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hal@finney.org wrote:

*>
*

*> Eliezer writes:
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*> > If the Goldbach Conjecture is true as a consequence of an infinite number
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*> > of independent mathematical facts, it would mean that there's a *specific*
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*> > problem that *no* Turing machine can solve.
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*>
*

*> That doesn't sound right, as you can always add the GC or some equivalent
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*> theorem to your axiom base. I don't see how a fact like the GC can be
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*> true only as a consequence of an infinite number of facts. It seems for
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*> that to be true that your fact would have to carry an infinite amount of
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*> information, like the value of Chaitin's Omega.
*

The reason why this keeps coming up for the Goldbach Conjecture is that,

by the time you get into large numbers, any given even number is the sum

of two primes in hundreds of thousands of different ways. The larger a

number, the more infinitesimal the probability that it's not the sum of

two primes; *that* can be demonstrated. But it's quite possible that -

even though one could demonstrate, using a finite amount of computation,

an arbitrarily small probability that the Goldbach Conjecture is false -

there's no way to prove that the Goldbach Conjecture is true. GC would be

true as a consequence of an infinite number of facts distributed over all

the even numbers and all the primes. Someone (I think Hofstadter) said

that arithmetic and multiplication don't relate well in this instance.

The higher you go, the more primes you get, so it's not surprising that

larger and larger even numbers can be expressed as the sum of two primes

in more and more ways. Nor is it surprising that the probability of an

exception gets smaller and smaller. But that doesn't mean that there's

any coherent rule relating primes to even numbers. So it's quite possible

that the Goldbach Conjecture happens to be one of the easiest sentences to

state that's true, and statistically "proveable", but mathematically

unproveable.

(But it should now be clear why the Fifth Postulate is a bad analogy, and

also why I talked about the Goldbach Conjecture rather than the Riemann

Hypothesis.)

-- -- -- -- --

Eliezer S. Yudkowsky http://singinst.org/

Research Fellow, Singularity Institute for Artificial Intelligence

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