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.
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.