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Maximality Principles in the Hyperuniverse Programme

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Abstract

In recent years, one of the main thrusts of set-theoretic research has been the investigation of maximality principles for V, the universe of sets. The Hyperuniverse Programme (HP) has formulated several maximality principles, which express the maximality of V both in height and width. The paper provides an overview of the principles which have been investigated so far in the programme, as well as of the logical and model-theoretic tools which are needed to formulate them mathematically, and also briefly shows how optimal principles, among those available, may be selected in a justifiable way.

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Notes

  1. As is known, V is recursively defined as follows: \(V_0={\emptyset }, V_{\alpha +1}={\mathcal {P}}(V_\alpha )\), where \(\alpha\) is a successor ordinal, and \(V_\lambda =\bigcup _{\alpha <\lambda } V_\alpha\), where \(\lambda\) is a limit ordinal.

  2. For instance, see Gödel’s oft-quoted remark in Gödel (1947), pp. 478–479: ‘..I am thinking of an axiom which (similar to Hilbert’s completeness axiom in geometry) would state some maximum property of the system of all sets [...]. Note that only a maximum property would seem to harmonize with the concept of set [...]’. For a review of maximality principles in set theory, see Incurvati (2017).

  3. At least, this is the form in which the principle can be stated if one subscribes to actualism (for which see Sect. 3 of this paper), a view that some scholars view as originating from Cantor’s conception of a maximal, inextensible absolute infinite (for which see, in particular, Cantor’s 1899 letter to Dedekind, in Ewald (1996), pp. 931–935), but the introduction of RP’s may also be justified using a potentialist conception of V (for which see, again, our section 3, or Tait (1998)). Incidentally, Gödel was also a major advocate of RP, to the point that he seems to have surmised that the axioms of set theory should be reduced to one single reflection axiom (see also Wang (1996), pp. 280–285, and Ternullo (2015), pp. 431–435).

  4. A full account of such limitations is in Koellner (2009).

  5. See, in particular, Arrigoni and Friedman (2013), Antos et al. (2015), Friedman (2016), Friedman and Honzik (2016), Ternullo and Friedman (2016) and Antos et al. (2020).

  6. What has been for some time known as the ‘(maximal) iterative concept of set’ is described by, among others, Wang (1974) and Boolos (1971). Further discussion is in Parsons (1983) and Maddy (1988).

  7. More specifically, Maddy’s argument is that, since \(V=L\) is inconsistent with the existence of \(0^{\#}\), then under ZFC+\(V=L\), there will be fewer isomorphism types in V. See Maddy, Maddy (1997), pp. 219–232. Recently, the alleged ‘restrictiveness’ of \(V=L\) has been challenged by Hamkins (2014), and further discussed by Incurvati and Löwe (2016).

  8. Or any two other theories whose consistency strength is, at least, that of ZFC+LC’s.

  9. Steel (2014), p. 159. But notice: this is true only up to the level of second-order arithmetic, that is, up to the level of \(\Pi ^{1}_{\omega }\) sentences.

  10. See Steel (2014), pp. 158–160. A theory T is consistency stronger than a theory U if T proves Con(U). It should be noted that the theories addressed by Steel need to have at least the same consistency strength as that of ZFC\(+\)‘there are infinitely many Woodin cardinals’.

  11. On this, see Bagaria (2005), and Woodin (2001).

  12. For the terminology and its meaning, see Gödel (1947) and Gödel (1964). For a discussion of the relevance of ‘extrinsic’ arguments, see Maddy (1996).

  13. The topic is fully explored by one of the authors and Neil Barton in Barton and Friedman (2020).

  14. Zermelo’s proof is discussed (and defended) at length in Martin (2001). A philosophical discussion of Zermelo’s conception is also in Tait (1998). Also see the more recent Linnebo (2017), pp. 179–182.

  15. Recall that an uncountable cardinal \(\kappa\) is inaccessible if and only if: (1) \(\kappa\) is regular (that is, its cofinality is \(\kappa\)) and (2) \(\kappa\) is a limit cardinal (that is, it is not the successor of any cardinal). \(\kappa\) is strongly inaccessible if, in addition, is: (3) strong limit, that is, if for all cardinals \(\lambda <\kappa\), \(2^\lambda <\kappa\).

  16. A different (and significantly more potentialist) construal of Zermelo’s normal-domain construction may be found in Hellman (1989) and Linnebo (2013), both of which take end-extensions of V to be interpretable in modal terms.

  17. However, we do not claim the preferability of the second-order axioms of set theory. What we, rather, claim here is the view that the ordinary first-order iterative concept of set is constrained by (and even derivable from) the second-order concept, in such a way as to imply that the first-order understanding of the extendibility of V should be seen as exclusively applying to the height of V itself.

