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Why is the universe of sets not a set?

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Abstract

According to the iterative conception of sets, standardly formalized by ZFC, there is no set of all sets. But why is there no set of all sets? A simple-minded, though unpopular, “minimal” explanation for why there is no set of all sets is that the supposition that there is contradicts some axioms of ZFC. In this paper, I first explain the core complaint against the minimal explanation, and then argue against the two main alternative answers to the guiding question. I conclude the paper by outlining a close alternative to the minimal explanation, the conception-based explanation, that avoids the core complaint against the minimal explanation.

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Notes

  1. I intend the expression ‘the universe of sets’ to be neutral with respect to the nature of the “totality” of all sets in the iterative hierarchy. As we will see in this paper, the universe of sets may be a proper class, a plurality, an incomplete totality, a potential hierarchy (in which case there does not actually exist a plurality of all sets), etc.

  2. The discussion can be generalized to other why-questions by formulating equivalent forms of the minimal explanation. So, for instance, the equivalent “minimal explanation” for the why-question about the “totality” of all ordinals would be: “The totality of all ordinals is not a set because the assumption that it is contradicts some axioms of ZFC.”

  3. See for instance Yablo (2004, pp. 152–155).

  4. See for instance Welch (ms), Lewis (1991), and Mayberry (1986).

  5. See for instance Boolos (1984), Burgess (2004), and Uzquiano (2003).

  6. The derivation requires some version of the axiom of Global Choice. See for instance Linnebo (2010, 162) for the derivation of a limitation of size principle from plural set theory.

  7. See for instance Fraenkel et al. (1973). For a thorough study of the history of the Limitation of Size Principle from Cantor to von Neumann and beyond, see Hallett (1984).

  8. Size-actualism is defended in Fraenkel et al. (1973) and by Georg Cantor, John Von Neumann, Dmitry Mirimanoff and other Limitation of Size-Theorists, as explained in Hallett (1984). More recently, Welch (ms, 9) assumes some version of the size-explanation. Arguably [and as Linnebo states in Linnebo (2010, 151, fn. 9)] size-actualists also include Burgess (2004) and Lewis (1991). Actualists who aren’t size-actualists include Peter Koellner and Hugh Woodin; they endorse some version of the minimal explanation.

  9. I take Linnebo’s statement of the Arbitrary Threshold Objection to be the clearest, but others make it too. See for instance Fine (2006, 23), Studd (2013, 700), and Yablo (2004, 152f., 155).

  10. See also Linnebo (2013, 206): “To disallow such a set [of all sets] would be to truncate the iterative hierarchy at an arbitrary level.”

  11. This definition requires the Axiom of Choice. Without Choice, one can use Dana Scott’s trick to define \(|A| = \{X \ | \ X \ \text {is of minimal rank s.t.} \ A \approx X\}\). Since the collection of sets of a given rank is always a set, cardinalities are always sets according to this definition.

  12. One can derive a contradiction from the assumptions that \(A \approx V\) and that A is a set by using for instance Replacement.

  13. Such entities are discussed for instance in Lévy (1976).

  14. For instance, with Von Neumann–Bernays–Gödel (Gödel 1940), Morse–Kelley (Kelley 1955) or Lévy–Ackermann set theories (Ackermann 1956) as theories of classes, or with the formalization by Burgess (2004) as a theory of pluralities.

  15. After raising the Arbitrary Threshold Objection, Linnebo goes on to raise another objection to the size-explanation that is perhaps best understood as a version of the Objection from the Explanatory Insignificance of Size. He states: “To probe further, consider the question why there are not more ordinals than ... [\(\varOmega \)] ...According to the view under discussion, the explanation is that ...[\(\varOmega \)] are too many to form a set, where being too many is defined as being as many as ...[\(\varOmega \)]. Thus the proposed explanation moves in a tiny circle. The threshold cardinality is what it is because of the cardinality of the plurality of all ordinals, but the cardinality of this plurality is what it is because of the threshold. I conclude that the response fails to make any substantial progress, and that the proposed threshold remains arbitrary” (Linnebo 2010, 153f.). Here in Sect. 3.2, I aim to clarify this type of objection, disentangle it from the Arbitrary Threshold Objection, and connect it to the minimal explanation.

  16. E.g. Cantor’s argument for the Aleph Theorem, which states that every cardinal is an aleph [see Hallett (1984, 169)], and other arguments cited in Hallett (1984, 165ff.).

