It occurred to me a little while ago, that there is a trichotomy in set theory that maps to the positive solutions to the problem of the regress of inferential reasons. Namely, well-founded sets map to foundationalism, looping sets to coherentism, and infinitely descending elementhood chains to infinitism. (The empty set maps to the empty ("skeptical") justification logic, J0.) What I gleaned from this conception was that, though it is possible to represent axioms of antifoundation, such axioms conflict with the purpose of axioms, which is to provide for well-founded justification. In other words, despite being logical possibilities, such principles are not otherwise justifiable (though, to be sure, nonwell-founded justification itself is possible, i.e. there are beliefs that can be coherentistically or infinitistically justified, including beliefs about nonwell-founded sets existing).
Now, I also have been assuming that the general-particular ordering is the original source of mathematical order. Let us refer to generality as "𝔽," and the ordering in question as the "𝔽 → F" order. The principal thing seems to be that "𝔽 → F" is transitive: if A is more general than B, and if B is more general than C, then A is more general than C. I have a file downloaded somewhere, of an incomplete copy of Zalta's(?) axiomatic metaphysics treatise, so I imagine reflections like this are present there, but otherwise I've never read of the transitivity of "𝔽 → F."
My question is this: does such a picture of axiomatic justification, rule out overly specific axioms? For example, in the SEP article on the Continuum Hypothesis, Koellner goes over an axiom that is stated like so: "Axiom (∗): ADL(ℝ) holds and L(P(ω1)) is a ℙmax-generic extension of L(ℝ)." But modulo 𝔽, this sounds way too particular to be sufficiently justified.
One might object that the question of generalized justification is not otherwise at issue in characterizing a higher set-theoretic axiom; but I think that the deep issue of justifying axioms does involve the generalization problem, anyway.