[FOM] Is Godel's Theorem surprising?
Harry Deutsch
hdeutsch at ilstu.edu
Mon Dec 11 12:52:53 EST 2006
Thanks for the reference to Kripke's proof. Let me rephrase what I
said: Formalization makes Godel's first theorem possible. There are
mathematicians who view the theorem as a not-so-surprising limitation
on formalization--bearing at best an indirect relation to intuitive
mathematical proof. HD
On Dec 7, 2006, at 7:12 AM, Charles Silver wrote:
> 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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