From: Ken Clements (Ken@Innovation-On-Demand.com)
Date: Thu May 30 2002 - 03:15:12 MDT
Damien Broderick wrote:
> 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
Logical systems are generally divided into statements that are axioms (or
postulates), and statements that are theorems. Axioms are taken to be true without
proof. All theorems must be derived (proven) from the axioms. If an axiom turns
out to be provable within the system (minus itself), it gets kicked down to theorem
status. Gödel showed that any system in which no false statement could be shown to
be a theorem would be missing some true statements from its set of all theorems (all
statements that could be proved true in the system). That is what is going on with
the `G is not a theorem of this formal number system' above.
-Ken
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