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Suppose that there is a set of all good things, and that it is well-founded. Then it would not be an element of itself, i.e. would not be a good thing. Maybe it would be hypergood, but maybe it would be neutral; for it to be evil seems unlikely/impossible.

At any rate, one might expect that the universal set of goodness should be good, too, so one might be led to conclude that such a set (if it exists or is even at all really conceivable) must be at least a Quine atom of some sort, or akin to those anyway. However, then the cardinality associated with goodness is not well-founded. If not well-founded, is it incommensurable with sets of numbers that are well-founded?{!} If incommensurable with the n, perhaps the "moral numbers" (or quantities, anyway) would admit of a different sense of their arithmetic, too; this might be what G. E. Moore subconsciously (or consciously) considered to explain the possibility of "organic unities" of intrinsic ethical values.

{!}Suppose there were a set of all sets that are commensurable with other sets. If indeed well-founded, such a set would not be commensurable with other sets. This, then, for a reason that is sort of an obverse of the above, yet still a testament to the existence of incommensurability in general (not just the obscurely abstract possibility of it, either, though note that many are moved to incommensurability/incomparability claims in ethics on intuitive grounds).

Kristian Berry
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  • Isn't there a transgression between types of things: the elements of the set themselves carry the property "goodness", but that doesn't mean that this property is applicable/transfers to the set of good things. This would have to be added separately, and if a set is not of the same nature as its elements, then we would have 2 kinds of goodness: one for good things, and one for sets of good things. – Frank Jan 13 '23 at 04:09
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    I agree with Frank, applying "good" to sets hardly makes any sense. A set of apples is not an apple and a set of red things is not red, so why should good be any different? If your "goodness" is just supposed to be cardinality (or some like function) then you are essentially forming the set of all sets and asking if its cardinality is "commensurable" with something. Well, it is not a set and it has no cardinality. – Conifold Jan 13 '23 at 08:52
  • @Conifold, the universal set/cardinal/w/e exists or doesn't exist or exists as a non-set or whatever, depending on how one tailors one's set theory. Apparently it is even possible to construct acceptable versions of set theory where there is a **largest** cardinal in the more normal way (Gitman/Hamkins, "What is the theory ZFC without powerset?"). Now sometimes a predicate is such that a set can share it with its elements, or would be expected to, e.g. a set of all true sentences would perhaps be a conjunctive sentence itself, then, and true to boot. So too with goodness, here, *perhaps.* – Kristian Berry Jan 13 '23 at 09:34
  • @Frank so I would have to argue, anyway, that goodness is a property that can be shared by a set and its elements, on account of something to do with goodness, the encompassing sets, etc. Hypergoodness (or theologians tended to call it something like "supergoodness," I think for Latin-related reasons) is a plausible alternative, though (a set of goods is itself supergood, a set of supergoods is superdupergood, etc. and blahblahblah). – Kristian Berry Jan 13 '23 at 09:38
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    I do not see anything to the "goodness" here beyond being cardinality-like, you might as well call it beauty or strength. If that is all there is to it then the answer is - whatever set theory can be "tailored" to, no other restrictions. – Conifold Jan 13 '23 at 10:48
  • Hasn't this kind of (foundational) issue been explored at length in various set theories, type theories, univalent foundations, etc etc? – Frank Jan 13 '23 at 16:03
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    @Frank my exposure to alternatives to the (strict version of) the foundation axiom began with the SEP article on Aczel's work. By now it seems, for me, as if "all that is solid melts into air" and I wonder if mathematics is founded on individual problems instead of an attempt at general simplification. – Kristian Berry Jan 14 '23 at 11:14
  • @Conifold so I do worry a lot about the "stability" of set theory, at this point. At any rate, it seems that incommensurability requires separate sequences/kinds of cardinality so I wonder here if the deontic separation can be accounted for by divergent well-foundedness relations. Then the incommensurability of moral values with each other could be modeled by further such different relations. – Kristian Berry Jan 14 '23 at 11:20
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    @KristianBerry I think it's "worse" and "better" - mathematics is ill-defined. You can choose axioms, and even the logic you use. One wonders what is really fixed in "mathematics". Maybe all that is left is a game with symbols. – Frank Jan 14 '23 at 17:47
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    Zeno the Stoic and Kant would judge the set of forms of (definite, certain) good be a Quine atom in modern set-theoretic jargon since the good are categorically imperative and all lead to eudaimonia, similar to Frege's the truth object. The partial ordering lattice may fit those uncertain good in practice assignable with either prior or updated posterior probabilities ideally collapsible to the set of real line segment [0,1] which itself is well-defined and morally neutral situated in an entirely rational realm... – Double Knot Jan 19 '23 at 02:04

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