**From:** Eliezer Yudkowsky (*sentience@pobox.com*)

**Date:** Sat Oct 09 2004 - 02:05:09 MDT

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Ben Goertzel wrote:

*> The problem with Penrose's argument is pretty simple. He argues as
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*> follows:
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*>
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*> Premise 1) FOR EACH computer, THERE EXISTS some problem which that
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*> computer can't solve Premise 2) NOT [ FOR EACH human, THERE EXISTS some
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*> problem which that human can't solve ] Conclusion) Therefore, humans are
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*> not computers
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*>
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*> The problem is that his premise 2 is false
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*>
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*> Now, I can't prove that his premise 2 is false. So I can't prove he's
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*> wrong.
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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
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*> all.
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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

**Next message:**Christian Szegedy: "Re: Human mind not Turing computable according to Eliezer?"**Previous message:**Marc Geddes: "'The Next Really Big Enormous Thing' by Robin Hanson"**In reply to:**Ben Goertzel: "RE: Human mind not Turing computable according to Eliezer?"**Next in thread:**Ben Goertzel: "RE: Human mind not Turing computable according to Eliezer?"**Reply:**Ben Goertzel: "RE: Human mind not Turing computable according to Eliezer?"**Reply:**Thomas Buckner: "Re: Human mind not Turing computable according to Eliezer?"**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ] [ attachment ]

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