[FOM] Is Goedel's Theorem Surprising?
Vann McGee
vmcgee at MIT.EDU
Mon Dec 11 23:00:10 EST 2006
I wonder if the following message could be posted to the FOM group,
in answer to the question, "Is Goedel's Theorem surprising?" Thank you very
much.
Regarding the question why the First Incompleteness Theorem was
surprising, I've always supposed - although I hasten to admit I haven't any
textual evidence for this - that, at the time the theorem was published, it
seemed highly probable that Dedekind's proof that any two "simply infinite"
systems are isomorphic could be deployed, if anyone took the time to fill in
the details, to show that Goedel's system P (which embeds second-order Peano
Arithmetic within the simple theory of types) is categorical, and hence
complete. The incompleteness of first-order PA came as no surprise, but the
first-order theory was widely regarded as an artificially circumscribed
fragment.
For first-order arithmetical theories, the fact that, for any
axiomatized theory we can recognize as true (in the standard model), there
are sentences the theory doesn't decide isn't surprising, but the fact that
there's an undecidable sentence that we can recognize as true is an
observation that continues to astonish, as evidenced by the continuing
discussions of the Lucas-Penrose argument.
Regards,
Vann McGee
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