Matthew P Wiener <weemba@sagi.wistar.upenn.edu> wrote: > >>>For example, an ultraproduct of PA models does not lead to any kind >>>of "P=1/2" statements. Truth is as black and white in the ultraproduct >>>as it was going in. > >>Truth is black and white in any model. Probability, or something akin >>to it, is an added feature that you are assuming in order to relate the >>models to evolution. Symmetries such as even/odd, positive/negative >>and so on will introduce fractional probability statements and so spoil >>any 0-1 law. > >This is ludicrous. I can talk about Q inside N, and then take an ultra- >power with its black-and-white truth. Talk about fractions *inside* a >theory doesn't force fractions to show up in the metatheory.
Maybe I said something unclear, but "talking about fractions" within a theory has nothing to do with anything I was saying.
> >>So, in your random graph metaphor, "has an odd number of edges" >>is a statement that any expressive enough theory can make, and which will >>evaluate to 1/2 under any notion of "probability". > >Of course not! How ridiculous. The 0-1 laws I'm talking about concern >*proofs* of statements, not the statements of probability themselves.
It's not ridiculous, and obviously true, for 0-1 laws of the finite model theory variety. Since your discussion all seemed to be in reference to the kind of 0-1 laws that do refer to probability statements, and hypothetical extensions of the latter, I assumed that's the kind you were using in your Goedel argument.
Since it turns out you're actually talking about something else, then yes, the business about symmetries is not relevant. However, it also means there are no longer any "prototypes" around to make credible the idea of a 0-1 law applying at all.
