[FOM] Is Godel's Theorem surprising?
Antonino Drago
drago at unina.it
Wed Dec 13 16:41:33 EST 2006
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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