From: Eliezer Yudkowsky (sentience@pobox.com)
Date: Sat Oct 09 2004 - 02:05:09 MDT
Ben Goertzel wrote:
> The problem with Penrose's argument is pretty simple. He argues as
> follows:
>
> Premise 1) FOR EACH computer, THERE EXISTS some problem which that
> computer can't solve Premise 2) NOT [ FOR EACH human, THERE EXISTS some
> problem which that human can't solve ] Conclusion) Therefore, humans are
> not computers
>
> The problem is that his premise 2 is false
>
> Now, I can't prove that his premise 2 is false. So I can't prove he's
> wrong.
It's quite easy to prove that his premise 2 is false. You cannot correctly
solve the problem "Will Ben Goertzel solve this problem by choosing the
answer 'no'?" Feel free to use an appropriate diagonalization lemma if you
object to the self-reference.
> So, it's not true that Penrose misinterprets Godel's Theorem. Not at
> all.
I posted this to wta-talk on July 28th:
****
I don't have the time to dissect Penrose's Godelian argument in detail, but
many, *many* others have done so; you can find it by Googling, e.g.,
Penrose refutation. I'd recommend the refutations found in PSYCHE:
http://psyche.csse.monash.edu.au/v2/psyche-2-23-penrose.html
A *fast* summary of Penrose's math error is this:
Godel actually did prove:
1) If and only if first-order arithmetic is consistent, a certain
sentence, the Godel sentence, is true but not provable in first-order
arithmetic.
The above statement is, itself, provable in first-order arithmetic - it is
*not* something that humans can see but machines can't!
2) Because the above statement *is* provable in first-order arithmetic, it
follows that a proof in first-order arithmetic of the consistency of
first-order arithmetic could be used to prove the Godel sentence in
first-order arithmetic. Therefore first-order arithmetic can prove the
consistency of first-order arithmetic if and only if first-order arithmetic
is inconsistent.
Which is to say:
1) If first-order arithmetic is consistent, it contains sentences which
must be either true or false (by the Law of the Excluded Middle, a theorem
of classical logic albeit not intuitionistic logic), but which cannot be
proved or disproved in the system (neither the sentence nor its negation
has a proof within first-order arithmetic).
2) If first-order arithmetic is consistent in the sense that the formal
system will never produce a theorem stating "P and not-P", first-order
arithmetic will never produce a proof of its own consistency. (Because the
proof of consistency could be used to prove the Godel sentence which states
that no proof of the Godel sentence exists, thus creating a contradiction
within the system, thus contradicting our beginning assumption that
first-order arithmetic was consistent.)
Or to summarize even further:
1) If first-order arithmetic is consistent there are true sentences that
cannot be proven within the system, such as Godel's sentence.
2) If first-order arithmetic is consistent, it cannot prove its own
inconsistency.
Penrose's Big Mistake:
Penrose assumes that humans know first-order arithmetic to be consistent,
in which case we could conclude Godel's sentence to be true, and we would
have access to a knowledge not provable in first-order arithmetic. Penrose
also assumes, completely without justification, that no program can share
the human outlook which leads us to this knowledge; but it is not needful
to go into that. Penrose's primary premise is wrong. Humans don't know
that first-order arithmetic is consistent. We just have faith in it
because it has always worked so far.
Humans *know* (can prove, as opposed to taking on faith) that first-order
arithmetic is consistent if and only if it cannot prove its consistency.
This statement is itself provable in first-order arithmetic.
Godel's Theorem does not establish that humans know anything not knowable
to first-order arithmetic - let alone establish that human intelligence is
beyond all mechanism, whatever that is supposed to mean.
****
Today I would add that, although it is possible to prove the consistency of
first-order arithmetic in ZF, whether you trust that proof depends on
whether you believe in the consistency of ZF - I think. If not, substitute
ZF (and the appropriate Godel statement) for "first-order arithmetic" in
all sentences above.
-- Eliezer S. Yudkowsky http://intelligence.org/ Research Fellow, Singularity Institute for Artificial Intelligence
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