In article <5obiom$2kk@drn.zippo.com>, daryl@cogentex (Daryl McCullough) writes: >weemba@sagi.wistar.upenn.edu (Matthew P. Wiener) says...
>>..the essense of my argument is that evolution and natural selection >>work with the physical, and ZFC is ridiculous overkill so far as the >>physical is considered.
>I agree that ZFC allows us to prove theorems (about infinite sets) that are >completely irrelevant to survival in the world. However, ZFC also does a >much better job of proving facts about *integers* than PA does. All around, >ZFC is a better theory for constructing models in than PA, and models are >generally useful for reasoning. The extravagant power of ZFC may be just >a side-effect of having a theory that is powerful enough to prove all the >physically significant arithmetic facts.
Come come. V_w^w, V_w_1, etc, would do this better job too.
>Suppose that the theory C is definable via a formula C(x). Let T(n,x) >be the formula in PA meaning "x is a code for a theorem of the r.e. theory >with index n". Let Con(n) be the formula saying in PA that n is the index >of a consistent r.e. theory. There is a big difference between the >following two claims:
> 1. For some n, A |- for all x, C(x) <-> T(n,x) and Con(n). >and > 2. A |- for some n, C(x) <-> T(n,x) and Con(n).
>When you talk about the "limit theory" or the "omega theory", I think >that you are in case 2, above. The limit theory is perhaps definable (and >provably consistent) in PA, but that *doesn't* mean that PA can determine >an index for the limit theory, and it doesn't mean that PA has a greater >consistency strength than the limit theory.
The evolutionary model I've described is quite constructive in its weeding out of inconsistencies. A TM-mind presumably doesn't balk at merely having A and ~A around, if it doesn't yet see that the two are contradictory. Only when a short proof of 0=1 comes up--if necessary, due to the appearance of intermediate steps as mutant axioms--*then* it balks.
As such, I expect the 0-1 laws to put me in case 1 above. It is not the case that the omega consistency is arrived at by an impenetrable fiat, as with your example (omitted) of a case 2, but that glaringly obvious inconsistencies are eliminated, and that given time, all inconsistencies become obvious. -- -Matthew P Wiener (weemba@sagi.wistar.upenn.edu)
