I guess it's in the air at the moment

Indeed, it seems relatively easy to make progress in this direction given the current state of things! (I would have been surprised if you had not been thinking along similar lines to my paper.)

In fact P( "axioms inconsistent" ) > 0 in our formalism.

Ah! Interesting. Why is that?

I did construct a distribution in which this is not the case, but it was not particularly satisfying. If (during the random theory generation) you block the creation of existential statements until an example has already been introduced, then you seem to get acceptable results. However, the probability distribution does not seem obviously correct.