> That's all well and good but what other people are talking about is the nature of reality which, in pursuit of, those crazy people, they are perfectly willing to doubt axioms.

Solomonoff induction does doubt and change axioms. It's a fundamental part of the whole process in fact.

> That doesn't make them true in the big sense of True, it makes them expedient, productive of theory, generative, a lot of wonderful things, maybe even strongly implied by all evidence, but not apriori true. They're dubitable.

Logic is used to make distinctions. Two theories with differing axiomatic bases will make different distinctions, but if they make the same predictions in all cases, then they are logically the same, ie. there is a fundamental isomorphism between them. In this case, it literally doesn't matter if one is "actually really true", and the other is a mathematical dual of some sort.

For instance, polar and Cartesian coordinates are completely equivalent. A theory cast in one might be easier for us to work with, but even if reality really used the other coordinate system, it quite literally doesn't matter.

In the case when the two theories do differ in their predictions, we should epistemically prefer one over the other, and Solomonoff Induction shows us how to do this rigourously.