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What is the most general form of the well-ordering theorem? I ask because I wanted to share the following:

  1. In its most general form the Well-Ordering Theorem is the following: Any collection of things can be well ordered. Where the words “collection” and “things” are primitive notions. And for any collection of things, it is “well-ordered” if and only if it has a first member.
  2. Thus, the Well-Ordering Theorem in Mathematics is a universal instantiation of 1.
  3. Thus, Aquinas’s Argument from Motion is a universal instantiation of 1
  4. Thus, my generalization of Aquinas’s Argument from Motion is a universal instantiation of 1 too.
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When it comes to applying specialized axioms from set theory to things like causal regresses, you might well be looking more for the foundation/regularity than the choice axioms. I say this because I used to subconsciously conflate the two, or think that the choice axiom was like a specification of the foundation one, but at any rate, I thought of questions like, "Should a set of axioms be well-ordered?" when what I really had in mind was, "Are the axioms themselves well-founded?" (in the sense of being non-deductively grounded in some meta-axiom that made for a "preferred sequence of consideration," e.g. as if we were to accept a certain number of axioms, and then we were to accept these one after another in a specific order).

And it's not like this is a hard mistake to make: see this MathSE question and this MathOF one.

(That a causal regress, with a first cause to its name, should be called "well-founded," is a proposal that can be justified by appeal to the role of the foundation concept (well- or ill-formed as it may be) in the analysis of grounding and fundamentality; as well as the programmatique of causal-set theory.)


But, if you still really want something like a most general well-ordering principle, that applies to all collections of things as primitively situated as such, then besides the type-theoretic choice axiom I mentioned in my comment, there's also the set-theoretic global choice axiom ((meta)scheme) to consider.

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  • thank you for reading the OP with charity and understanding rather than with derision.
    – AUTIST INC
    Commented Jun 23 at 10:12
  • Also, I don't think you were conflating the two. I think you were using the transfer principle whether unconsciously or implicitly. For more information as to what the transfer principle is, see the following: en.wikipedia.org/wiki/Axiom_of_choice#Equivalents
    – AUTIST INC
    Commented Jun 23 at 10:14
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    @AUTISTINC at the time, I was even less familiar with the determinacy axioms (as alternatives to choice) than I am now, but so I think that I could've held that either choice or determinacy can be "grounded in" foundation "after the fact." (Either of those, or maybe something else, even...) That is, the vague notion I had of "emerging from foundation as a theme of the axioms" could support multiple specifications at this juncture. But by now I'm not so exercised to find such convenient groups of axioms (the multiverse did away with that...). Commented Jun 23 at 10:25

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