Le 05-juil.-07, à 22:14, Jesse Mazer a écrit :
> His [Penrose] whole Godelian argument is based on the idea that for
> any computational
> theorem-proving machine, by examining its construction we can use this
> "understanding" to find a mathematical statement which *we* know must
> true, but which the machine can never output--that we understand
> it doesn't. But I think my argument shows that if you were really to
> build a
> simulated mathematician or community of mathematicians in a computer,
> Godel statement for this system would only be true *if* they never
> made a
> mistake in reasoning or chose to output a false statement to be
> and that therefore there is no way for us on the outside to have any
> confidence about whether they will ever output this statement than
> they do
> (and thus neither of us can know whether the statement is actually a
> true or
> false theorem of arithmetic).
I think I agree with your line of argumentation, but you way of talking
could be misleading. Especially if people interpret "arithmetic" by
If we are in front of a machine that we know to be sound, then we can
indeed know that the Godelian proposition associated to the machine is
true. For example, nobody (serious) doubt that PA (Peano Arithmetic,
the first order formal arithmetic theory/machine) is sound. So we know
that all the godelian sentences are true, and PA cannot know that. But
this just proves that I am not PA, and that I have actually stronger
ability than PA.
I could have taken ZF instead (ZF is Zermelo Fraenkel formal
theory/machine of sets), although I must say that if I have entire
confidence in PA, I have only 99,9998% confidence in ZF (and thus I can
already be only 99,9998% sure of the ZF godelian sentences).
About NF (Quine's New Foundation formal theory machine) I have only 50%
Now all (sufficiently rich) theories/machine can prove their own
Godel's theorem. PA can prove that if PA is consistent then PA cannot
prove its consitency. A somehow weak (compared to ZF) theory like PA
can even prove the corresponding theorem for the richer ZF: PA can
prove that if ZF is consistent then ZF can prove its own consistency.
So, in general a machine can find its own godelian sentences, and can
even infer their truth in some abductive way from very minimal
inference inductive abilities, or from assumptions.
No sound (or just consistent) machine can ever prove its own godelian
sentences, in particular no machine can prove its own consistency, but
then machine can bet on them or "know" them serendipitously). This is
comparable with consciousness. Indeed it is easy to manufacture thought
experiements illustrating that no conscious being can prove it is
conscious, except that "consciousness" is more truth related, so that
machine cannot even define their own consciousness (by Tarski
undefinability of truth theorem).
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