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Abolishing Platonism in Multiverse Theories

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Abstract

A debated issue in the mathematical foundations in at least the last two decades is whether one can plausibly argue for the merits of treating undecidable questions of mathematics, e.g., the Continuum Hypothesis (CH), by relying on the existence of a plurality of set-theoretical universes except for a single one, i.e., the well-known set-theoretical universe V associated with the cumulative hierarchy of sets. The multiverse approach has some varying versions of the general concept of multiverse yet my intention is to primarily address ontological multiversism as advocated, for instance, by Hamkins or Väätänen, precisely for the reason that they proclaim, to the one or the other extent, ontological preoccupations for the introduction of respective multiverse theories. Taking also into account Woodin’s and Steel’s multiverse versions, I take up an argumentation against multiversism, and in a certain sense against platonism in mathematical foundations, mainly on subjectively founded grounds, while keeping an eye on Clarke-Doane’s concern with Benacerraf’s challenge. I note that even though the paper is rather technically constructed in arguing against multiversism, the non-negligible philosophical part is influenced to a certain extent by a phenomenologically motivated view of the matter.

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Notes

  1. I point out, however, that in adding ‘almost’ next to ‘nothing to do’ I reserve a clue to a possible interpretational connection between the mathematical and physical versions of universe - multiplicity of universes in this section (par. 5).

  2. The generalized version of the Continuum Hypothesis will be abbreviated in the text as GCH.

  3. Gödel’s axiom \(V=L\) essentially identifying the set-theoretic universe V with the constructible universe L is generally thought to be restrictive in the sense of imposing the predicative formation of sets across ordinals in L to the universe V. Notably \(V=L\) has been proved consistent with ZFC theory if ZFC is. See for details Kunen (1982), Ch. VI.

  4. By eidetic laws or eidetic attributes in the world of phenomena one can roughly communicate to a non-phenomenologist what on subjective grounds holds of the existence of objects or states-of-affairs as regularities by essential necessity and not by mere facticity. One may consult a propos E. Husserl’s Ideas I: Husserl 1983, Engl. transl., pp. 12–15.

  5. A kind of mathematical platonism is often attributed to Gödel, yet even if this has a solid base it is also true that Gödel especially in his later years was allured by Husserl’s phenomenology and this was reflected in a certain sense in the view he held of mathematical objects as implying a special kind of intuition we have of them forced by mathematical objects upon us. For more details on this matter the reader may see Livadas (2019).

  6. See Feferman, S: ‘The Continuum Hypothesis is neither a definite mathematical problem nor a definite logical problem’, p. 2, a revised version of Feferman (2011).

  7. Gödel had noted that there is no element of randomness in the definition of ordinals and hence neither in sets defined in terms of them. He found this particularly clear in von Neumann’s definition of ordinals insofar as it is not based on any well-ordering relations of sets which may well involve some random element as applied to various ranks of infinity. See (Gödel 1965, p. 87).

  8. Steel’s multiverse (MV) axioms in a two-sorted multiverse language are found in Steel (2014, p. 165).

  9. A set is ordinal definable, OD, if and only if it is definable over the universe of sets from ordinal parameters, and is hereditarily ordinal definable, HOD, in the case that itself and all members of its transitive closure are ordinal definable. The precise statement of a version of axiom H is found in Steel (2014, p. 171).

  10. The reader who wants to avoid Woodin’s extremely technical proofs may well skip the Appendix without losing anything of the general picture.

  11. I use the term standard in quotation marks to refer to certain notions in set-theoretical constructions, like the transitivity of \(\in\)-inclusion or the well-foundedness, having some direct or indirect relation to the concept of absoluteness or to other concepts linked with natural intuition, to distinguish from the conventional term standard as used in non-standard mathematics.

  12. The case of forcing extensions on the set-theoretic universe V as presumably legitimate universes in their own right can be addressed by universism, on the one hand, as simply model-theoretic representations within V with nothing non-trivial added to the semantics of V, and, on the other hand, by easily accommodating this situation by appealing to the reflection theorem and the downward Skolem-Löweheim Theorem to define a countable model on which to implement forcing (Barton 2016, p. 4).

