Matthew P Wiener <firstname.lastname@example.org> wrote: > >>If you mean some other type of 0-1 law, please say so -- it would >>mark the references to Fagin's theorem, etc, as just so much red >>herring. > >The reference was called a prototype for a reason. Obviously first >order finite graph theory is not enough for physical mathematics.
The question was whether you meant a syntactic 0-1 law: under some evolution(s), every statement of some theory gets probability 0 or 1. The even/odd problem says that there is no such law for your purposes, whatever the evolutionary model.
> >>>The original theory can thus talk about the theory that consists of the >>>evolved probability 1 statements. > >>No, it cannot. That would require a blueprint for evolution itself. > >How so? The whole point is we do not have any blueprint. If we had a known >accepted blueprint, I would not have to appeal to generic 0-1 behavior. In >other words, _if you fix a recursive blueprint for evolution_, then a specific >0-1 law would apply and be used by the alpha theory to talk about the omega >theory.
Well, that raises the same suspicion as a physical argument that strongly relies on some continuous parameter being rational. Probability 0 premises being invoked crucially in an argument about what happens with probability 1 make it rather brittle.
> >>These are consistent and definable in the original TM-mind. > >>No, with probability 1 they are not definable. > >Define them collectively as the probability 1 sentences in the omega limit.
IF you have a blueprint, and do a lot of handwaving. Big nonrobust "IF" there, but OK, let's assume it along with a 0-1 law as defined above.
> >>>So by the Goedel theorem, it's just the original system! > >>No, Goedel's theorem doesn't apply, with probability 1. > >Not _the_ Goedel's theorem, correct, but the obvious variants.
The argument gets very hazy at that point, and I don't buy it at face value (specifically, consistency-of-fragments for the limit need not come for free, and a 0-1 law almost certainly enables a full consistency proof for the base theory). Could you explain it in more detail? The excuses about sneering AI types don't apply here, as nobody is asking you for a mechanism for evolution or the 0-1 law.