If you’re so interested in logical induction, aren’t you already assuming that classical mathematics is The One True Logic? Why is that? Why not look at ordinary mathematics internal to a topos and then ask what logical induction looks like for that?
And as for reflection, a topos with a NNO has (higher order) primitive recursion so your claim about not having reflection is confusing.
And lastly, your title doesn’t match your thesis. All you show is that you can’t directly do probability in toposes. Category theory is extraordinarily useful for many areas of mathematics in general, and is more than just a language. See Beck’s monadicity theorem, the adjoint functor theorem, the small object argument, Gabriel Ulmer duality, and so on for nontrivial results in category theory.
Maybe you shouldn’t base your entire identity around doing probability theory. At the very least, epistemology spills far beyond the purview of probability theory.
Yes. That is the logical induction I was talking about.