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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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Kristian Berry
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  • Suggested p-o-math tag. – J D Dec 11 '21 at 14:37
  • Normally I would agree, but I think this plays into the axiomatic method in general, including metaphysical and metaethical cases. It is just that in mathematics the question is more apparent. – Kristian Berry Dec 11 '21 at 15:21
  • I found an application of the "technique" first in a set-theoretic context but it quickly proved applicable to ethics in a surprising and weird way. – Kristian Berry Dec 11 '21 at 15:25
  • I'll post that more in chat, though. – Kristian Berry Dec 11 '21 at 15:25
  • You're viewing the decision-making in terms of conceptual analysis. My suggestion is predicated on the idea that someone, like me, might have a general interest in filtering tags broadly for categorization instead of having loads of more specific tags on the interest list. The volume on this site is enough for me to have a tag for p-o-science, p-o-mathematics without listing 50 other tags that constitute a more specific mechanism. – J D Dec 11 '21 at 15:25
  • Please. My knowledge of the philosophy of set theory doesn't include these specific forms of justification. – J D Dec 11 '21 at 15:26
  • Let us [continue this discussion in chat](https://chat.stackexchange.com/rooms/132245/discussion-between-j-d-and-kristian-berry). – J D Dec 11 '21 at 15:26
  • Rule out on what grounds? If you treat an axiom as if it were true, and that helps you solve mathematical and physical problems, what is the problem? Elaine Landry: "*I further use my reading of Plato to develop what I call as-if-ism, the view that, in mathematics, we treat our hypotheses as if they were first principles and we do this with the aim of solving mathematical problems. I then extend this view to modern mathematics wherein the method of mathematics becomes the axiomatic method, noting that this engenders a shift from as-if hypotheses to as-if axioms*". – J Kusin Dec 11 '21 at 16:11
  • @JKusin, but what is an axiom? I suppose the standard definition is, "A premise that is not a conclusion to any higher/deeper argument." However, I have come to reject that definition, since I think axioms can be inferred, but the mechanism of inference is a function of erotetic logic, i.e. an axiom is an assertoric function that is inferred from an erotetic function, so to say. At any rate, that is how I use the word "axiom" in my writing, now. The only other characterization I have is, "An axiom is a premise that well-founds the order of inference." – Kristian Berry Dec 11 '21 at 22:34
  • So since coherentism and infinitism are not well-founded inferential orders, it would be absurd/bizarre to posit axioms of coherentism and infinitism as such. This transfers to set theory in a rejection of the axiomatic method for looping and descending hypersets, but *not* a rejection of non-axiomatic methods of justification for hyperset claims. – Kristian Berry Dec 11 '21 at 22:35

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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.

"

  • 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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  • But how do axioms solve mathematical problems, besides by providing a factive well-foundation from which to derive the solutions to those problems? We don't adopt CH or ~CH as an axiom just because that adoption "solves" the Continuum problem (directly, by fiat). Instead, in "actual mathematical practice" we look for more general axioms from which the particular answers desired can be inferred. – Kristian Berry Dec 12 '21 at 00:02
  • But you deserve an upvote, nonetheless! – Kristian Berry Dec 12 '21 at 00:05
  • But one more thing, too: there is something paradoxical about foundationalism, not necessarily in itself, but historically, in that people *argue for* it. You could look at foundationalism as the conclusion of a simple disjunctive inference: Either foundationalism, coherentism, infinitism, or skepticism; ~(~foundationalism); therefore, foundationalism. But this quasi-paradox is resolved by the erotetic logic of the system: the disjunction is an erotetic function first (and in fact, no single solution is inferred, but all of them are). – Kristian Berry Dec 12 '21 at 00:35
  • @KristianBerry I agree we tend to look for more general axioms first. But why do we look for more general axioms first? Because they are human-understandable? Because they more fit our intuitions about sets and the real world? And Is "solving problems" really a foundation? If it is, it is the most conservative of foundations. I'm afraid I can't keep up with these conversations so hopefully someone else will come along :) They are way beyond my depth, and I am probably already diluting them. – J Kusin Dec 12 '21 at 01:02