[FOM] Is Godel's Theorem surprising?
silver_1 at mindspring.com
Thu Dec 7 08:12:42 EST 2006
Why should it be so surprising that PA is incomplete, and even (in a
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
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?
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