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paulfchristiano comments on Reflection in Probabilistic Logic - Less Wrong
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This is right. (I think we point out in the paper that the model necessarily contains non-standard infinitesimals.) But an interesting question is: can this distribution assign any model a positive probability? I would guess not (e.g. because P("G") is uniformly random for some "G").
I agree with your guess and I think that I see a way of proving it.
Let us make the same assumptions as in Theo. 2 and assume that T contains PA.
Let G be any sentence of L'. We can build a second sentence G2 such that "G2 <=> ( G and (Pr(G and G2)< p))" for P(G)/3< p <2P(G)/3, using diagonalization. From this and the reflection scheme, it should be possible to prove that P(G and G2) and P(G and not G2) are both smaller than (2/3)P(G).
Repeating the argument above, we can show that any complete theory in L' must have vanishing probability and therefore every model must also have vanishing probability.