  17. In a recent paper, Christopher Menzel argues that the iterative conception of sets is intuitively consistent with the existence of a proper-class sized set of ur-elements, and proposes a modification to Replacement and Powerset to accommodate these “wide” sets (Menzel 2014). Menzel (2014) may thus be seen as providing an alternative argument for the insignificance of size in explaining why the universe of sets is not a set: The universe of sets is still not a set according to Menzel’s modified formalization of the iterative conception, but this has nothing to do with the “size” of the universe of sets.

  18. See for instance Jané (1995), Hellman (1989) and Zermelo (1930). The other potentialists are cited below.

  19. Parsons also says that the totalities that don’t form sets are “merely potential,” where “one can distinguish potential from actual being in some way so that it is impossible that all elements of an inconsistent multiplicity should be actual,” and then goes on to say that “where there is an essential obstacle to a multiplicity’s being collected into a unity, this is due to the fact that in a certain sense the multiplicity does not exist” (Parsons 1977, 345, my emphasis). So, for Parsons, the fact that the universe of sets is not a set is “due to” its potential nature.

  20. In his recent defense of potentialism, Studd similarly takes potentialism to provide an answer to the why-question; he states that potentialism is supposed to answer the “difficult question” I cited above on page 2, namely the question: “What is it about the world that allows some sets to form a set, whilst prohibiting others from doing the same?” (Studd 2013, 699, emphasis removed).

  21. Studd calls this ‘the Maximality thesis’ (Studd 2013, 699), Linnebo calls it ‘(C)’ (Linnebo 2013, 219), and Parsons introduces it in Parsons (1977, 357).

  22. See for instance Linnebo (2013, 214) and Studd (2013, 710).

  23. Note that, although Fig. 2 suggests otherwise, the accessibility relation in S4.2 is not a linear relation; Fig. 2 is merely intended to provide a rough visual idea of the potential hierarchy of sets.

  24. In particular, Studd’s modalization is different from Linnebo’s in that it has two basic necessity operators ‘\(\Box _<\)’ and ‘\(\Box _>\)’, and derived ones ‘\(\Box \)’, ‘\(\Box _\le \)’ and ‘\(\Box _\ge \)’, corresponding respectively to S5 and S4.3 (for the last two). Here I set aside Studd’s formalization without loss of generality for my arguments in Sects. 4.2, 4.3.

  25. See for instance Linnebo (2013, 220ff.), Parsons (1983, 318ff.), and Studd (2013, 712ff.).

  26. In particular, Linnebo maps his claim (C) (see fn. 21) that any sets (any zero or more sets) can form a set onto the claim that for any stage \(\alpha \), and any subset of \(V_\alpha \), there is a later stage \(\beta \) such that all later stages \(\gamma \) contain a set containing all and only those things in the subset of \(V_\alpha \) (Linnebo 2013, 224).

  27. See also Studd (2013, 698f.) where Studd says of Boolos’ stage theory (Boolos 1971) that it provides a good motivation for the axioms of set theory but is not able to preserve the claim that every sets can form a set nor answer the why-question. Studd then proposes modal set theory as an alternative to stage theory.

  28. See for instance Parsons (1977, 345), quoted in fn. 19. Or, for instance, see Linnebo (2010), where Linnebo argues that there is no set of all sets because there is no plurality of all sets, which is because the universe of sets is potential in nature. He thus endorses what he takes to be the view of Yablo (2004, 152): “How can there fail to be a determinate pool of candidates [i.e. a plurality of all sets]? According to Yablo, the answer has to do with the iterative conception of sets. The universe of sets is build up in stages. At each new stage we introduce all the sets that can be formed from the objects available at the preceding stage. But there is no stage at which this process of set formation is complete. At any one stage it is possible to go on and form new and even larger sets. This very suggestive answer needs to be spelled out. I will now propose a way of doing so, based on the idea that the hierarchy of sets is a potential one, not a completed or actual one” (Linnebo 2010, 155).

  29. See for instance Linnebo (2010, 158), Studd (2013, 706), Linnebo (2013, 226), Fine (2006, 31), and Parsons (1983, 328).

