[FOM] Proof "from the book"
Arnon Avron
aa at tau.ac.il
Mon Aug 30 06:15:02 EDT 2004
Here are short replies to two reactions to my message (I hope this
is my last reply, since I dont see much point in entering long debates here).
1) Torkel Franzen writes:
"Godel's proof does not establish that the Godel sentence of a theory
is true. It would be unfortunate if it did, since the Godel sentence
of a theory is sometimes false."
I did not bother to give the exact conditions in my original short
message, assuming that they are well known. A Godel sentence (NOT "the"
Godel sentence) for a consistent extension of Q is true, and
Godel's proof does show this (after all, Franzen strongly claimed here
in the past that a Godel sentence for T "says about itself" that it is
not provable in T, and indeed it is not provable in T. So how can
it be false?). As for other theories, if a theory is inconsistent
than it proves a false sentence, while if it not even an extension of Q
then (by definition) there is a true sentence it does not prove. So
I do apologize that I have ignored such theories in my original message.
2) Marcin Mostowski writes:
"Arnon Avron is wrong in repeating the argument describing Tarski's proof
mentioned by Martin Davis - as nonconstructive in the sense that it does
not give an example of an independent sentence."
The proof of Godel's theorem Martin Davis was refering to (at least the
one I thought he was refering to) *uses* Tarski's theorem (not
Tarski's proof of his theorem!) to derive what I take as a weak form of
Godel's theorem. This fact is what makes it short and elegant. What
Mostowski describes in his message is something completely different:
transforming Tarski's proof of Tarski's theorem into a proof
of Godel's theorem which provides
an independent sentence. That this can be done is obvious. Almost as obvious
is the fact that by doing this we practically get back to
Godel's original proof (as Mostowski homself implicitly noticed
in his message). I indeed
believe that Godel's original proof is still the best, and I would
have certainly put it in my private book of celebrated proofs!
(Historically, Tarski's proof was a reproduction of Godel's
proof, not the other way around...).
Arnon Avron
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