If you let X be incomplete like twanvl suggests, then you pretty much agree with my position of using probability as a useful tool, and disagree with Bayesian fundamentalism.

Getting into finer points of what is constructible or provable in what language is really not a kind of discussion we could usefully have within confines of lesswrong comment boxes, since we would need to start by formalizing everything far more than we normally do. And it wouldn't really work, it is simply not possible to escape Goedel's incompleteness theorem if you have something even slightly powerful and non-finite, it will get you one way or another.