Suppose there existed a natural probability (or density, or weakening of such) on the “set” of all models of the Zermelo-Fraenkel set-theory axioms. Suppose some “natural” mathematical statement had the property of having positive probability, different from 1, of holding in a random model. How should we interpret such a situation? Say, if the Continuum Hypothesis has probability 6/π2, of being false?
And if some natural statement P was shown to be a consequence of two other statements, having probability p and q, respectively, of holding in a random model, with p+q>1… so that the existence of a model where P holds would follow in highly non-constructive fashion… What would you think, philosophically or intuitively, of the “truth” of that statement?