From: Damien Broderick (d.broderick@english.unimelb.edu.au)
Date: Thu May 30 2002 - 01:24:58 MDT
At 11:54 PM 5/29/02 -0700, Hal wrote:
>> If a statement is declared a theorem (as I understand it), it must be true,
>> by fiat definition, *for those limited logical purposes*.
>No, I'm afraid you've got it backwards. What you have described would
>be more like an axiom.
My bad, then. I was thinking of this summary I'd made of the discussion in
*Gödel, Escher, Bach* (pp. 271-2):
< Hofstadter posits a string of numbers called, briefly, `G', which can be
interpreted through a Gödel matching, or isomorphism, to read: `G is not a
theorem of this formal number system.' We may ask at once: is the
number-string G itself a theorem?
No theorem may utter a falsity, or its statement would not be a theorem.
G, therefore, is not a theorem . But that, after all, is what it tells us
`about itself'. . . so G must be expressing a truth. Hence, that formal
system does not contain as theorems all the true statements of arithmetic,
for G is one such truth. >
I assume I've misunderstood, or misapplied, Hofstader's account of theorems
and falsity?
Damien Broderick
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