From: Eliezer Yudkowsky (sentience@pobox.com)
Date: Wed Aug 18 2004 - 03:41:14 MDT
Christian Szegedy wrote:
>>
>> 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.
>
> I think that the mathematical statements have different degrees of
> "absoluteness" (some kind of relevance to the existing world).
My attitude toward mathematics is strongly dependent on relevance -
specifically, relevance to an AI theory. It is not exactly a theorem of
Yudkowskian Really Powerful Optimization Process FAI theory that flipping
any bit in the RPOP has an expected (probabilistic) effect on the RPOP's
real-world actions. It's not a theorem, but it's a good guideline to the
theory. (In practice, you'd want to code information in less fragile form;
but this doesn't change the *probabilistic* effect.) Suppose the RPOP
believes in the axiom of choice. Now suppose we flip a bit so that the
RPOP does not believe in the axiom of choice. As far as I can tell this
makes absolutely no difference to the RPOP's actions, not even a
probabilistic difference, which places the Axiom of Choice entirely outside
the conceptual universe and ontology of an RPOP - you can't even translate
the question into RPOP-language. If it's a Collective Volition RPOP, the
RPOP can ask whether a human is likely to believe in the Axiom of Choice,
but not whether the Axiom of Choice is "true"; it's like asking about
whether free will exists. You can ask whether humans think "free will"
exists, but not ask whether free will actually exists.
Anyway, the upshot is that I acknowledge the existence of only those math
questions which can affect an RPOP action. This is the hidden, underlying
motivation behind my infinite set atheism.
> For example:
>
> "1+1=2" is an absolute statement which is indisputable. One reason for
> that, that it does not include quantors.
>
> "There is an odd perfect number" has a high relevance, since it can be
> refuted by giving an example. Its independence to any reasonable
> axiom-system would prove its validity. This shows that this statement is
> somewhat absolute.
>
> "There are infinitely many prime numbers" is less absolute, but it is
> still quite absolute, since you can show a concrete Turing machine
> generating them.
>
> "Every Goodstein sequence stops" is even less absolute since you can
> neither refute it nor exhibit some sequence of objects supporting it.
"Every Goodstein sequence stops" is relevant to an RPOP. If the RPOP for
some reason needs to compute whether a physically instantiated Goodstein
sequence halts, the RPOP must separately prove that Goodstein stops for
each relevant N, if the RPOP is using Peano axioms. While if the RPOP uses
ZFC, the RPOP will be able to decide once and for all that every Goodstein
sequence stops, then just retrieve the fact from memory on each relevant
occasion. This is not only more efficient, it makes ZFC a more *compressed
explanation* for the behavior of natural numbers (and hence, physical
processes) than Peano Arithmetic. This is a reason for adopting ZFC as an
"explanation" for observed math, by the law of Occam's Razor. Adopting ZFC
would then be like adopting a physical hypothesis that enables us to deduce
conservation of momentum; as opposed to PA recording the history of every
known finite physics experiment in which momentum has been conserved,
without allowing us to generalize a universal quantifier.
I wish I knew exactly *which* axioms of ZFC allow mathematicians to
conclude that every Goodstein sequence stops! Which specific aspect of ZFC
makes ZFC a more compressed explanation for real-world physical behaviors
than PA? I doubt the Axiom of Choice has anything to do with it. But I
would be forced to believe in infinite sets if they were the simplest
explanation for the behavior of the Goodstein sequence, and there was no
simpler explanation that did not postulate infinite sets.
> The axiom of choice is even less absolute. It cannot be refuted or
> supported by any examples. I don't know of any interpretation which
> holds any information about the "real world".
On this, of course, we are in agreement.
-- Eliezer S. Yudkowsky http://intelligence.org/ Research Fellow, Singularity Institute for Artificial Intelligence
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