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

**Date:** Mon Aug 16 2004 - 09:18:23 MDT

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Christian Szegedy wrote:

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*> You may be able to reduce the notion of physical experiment so skilfully
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*> that the outcome of each experiment can be decided by the axioms alone.
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*> Of course, in this case you can tell using experiments in which model
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*> you are. Otherwise the theory is not complete: it does not describe the
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*> physical laws completely.
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I am told there are natural-seeming statements about the natural numbers

which can be formulated in Peano Arithmetic, cannot be proved in Peano

Arithmetic, and can be proved in ZFC set theory. This is, as I see it, the

strongest argument against infinite set atheism.

http://www.maths.uq.edu.au/~krm/goodstein.html

http://www.ltn.lv/~podnieks/gta.html

So Peano arithmetic determines, e.g., the result of any specific instance

of the Goodstein sequence, because a Goodstein sequence is

straightforwardly computable; but the fact that all Goodstein sequences

terminate at zero after finite time is provable in ZFC but not PA. So the

question is whether the halting property of Goodstein sequences is

"decided" by the Peano axioms - the complete behavior of any actual

sequence is determined by the Peano axioms; and the halting property is

describable in Peano sentences, and provable in ZFC, but not provable from

the Peano axioms.

-- Eliezer S. Yudkowsky http://intelligence.org/ Research Fellow, Singularity Institute for Artificial Intelligence

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