Multiverse conceptions in set theory

We review different conceptions of the set-theoretic multiverse and evaluate their features and strengths. In Sect. 1, we set the stage by briefly discussing the opposition between the ‘universe view’ and the ‘multiverse view’. Furthermore, we propose to classify multiverse conceptions in terms of their adherence to some form of mathematical realism. In Sect. 2, we use this classification to review four major conceptions. Finally, in Sect. 3, we focus on the distinction between actualism and potentialism with regard to the universe of sets, then we discuss the Zermelian view, featuring a ‘vertical’ multiverse, and give special attention to this multiverse conception in light of the hyperuniverse programme introduced in Arrigoni and Friedman (Bull Symb Logic 19(1):77–96, 2013). We argue that the distinctive feature of the multiverse conception chosen for the hyperuniverse programme is its utility for finding new candidates for axioms of set theory.

1. For the distinction and its conceptual relevance within the philosophy of mathematics, see Shapiro’s introduction to Shapiro (2005) or Shapiro (2000).

2. See, for instance, the following oft-quoted passage in his Cantor paper: “It is to be noted, however, that on the basis of the point of view here adopted [that is, the ‘platonistic conception’, our note], a proof of the undecidability of Cantor’s conjecture from the accepted axioms of set theory (in contradistinction, e.g., to the proof of the transcendency of \(\pi \)) would by no means solve the problem. For if the meaning of the primitive terms of set theory as explained on page 262 and in footnote 14 are accepted as sound, it follows that the set-theoretical concepts and theorems describe some well-determined reality in which Cantor’s conjecture must be either true or false.” (Gödel (1947), in Gödel (1990), p. 260). For an account of the development of Gödel’s conceptions, see, for instance, Wang’s books, Wang (1974, 1996) and also van Atten and Kennedy (2003).

3. For the full characterisation of Maddy’s ‘thin’ realist, see Maddy (1997, 2011). A possible middle ground between a realist and a non-realist universe view has been described by Putnam in his (1979). A ‘moderate realist’, as featured there, would be someone who does not buy into full-blown platonism but who, at the same time, still believes to be able to find evidence in favour of some sort of ‘ultimate’ universe.

4. See, especially, Balaguer (1995, 1998).

5. Incidentally, it is not clear whether a realist in truth-value is best accommodated to the universe view. For instance, take Hauser, who seems to be only a realist in truth-value. He says: “At the outset mathematical propositions are treated as having determinate truth values, but no attempt is made to describe their truth by relying on a specific picture of mathematical objects. Instead one seeks to exhibit the truth or falsity of mathematical propositions by rational and reliable methods” (Hauser 2002, p. 266). From this, it is far from clear that one single picture of sets would have to be found anyway, if not at the outset, at least in due course, presumably after the truth-value of such statements as CH has been reliably fixed. Hauser has also addressed truth-value realism and its conceptual emphasis on objectivity rather than on objects in Hauser (2001). See also Martin (1998).

6. Hamkins epitomises this conception through the adoption of the naturalistic maxim ‘maximise’, by virtue of which one should not place “undue limitations on what universes might exist in the multiverse. This is simply a higher-order analogue of the same motivation underlying set-theorists’ ever more expansive vision of set theory. We want to imagine the multiverse as big as possible” (Hamkins 2012, p. 437).

7. ZF+ the negation of the Axiom of Choice.

8. Hamkins defines this interpretation of forcing ‘naturalist’, as opposed to the ‘original’ interpretation of forcing, whereby one starts the construction with a countable transitive ground model M and extends it to an M[G], by adding an M-generic filter G.

9. See Koellner (2009b), especially Sect. 2.

10. This is, for instance, Koellner’s line of attack in the aforementioned paper.

11. We say ‘mild’, as he seems to want to deny to be a fully committed (an ‘extreme’, in his words) formalist. He says: “..I reject also the extreme formalistic attitude which says that we just scribble symbols on paper or all consistent set theories are equal” (Shelah 2003, p. 212).

12. These views have been stated by the author several times. See, in particular, Feferman (1999), Feferman et al. (2000) and the more recent Feferman (2014). A careful response to Feferman’s concerns is in Hauser (2002).

13. Details on these can also be found in Woodin (2011a).

14. For the definition of Woodin cardinals, see Kanamori (2003, p. 360).

15. In fact, this result, as Steel clarifies, only holds for ‘natural’ theories, that is theories with ‘natural’ mathematical axioms, not quite like, for instance, the ‘Rosser sentence’.

16. For instance, Steel’s result on p. 7 only holds in \(L(\mathbb {R})\).

17. For more accurate mathematical details, we refer the reader to Steel’s cited paper.

18. A crucial reading on this is Hallett’s book on the emergence of the limitation of size doctrine, Hallett (1984). The distinction between actualists and potentialists may be construed as the result of different interpretations of Cantor’s absolute infinite. One of the most exhaustive articles on Cantor’s conception of absoluteness and of its inherent tension between actualism and potentialism is Jané (1995). For a discussion of actualism and potentialism, with reference to the justification of reflection principles, see Koellner (2009a). For an accurate overview of several potentialist positions, see Linnebo (2013).

19. The distinction between ‘height’ and ‘width’ of the universe is firmly based on the iterative concept of set: the length of the ordinal sequence determines the height of the universe, while the width of the universe is given by the powerset operation.

20. Also Reinhardt’s theory of legitimate candidates (see Reinhardt (1974)) seems to follow Zermelo’s account. Finally, Hellman (in Hellman (1989)), develops a structuralist account of the universe in line with Zermelo’s concerns, but based on modal assumptions.

21. The proof is in Friedman et al. (2008).

22. The use of the Löwenheim–Skolem theorem, while completely legitimate, is actually optional: if one wishes to analyse the outer models of V without ‘going countable’, one can do it by using the V-logic introduced below. However, there is a price to pay: instead of having the elegant clarity of countable models, one will just have to refer to different theories. This has analogies in forcing: to have an actual generic extension, one needs to start with a countable model; if the initial model is larger, one can still deal with forcing syntactically, but a generic extension may not exist (see, for instance, Kunen (2011)). A more relevant analogy in our case is that the Omitting Types Theorem (which is behind V-logic) works for countable theories, but not necessarily for larger cardinalities.

23. Full mathematical details are in Barwise (1975). We wish to stress that the infinitary logic discussed in this section appears only at the level of theory as a tool for discussing outer models. The ambient axioms of ZFC are still formulated in the usual first-order language.

24. Again, for more details we refer the reader to Barwise (1975).

25. This is in clear analogy to the treatment of set-forcing, see Footnote 22. However, note that unlike in set-forcing, where the syntactical treatment can be formulated inside V, to capture arbitrary outer models, we need a bit more, i.e. \(\mathrm {Hyp}(V)\).

Authors and Affiliations

1. KGRC, Vienna, Austria

   Carolin Antos, Sy-David Friedman, Radek Honzik & Claudio Ternullo

2. Charles University, Prague, Czech Republic

   Radek Honzik

Corresponding author

Correspondence to Claudio Ternullo.

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