Matthew P Wiener <weemba@sagi.wistar.upenn.edu> wrote: > >>>Now a 0-1 law kicks in. > >>No, it does not. Logical 0-1 laws of the kind you've been talking >>about apply to very weak theories only, not anything as strong as PA. > >I'm sorry, I've been throwing around two theories (the vague "physical >mathematics" and the precise PA), and not stating which one the argument >actually applies to, and which one is better known and simply in the >right general direction, for lazyman expository purposes only.
Actually, that much was clear. You were leaning on the surprising adequacy of PA for much of analysis, to motivate the supposed ability of "PM" to prove 0-1 laws, but applying the argument to PM.
However, that brings up another problem: your argument used nothing about the initial theory other than the ability to implement 0-1 arguments and Goedel's theorem. So start the argument at a weaker subsystem of PM that shares these properties, PA perhaps, and prove that we don't even get as high as PM. To get around this you'd have to give some handwaving about not all types of evolution being created equal for 0-1 purposes, which defeats the purpose of the argument (to appeal to the supposed robustness of 0-1 laws).
> >The 0-1 laws from finite model theory that I've alluded to before apply >to theories with finite models. [...] But I instinctively do not believe >that the known 0-1 phenomena disappear out at infinity--it's like >Ramsey's theorem, only more sophisticated.
I'm willing to assume the finite/infinite business can be fudged somehow. The real problem is that an ability to talk about "even" and "odd" leads to P=1/2 statements, and so on.
