From: Christian Szegedy (szegedy@or.uni-bonn.de)
Date: Wed Aug 18 2004 - 07:10:07 MDT
Eliezer Yudkowsky wrote:
> 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.
It is a valid point and I completely agree.
Just a side remark: I never stated that the goodstein sequence or P!=NP
has no relevance to the physical reality as we know it. Far from that!
I just think that a lot of people including most mathematiciains do
not differenciate between different type of existences. If I write down an
logical expression starting with a quantor it does not always imply the
(possible) existence or nonexistence of objects in the "real physical
world".
There are a lot of degrees of existence including constructible finite
structures,
potentially infinite but recursive objects, countable but not omputable
objects, unreachable cardinals and a lot of other complicated possibilities.
I don't say I believe or not belive in any of them, I just say that one
should differentiate between those type of "existences".
Another point: the Axiom of Choice is also a useful "tool of logic". In
algebra
it is common that statements can be proved for most practically relevant
cases, but the general case requires the Axiom of Choice. (The existence
of the algebraic closure of fields is a prominent but not singular example)
In this sense the Axiom of Choice also yields a more "compressed
explanation".
This does not make me believe in the existance of the cardinal numbers and
similar stuff. However my intuition is that a PROP using AOC would be more
efficient than another one not considering it at all.
> 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.
In fact, you don't need the Axiom of Choice. Operations on
countable cardinal numbers are needed. I could not tell whether
you need the full ZFC or not. My guess is that you need it.
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