  30. Examples of the former include Linnebo (2013) and Parsons (1977). Linnebo considers the latter option in Linnebo (2010, 159). Some potentialists simply propose to forgo specifying the modality: “A full-fledged explanation of the modal notions will have to await another occasion” (Linnebo 2013, 207). See also Studd: “There is a great deal more to be said about each of these views, but it would take us too far afield to say it here. Rather, safe in the knowledge that taking the tense more seriously than usual need not commit us to taking it literally, I shall continue to elaborate on this view in general, leaving it open (within the bounds of LST) how the modality is to be interpreted” (Studd 2013, 707).

  31. See Linnebo (2013, 222f.) for one example of this kind of “design.”

  32. One further result deserves mention here besides the one in Sect. 4.1. If \(\varphi \) is a formula all of whose quantifiers are modalized, then modal set theory proves that \(\varphi \), \(\Box \varphi \) and \(\Diamond \varphi \) are equivalent [see for instance Studd (2013, 709) and Linnebo (2013, 213)], meaning that it doesn’t matter at which world a full modalized formula is evaluated, and which is “another reason why the implicit modalities that I have postulated in the set-theoretic quantifiers do not surface in ordinary set-theoretic practice” (Linnebo 2010, 164).

  33. See also Linnebo (2013, 207f.).

  34. See Koellner (2011) for more on the Large Cardinal Hierarchy.

  35. There are many other problems facing a radical constructivist view on which the existence of sets depends on whether set theorists define them or think about them. For one, it is widely accepted that such a radical constructivist approach will only sanction a much weaker and non-classical set theory. But actualists and potentialists don’t want to be revisionists—as we have seen, they want to “recover” most of the axioms of ZF. So I set aside this kind of revisionist option in this paper. For a recent discussion and summary of the arguments against constructivism, see for instance Incurvati (2012).

  36. This way of making sense of expandability fits with Tait (2005) and Parsons (1974).

  37. For exposition of set-theories with universal sets, see Forster (1992), Holmes (1998) and Holmes (2001).

  38. NFU is consistent (relative to PA) and \(\text {NFU}^+\) is consistent relative to Zermelo set theory with only \(\Delta _0\)-Comprehension (Jensen 1969). \(\text {NFU}^+\) + the Axiom of Cantorian Sets (which says that all Cantorian sets are strongly Cantorian) proves the existence of inaccessible cardinals, and n-Mahlo cardinals for each n. A Cantorian set is a set A such that \(|A| = |\mathscr {P}_1(A)|\), where \(\mathscr {P}_1(A)\) is the set of all one-element subsets of A, and a set A is strongly Cantorian if the class map \((x \mapsto \{x\}) \upharpoonright A\) is a set.

  39. Potentialists usually make clear that their focus is on the iterative concept of set, so this kind of response is not implausible for a potentialist. See for instance Linnebo (2010, 144).

  40. In the literature on grounding, people call this principle ‘the entailment principle’ (Rosen 2010). Eli Chudnoff (ms) and Dancy (2004) are among philosophers who deny this principle.

  41. See also Studd (2013, 698).

  42. This argument can be found for instance in Maddy (1988, 485).

  43. Aside from Boolos (1971), see also for instance Shoenfield (1977, 327).

  44. Potentialists themselves might agree with this point. Indeed, Linnebo himself simply assumes Foundation in his modal set theory (Linnebo 2013).

  45. For more on the conception of sets based on the Limitation of Size Principle, see for instance Maddy (1988, 484f.) or Hallett (1984).

  46. For simplicity, I assume here that the concept of set is the meaning of our expression ‘set’.

  47. There is a long line of arguments against the existence of conceptual or analytic truths and/or of their epistemological consequences, including Quine (1951), Putnam (1962) and the most recent Williamson (2007).

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Acknowledgements

I would like to thank Olivia Bailey, Neil Barton, Selim Berker, Douglas Blue, Ekaterina Botchkina, Warren Goldfarb, Ned Hall, Jens Kipper, Doug Kremm, Elizabeth Miller, Bernhard Nickel, Charles Parsons, Chris Scambler, Kate Vredenburgh, Stephen Yablo, three anonymous referees for this journal, audiences at Harvard University and at the Kurt Gödel Research Center, and Peter Koellner, especially, for very helpful discussions and comments on various versions of this paper.

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    Zeynep Soysal

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Soysal, Z. Why is the universe of sets not a set?. Synthese 197, 575–597 (2020). https://doi.org/10.1007/s11229-017-1513-x

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