[FOM] Is Godel's Theorem surprising?
Antonino Drago
drago at unina.it
Mon Dec 11 17:31:30 EST 2006
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). In
Goedel theorem we are dealing with a property concerning the totality of
theorems of arithmetics
which moreover are considered as deduced in all possible deductive ways (how
many proofs has an arithmetics theorem?). I cannot see a different
qualification of this subject than a philosophical
one.
Then what is surprising for me in Goedel theorem are the following features:
1) an essentially philosophical subject is captured by a seemingly strict
deduction. In other
terms, is it possible to exclude that a philosophical hypothesis is hidden
inside
Goedel proof?
2) the surprise is enhanced when one considers that Goedel offered no one
specific technique for capturing this new
philosophical
problem in mathematical terms; on the contrary,
he suggested to reduce the power of Hilbert's metamathematics to
recursive mathematics only.
More in general, are we sure that Hilbert did not misled mathematicians
by proposing some essentially philosophical problems?
I suppose that Brouwer would have answered negatively to the above two
questions.
Actually what was surprising for me in studying the history of Goedel
theorem is why intuitionists never commented in an attentive way this
theorem (I was capable to discover some remarks only in A. Heyting:
Intuitionnisme, Gauthier-Villars) although they could proclaimed it as their
indisputable victory. Do they thinked that this theorem was trivial in an
intuitive sense? If yes, which is this intuitive sense and in which way it
can be possibly formalised in present intuitionism?
By summarising, from both the semi-intuitionist (Poincaré) point
of view and intuitionist one the question (Is Goedel theorem surprising?) is
trivial; but the question is interesting because it suggests to investigate
upon
the odd intuitionists silence about Goedel theorem.
Best regards
Antonino Drago
----- Original Message -----
From: "Charles Silver" <silver_1 at mindspring.com>
To: "Foundations of Mathematics" <fom at cs.nyu.edu>
Sent: Thursday, December 07, 2006 2:12 PM
Subject: [FOM] Is Godel's Theorem surprising?
> Why should it be so surprising that PA is incomplete, and even (in a
> sense) incompletable?
> Or put the other way, why should we have thought PA (or, for Godel,
> the system of Principia Mathematica and related systems) would have
> to be complete? It has been alleged, for example, that at the time
> of Godel's proof John von Neumann had been working on proving
> *completeness* for PM or some related system? Why would von Neumann
> have thought *intuitively* that the system could be proved complete?
> I'm not intending the above to be questions of mathematical fact.
> I'm just wondering what accounts for the shock so to speak of Godel's
> Theorem. One answer I've read is to the effect that everyone at
> the time thought PM was complete. But for me, that's not
> satisfactory. I'd like to know *why* they should have thought it was
> complete. Did they have *intuitions* for thinking it had to be
> complete?
> I'm also wondering, though this is a separate point, whether today
> the theorem is not only not surprising, but perhaps even intuitively
> obvious. One unsatisfactory answer would be that incompleteness is
> now not surprising because we now know it holds. But do we now have
> distinctly different *intuitions*, aside from the proof itself
> (though of course the proof can't entirely be discounted), that
> establish, let's say the "obviousness" of the result?
>
>
> Charlie Silver
> _______________________________________________
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> FOM at cs.nyu.edu
> http://www.cs.nyu.edu/mailman/listinfo/fom
>
>
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