, which however essentially belongs to this 
> totality
> (according to van Heijnoorth From Frege to Goedel, Hilbert never answered
> to this criticism).

Panu Raatikainen praatika at mappi.helsinki.fi answered:
Hilbert did respond that the alleged consistency proof for the infinistic
mathematics (with unrestricted induction), which is to be carried out in
finitistic (meta-)mathematics, only uses restricted induction, and
therefore is not circular.

I have not the book at the hand, but I remember well that van Heijnoorth 
remarks that just this restricted induction never was qualified by Hilbert.

Panu Raatikainen praatika at mappi.helsinki.fi added:
Goedel had some critical remarks about this, arguing that the notion of
proof involved is insufficiently clear. And I am inclined to think that
the situation poses more problems for those contemporary intuitionist who
give a central role for intuitionistic logic than is generally recognized.
(I mean philosophical problems; I am not suggesting that it provides a
strict refutation of such intuitionism.)

Similarly Peter Aczel petera at cs.man.ac.uk commented:
Two notions of completeness for a sound (i.e. formally provable
 implies true) formal language need to be distinguished from the
 constructive point of view, although they are equivalent from the
 classical point of view:
1) Every sentence of the formal language is formally provable or
   formally refutable (i.e. its negation is formally provable).
2) Every true sentence of the formal language is formally provable.

I think that in intuitionistic logic a third notion of completeness can be 
added :
3) Every true sentence cannot be disproved, or, equivalently, every true 
sentence can be proved by a weak ad absurdum proof  (i.e. an ad absurdum 
proof whose final sentence is the double negation of the thesis; which in 
the following theory is used as an heuristic principle).
I seem that in some theories scientists made use of this notion of 
completeness: Sadi Carnot' theorem on the efficiency of the heat engines and 
Lobachevsky (in the propositions 19-22 of the work printed as an Appendix to 
R. Bonola: Non-Euclidean geometry, Dover).
In these two historical cases I cannot see in which way Goedel theorem 
applies.

Best regards
Antonino Drago 


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