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If it could be determinate, how many things don't exist, i.e. if there could be a set of nonexistent things, would the existence of other things follow "mechanically"?

  1. If it's not determinate that k-many things don't exist, then there is a proper class of nonexistent things, and there is not a proper such set.
  2. If there weren't a proper set of nonexistent things, then nothing would not exist (supposing that we don't properly quantify over things except by considering them to be in the range of a proper set; as if it's such that if they're in the range of a proper class, they're not quantification-determinate).
  3. If there is proper set of nonexistent things, then it must be of some size k.
  4. The set-sized existential complement of the set of size k is the rest of some superset of k as such.
  5. So the number of things that exist must be equal to or larger than the number of things that don't.
  6. So "there must be something rather than only nothing." QED∀

Historically, à la Fregeanism or the proposition V = WF, the empty set is styled as the set of things that don't exist in the theory (true contradictions for Fregeanism, unwell-founded sets for the exclusivistic axiom of foundation). However, if that is what was being meant in (1) through (6), the implication would be that the set of things that don't exist is empty, and so still everything somehow exists. And yet on another hand, if we are trying to answer, "Why is there something rather than nothing?" (at least for concrete somethings, if abstracta are necessarily existent) then have we yet left, "Why is there a set of nonexistent things instead of not there being such a set?" unanswered? (Note that, if there must be a proper set of nonexistent things, and a proper set of existent things, then the remaining proper class (the rest of the upward/outward expanse of sets) is such as neither exists nor doesn't exist, which conclusion might bring us some pause.)

∀Or, then, "If nothing is nontrivially absolutely true, then, 'Nothing exists,' is not nontrivially absolutely true, i.e. there is a substantive exception to, 'Nothing exists.' If there is a substantive exception to, 'Nothing exists,' then a substantive something exists. Ergo..." Perhaps my sense of these questions has been poisoned by the ontological neutralism of analytic philosophy of logic, but that (the preceding argument) sounds like trickery, however.

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  • Is there a Meinongian set theory with existence predicate? I never came across such a thing, but if there was I'd imagine comprehension and cardinality will not work there as in the ordinary quantificational theories. You seem to be combining results of the latter with existence predicate, which cannot be there by design. Such a chimera may well be inconsistent and entail anything by explosion.
    – Conifold
    Aug 27 at 8:27
  • @Conifold perhaps it's an inconsistency hidden in a generic or naive idea of the set-vs.-class distinction? We might ask about "the proper class of things that are not quantified over," whereby "is quantified over" is substituted for "exists" as a property/predicate (or: is "is quantified over" a predicate after all???). The only other thing I'm reminded of is Maddy's remark, concerning Reinhardt's set-class-superclass-superduper-class... "problem" (she thought at one point that it would be problematic to try differentiating such things rather than collapse them all to higher set-types).
    – Kristian Berry
    Aug 27 at 13:03
  • We know there is inconsistency (not so much) hidden in the unrestricted comprehension of naive set theory, and in the omnivorous truth predicate of natural language, so why should equally omnivorous "existence" be any different? The challenge is to carefully sort the wheat from the chaff enough to get a coherent theory, but not too much so that the intuitive prototype still comes through. It can be done in a number of ways, including with an existence predicate, I am guessing, but I am not sure if somebody actually did the work.
    – Conifold
    Aug 27 at 20:21
  • To answer the question, I don't believe there is anything wrong with having a universe which only contains the empty set, so I would presume that no. Having a set of nonexistent things, does not ensure the existence of any other sets. In this case, there would be 1 object which exists, and 0 things which don't exist. The question "why is there something rather than nothing", in this context reduces to: Can we prove the Set Existence Axiom, as a theorem? What kinds of frameworks would that be possible in?
    – Michael Carey
    Aug 31 at 16:59

1 Answer 1

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Classes can have infinite members, including classes which are parts of other classes including proper classes. Consider the real numbers and the surreal numbers.

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3
  • Traditionally, the class of real numbers is not thought to be proper, but is a proper set instead (R. Matthews has a fairly 'weird' forcing world where the real numbers are inflated to the size of a proper class, though). The surreals are usually given over to No, which is a proper class, though. Proper classes are infinite, but not determinately so (or: their infinity is determined generically, unlike sets for which more specific descriptions of their infinity are possible to the point of unique identification).
    – Kristian Berry
    Aug 26 at 22:22
  • Now, I went many months not even believing in proper classes, but I had an "anticlass principle" in play that said, "If all elements of type A are elements of some class, then so are all elements of every other type B," so it would be impossible to have a class of "all and only elements of type A." However, when working on an axiom of multifoundation, I found a way to use the all-and-only quantifier consistent with the related approach to Russell's paradox, so now I'm not sure again about the existence of proper classes.
    – Kristian Berry
    Aug 26 at 22:27
  • I should add that I'd prefer to block |ORD| = |CARD|, since from what I read in Hamkins' writings, that equality would make every set well-ordered by default, whereas I prefer working in a set world with at least some choiceless sets. And without the anticlass principle, it seems like the system I work in would make that equality true, whereas with the principle in play, the equality becomes intrinsically indeterminate. I'm not sure that's a successful maneuver in the long run, though.
    – Kristian Berry
    Aug 26 at 22:42

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