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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 "," and the ordering in question as the " → F" order. The principal thing seems to be that " → 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."

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 , 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.

J D
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I'm out of my depth but maybe this is helpful from https://youtu.be/j4dlamySLuE?t=379. It seems like the presenter Elaine Landry disagrees with your "the purpose of axioms...is to provide for well-founded justification". She would seem to say the purpose of axioms is to solve mathematical and physical problems.

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  • I begin first with Plato to show that much philosophical milk has been spilt owing to our conflating the method of mathematics with the method of philosophy .
  • I further use my reading of Plato to develop what I call as-ifism , the view that , in mathematics , we treat our hypotheses as if they were first principles and we do this with the purpose of solving mathematical problems not philosophical ones .
  • I next extend as-ifism to modem mathematics wherein the method of mathematics becomes the axiomatic method , noting that this engenders a shift from as-if hypotheses to as-if axioms , and axioms as implicit definitions .
  • Again , I pause to note that the conflation of the method of mathematics with the method of philosophy , witnessed well by the Frege-Hilbert debate , has led to the continued confusion of mathematics with metaphysics.
  • Finally , I use a methodologically interpreted as-ifism to break Benacerraf's dilemma by showing that there are two types of existence at play . My overall lesson is this : when we shift our focus from solving philosophical problems to solving mathematical ones , thereby avoiding the conflation of mathematical and metaphysical considerations , we see that a methodologically interpreted structural as-ifism can be used to provide an account of both the practice and the applicability of mathematics
  • My overall lesson is this : when we shift our focus from solving philosophical problems to solving mathematical ones , thereby avoiding the conflation of mathematical and metaphysical considerations , we see that a methodologically interpreted structural as-ifism can be used to provide an account of both the practice and the applicability of mathematics

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J Kusin
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