  18. See footnote 3.

  19. Properties are often formulated using higher-order quantification. Let M be a class. We say that a variable x is 1-st order (or of order 1) if it ranges over elements of M. In general, we say that a variable R is \(n+1\)-st order (or of order \(n+1\)), \(0<n<\omega\), if it ranges over \({\mathcal {P}}^n(M)\), where \({\mathcal {P}}^n(M)\) denotes the result of applying the powerset operation n times to M. A formula \(\varphi\) is \(\Pi ^n_m\) if it starts with a block of universal quantifiers of variables of order \(n+1\), followed by existential quantification of variables of order \(n+1\), and these blocks alternate at most \(m-1\) times; the rest of the formula can contain variables of order at most \(n+1\), and quantifications over variables of order at most n. \(\Sigma ^n_m\) is obtained by switching the words universal and existential.

  20. As an aside, it is worth noting that if formulated with third-order parameters, third-order reflection is in fact inconsistent! For instance, for a third-order parameter \({\mathcal {R}}\), i.e. a collection of classes, one is tempted by the following natural-looking principle:


    Third-order Reflection If \(\varphi ({\mathcal R})\) is true in \((V,{{\mathcal {R}}})\), then for some \(\alpha\), \(\varphi (\bar{{\mathcal {R}}})\) is true in \((V_\alpha ,\bar{\mathcal R})\), where \(\bar{{\mathcal {R}}}=\{R\cap V_\alpha \mid R\in {\mathcal R}\}\).


    But such a principle will fail if \({\mathcal {R}}\) consists of all bounded subsets of the ordinals (viewed as a collection of classes), and \(\varphi ({{\mathcal {R}}})\) simply says that each element of \({\mathcal {R}}\) is bounded in the ordinals. Therefore when discussing third-order reflection it is customary to only allow second-order, and not third-order, parameters.

  21. See Section 2.2 of Friedman and Honzik (2016).

  22. See Sect. 5.3.2

  23. The proof is in Friedman and Honzik (2016), p. 11.

  24. See the discussion of this in Section 2.1 of Friedman and Honzik (2016), and also Welch (2019).

  25. Note that IMH is also known to consistently hold for some choice of little-V. See Friedman et al. (2008).

  26. For further on this analogy, see Barton and Friedman (2020).

  27. V-logic is an extension (reformulation) of Barwise’s \({\mathfrak {M}}\)-logic, for which see Barwise (1975).

  28. The completeness result for little-V-logic is equivalent to the completeness result for countable fragments of Barwise’s \({\mathfrak {M}}\)-logic. See Barwise (1975), p. 89.

  29. See the paragraph immediately below Definition 1.

  30. Weak \(\#\)-generation is indeed strictly weaker than \(\#\)-generation for countable models. Suppose that \(0^\#\) exists and choose \(\alpha\) to be least, so that \(\alpha\) is the \(\alpha\)-th Silver indiscernible (\(\alpha\) is countable). Now let g be generic over L for Lévy collapsing \(\alpha\) to \(\omega\). Then by Lévy absoluteness, \(L_\alpha\) is weakly \(\#\)-generated in L[g], but it cannot be \(\#\)-generated in L[g] as \(0^\#\) does not belong to a generic extension of L.

  31. For further details on all of these, see Friedman (2016) and Friedman and Honzik (2016).

  32. It should be noted that, although the hyperuniverse, introduced in Sect. 5.3, may be construed as a collection of mutually alternative ‘universes’ of set theory, and may thus qualify as a multiverse, in this paper we do not automatically subscribe to, or wish to defend, a multiversist view, since, as also explained in the present section, several ontological conceptions are compatible with HP’s maximality protocol. In fact, as stated in Sect. 5, the ‘reduction to the hyperuniverse’ can just be seen as a convenient way to address the expression of maximality principles semantically, rather than as a way of advocating the actual existence of a collection of mutually alternative universes of set theory. We thank one anonymous referee for prompting us to clarify this point.

  33. For instance, take P to be the combination of a well-known axiom such as \(V=L\) and \(\#\)-generation. Suppose, however implausible it may seem, that one chose P as the ultimate, optimal maximality principle. One would, then, realise that such a principle would still be compatible with many universes of the form \(L_\alpha\), where \(\alpha\) is a limit of Silver indiscernibles. Notice, though, that the identification of P as an optimal maximality principle would, at least, narrow the range of universes of interest in the multiverse, as presumably only some universes will satisfy it.

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Authors and Affiliations

  1. Kurt Gödel Research Center for Mathematical Logic, University of Vienna, Kolingasse, 14-16, Wien, 1090, Austria

    Sy-David Friedman

  2. Department of Mathematics and Computer Science, Universitat de Barcelona, Gran Via de les Corts Catalanes, 585, 08007, Barcelona, Spain

    Claudio Ternullo

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  1. Sy-David Friedman
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The author would like to thank the FWF (Austrian Science Fund) for its support through the ‘Hyperuniverse Project’ (Grant P-28420).

The author wishes to thank the AGAUR for its support through the Beatriu de Pinós Post-Doctoral Fellowship 2018 BP 00192: ‘Mathematical and Philosophical Aspects of a Multiversist Foundation of Set Theory’.

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Friedman, SD., Ternullo, C. Maximality Principles in the Hyperuniverse Programme. Found Sci 28, 287–305 (2023). https://doi.org/10.1007/s10699-020-09707-8

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