In article <email@example.com>, kubo@abel (Tal Kubo) writes: >Matthew P Wiener <firstname.lastname@example.org> wrote:
>>>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.
>>It does? I've been thinking about this for a few days now, and I give up.
>>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.
>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.
> You imagined some PA >analogue of Wiener measure for random paths? No problem -- ask whether the >path ends up above or below a line. Wiener measure for paths with drift? >No problem again: ask whether one random path lies above or below another.
Your "no problem" is correct. The intent is for PA-like theories to prove things like "paths are above such a line with probability p" with probability 0 or 1. That's all. -- -Matthew P Wiener (email@example.com)