From: Hal Finney (hal@finney.org)
Date: Thu May 30 2002 - 10:06:42 MDT
Damien Broderick writes:
> < 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?
What he is really saying here is that theorems must be true. But this
is not because they are assumed to be true; rather, it is because
they are proven true. In this case we can look at the peculiar,
self-referential form of G and deduce directly that G cannot be true.
Hence we know that no matter how long we search, we will never find
a proof of G as a theorem. (Or rather, if we ever did find one, that
would be a contradiction, meaning that our axiom system is inconsistent.
That would be bad. Ghostbusters bad.)
Hal
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