  13. The naturalist account of forcing in Hamkins’ approach is closely related to the Boolean valued model approach to forcing primarily in the sense that one can implement forcing entirely within ZFC without restriction to the kind of models, in particular the countable transitive models in the classical forcing techniques, or without the need to appeal to the metatheory in the proof-theoretic machinery.

  14. See (Hamkins 2012), p. 423.

  15. Scott’s trick is a method for giving a definition of equivalence classes in a proper class by referring to the levels of the cumulative hierarchy \(V_{\alpha \in ON}\). It is basically a way of assigning representatives to cardinal numbers in ZF theory without the Axiom of Choice using the fact that for every set A there is a least rank in the cumulative hierarchy when some set of the same cardinality as A appears. As such Scott’s trick makes an essential use of the Axiom of Foundation (Axiom of Regularity).

  16. See lemma 4.2, p. 124 in Kunen (1982).

  17. See for details: (Van Atten et al. 2002, pp. 206–210).

  18. See for details (Livadas 2019).

  19. Hamkins refers to the dream solution template in the following sense: Step 1: Produce a set-theoretic assertion \(\Phi\) expressing an ‘obviously true’ set-theoretical principle and Step 2: Prove that \(\Phi\) determines CH, i.e., \(\Phi \Longrightarrow\) CH or \(\Phi \Longrightarrow\) \(\lnot\) CH (Hamkins 2012, p. 430).

  20. See for instance the way by which the forcing conditions consisting of functions \(\text{ FN } (I, J)\), with I arbitrary and J countable sets have, by the application of the Delta Lemma, the countable chain condition (CCC) so as to preserve cardinals among the ground and forcing model in the proof of the negation of CH (Kunen (1982), pp. 205–206).

  21. These concepts are discussed in more detail and in a phenomenological motivation in Livadas (2013); Tieszen (2005) and (Tieszen 2011).

  22. In team semantics the basic concept is not that of an assignment s satisfying a formula \(\varphi\) in a model M, but of a set \((\mathcal {S})\) of assignments, called a team, satisfying the formula \(\varphi\). See Väänänen (2014, p. 197).

  23. The Benacerraf problem, known subsequently also as the Benacerraf-Field challenge, was initially presented in Benacerraf’s article on mathematical truth, (Benacerraf 1973), in which Benacerraf claimed to be in favor of “a causal account of knowledge on which for (auth. add.: a subject) X to know that S is true requires some causal relation to obtain between X and the referents of the names, predicates, and quantifiers of S”. By this measure and on the subjectively based principle of the knowing person, Benacerraf argued that in view of the ‘asymmetry’ between the truth conditions of a proposition p put in formal terms and the grounds on which p is said to be known, e.g. in terms of reliability in connection with general mathematical or other knowledge “[..] makes it difficult to see how mathematical knowledge is possible”. See: (Benacerraf 1973, pp. 671–673).

  24. Clarke-Doane seeks to show through various subsumed interpretations of Benacerraf-Fields’ challenge, e.g., indispensability, counterfactual persistence, etc., that the pluralists do not have the edge over universists in all such cases. In his own words “if there is a reason to be a set theoretic pluralist, then it is not related to the challenge to establish a causal, explanatory, logical or even counterfactual dependence between our set-theoretic beliefs and the truths” (Clarke-Doane 2019, p. 13).

  25. In the footnote 20 of the article Is Mathematics Syntax of Language? Gödel offered as an example of a transfinite (i.e., non-constructive) concept the phrase ‘there exists’, if this phrase “means object existence irrespective of actual producibility” (Gödel 1995, p. 341).

  26. See for details: Theorem 7.25, p. 101 in Woodin (2017).

  27. See for details: Theorem 7.28, p. 103 in Woodin (2017).

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    Stathis Livadas

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Livadas, S. Abolishing Platonism in Multiverse Theories. Axiomathes 32, 321–343 (2022). https://doi.org/10.1007/s10516-020-09526-3

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