weemba@sagi.wistar.upenn.edu (Matthew P. Wiener) says...
>>...it seems to me that to apply the Goedelian argument, you >>have to assume that the limit theory is essentially the same >>as the starting theory. > >I don't see that assumption at all.
As has been pointed out before, if a theory is defined in such a way that it is necessarily consistent (because the inconsistencies are weeded out by some process), then it is no trick to prove that it is consistent. Such "by-definition" consistent theories can be of *arbitrary* strength. Goedel's theorem doesn't say anything about the relative consistency strength of the limit theory C versus the starting theory A. In order to apply Goedel, it is not enough to have a definition of the limit theory, there must also be an *index* for it. More specifically, for any statement X, if C proves X, then must be that A proves that C proves X. Why do you think that holds? It seems to me that your answer was that C is essentially equal to A, (and A can prove it). But if you know that, you don't need Goedel.
>>Actually, if it is true that inconsistencies are weeded out, then >>C is not just consistent relative to A, but is *absolutely* >>consistent. > >C's inconsistencies are weeded out. A's would be put back in.
Why????? That doesn't make a bit of sense. If the starting theory is inconsistent (and inconsistencies are bad, evolutionarily) then mutation plus natural selection would eventually get rid of the inconsistencies.
>>I don't know what you mean here. In order for you to conclude that >>theory A has a consistency strength greater than C, it must be that >>A can prove con(n) where n is an index for C. > >But it need not know n itself.
I don't see why not.
>>>>Sorry for not being clear. What I meant was this: If there is a >>>>design for a TM mind that is (a) adequate for survival, and (b) >>>>simple enough to have evolved in the first place, and (c) incapable >>>>of doing set theory, then I expect that that is what would have >>>>evolved, instead of a TM mind that is capable of doing set theory. > >>>Me too. Therefore, I suspect that we are not TM minds. > >>But replace the phrase "TM" by "non-TM" and it makes no >>difference. > >Sure it does. My argument does not apply to "non-TM".
Right, but only because you have astep that invokes Goedel unnecessarily.
>>>..our ability to do ZFC is a side effect. That's one of the central >>>points of my arguments. > >>Well, it could just as well be a side effect for a TM mind. > >No, not just as well. That is the point of my argument.
I know, but my point is that your argument doesn't work (or at least I don't see how it does.)
>>I'm not sure exactly what you mean. My point is that a TM that was >>adequately able to survive in the world might, as a side effect, be >>able to do ZFC. *Not* that doing ZFC would help it to survive. > >I am aware of your point. The question is how is it going to get there? >Based on fifty years of dismal, rather pathetic, failure, I consider the >short cut magic programming hypothesis for doing higher mathematics to be >implausible on its face.
Well, that's a different argument, that has nothing to do with Goedel or 0-1 laws.
