Abstract
Multiverse Views in set theory advocate the claim that there are many universes of sets, no-one of which is canonical, and have risen to prominence over the last few years. One motivating factor is that such positions are often argued to account very elegantly for technical practice. While there is much discussion of the technical aspects of these views, in this paper I analyse a radical form of Multiversism on largely philosophical grounds. Of particular importance will be an account of reference on the Multiversist conception, and the relativism that it implies. I argue that analysis of this central issue in the Philosophy of Mathematics indicates that Radical Multiversism must be algebraic, and cannot be viewed as an attempt to provide an account of reference without a softening of the position.
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Notes
- 1.
In order to disambiguate, throughout this paper I speak as if the Hamkinsian Multiversist is female and her opponents are male (with the exception of Hamkins himself).
- 2.
We should be mindful here of the fact that the idea of the Iterative Conception underpinning ZFC set theory is quite controversial in itself. There is an extensive literature on the topic, for a small selection see Maddy (1988) and Potter (2004). I do not address the question further here, and take it as assumed that the Iterative Conception is the justificatory resource to which the Universist appeals.
- 3.
It should be noted that there is the scope to hold that there is just one Universe of sets but that it is indeterminate in some sense (say because, some of its properties are indeterminate). For just such a view, see Feferman (2011).
- 4.
This fact will turn out to be important for assessing Hamkins’ view, and is discussed in more detail in Sect. 3.
- 5.
- 6.
- 7.
- 8.
In particular, this strategy seems attractive for the reason that often use of the term ‘V’ is patently an abuse of notation, designed to underscore the fact that any countable transitive model will suffice for the construction. See Koellner (2013) for discussion of some of these issues.
- 9.
For details, see Hamkins and Seabold (2012).
- 10.
Again, some of these issues are given consideration in Koellner (2013).
- 11.
See, for example, Foreman (2010).
- 12.
- 13.
It should be noted that the analogy is not total. For instance, a Universist is likely to deny the existence of V as a set for reasons of paradox. On Hamkins’ view, however, any particular V is a set in a taller model. Nonetheless, one can see the similarity, both assert that the subject matter of mathematics is constituted by mind-independent entities.
- 14.
Exactly which models of first-order ZFC is a subtle issue, and one I shall not consider here. Hamkins specifies (in addition to his broad picture) a list of Multiverse Axioms, designed to axiomatise movement within the Multiverse. Of note is that (within a particular V) the collection of all models of ZFC does not satisfy the Multiverse Axioms, though the collection of all computably saturated models of ZFC does: see Hamkins (2012a) and Gitman and Hamkins (2011) for details. These subtleties are unimportant for the discussion here, no matter what the axioms taken to characterise the Multiverse are, the arguments carry over immediately.
- 15.
Or, indeed, any substantial increase in expressive resources in the metalanguage (e.g. ancestral logic).
- 16.
This is not to say that second-order set theory is meaningless for the Hamkinsian. Rather, the interpretation of the second-order variables is dependent upon background concept of set, and so indeterminacy in the first-order language of ZFC carries over to the second-order language.
- 17.
For example, the Löwenheim–Skolem Theorems entail that if a theory has an infinite model, then it has (clearly non-isomorphic) models of every infinite cardinality. The existence of non-well-founded models (an immediate consequence of the Compactness Theorem for first-order theories) is another good example of clearly non-isomorphic models of theories that are the same with respect to first-order satisfaction.
- 18.
It should be noted that he states that this is only ‘often’ how reference is achieved. However, nowhere else is he explicit about exactly how reference occurs.
- 19.
See, for example, his presentation in Hamkins (2012b).
- 20.
This is known as the Well-foundedness Mirage. See Hamkins (2012a), p. 439.
- 21.
Hamkins (2012a), pp. 438–439.
- 22.
Examples of this sort include the Mostowski Collapse Lemma, Compactness Theorem for first-order theories, and Ultrapower Construction (the former implies the existence of non-well-founded models indirectly, the latter two by explicit construction).
- 23.
The astute reader will notice that the explanation given here is riddled with ‘scare quotes’. There is good reason for this, even the notion of standardness itself is relative for the Hamkinsian Multiversist.
- 24.
Many of these considerations, and in particular the above theorem, are discussed in Shapiro’s seminal (Shapiro 1991). For a discussion of the similarities and differences between indeterminacy in the notion of finiteness and the explicitly set-theoretic case, the reader is directed to Field (2003).
- 25.
Certainly, at least since, Shapiro (1991).
References
Balaguer, M. (1998). Platonism and anti-platonism in mathematics. Oxford: Oxford University Press.
Benacerraf, P. (1973). Mathematical truth. The Journal of Philosophy, 70(19), 661–679.
Feferman, S. (2011). Is CH a definite problem? Retrived April 13, 2014, from http://math.stanford.edu/feferman/papers.html
Field, H. (2003). Do we have a determinate conception of finiteness and natural number? In M. Schirn (Ed.), The philosophy of mathematics today (pp. 99–130). Oxford: Clarendon Press.
Foreman, M. (2010). Ideals and generic elementary embeddings. In A. Kanamori & M. Foreman (Eds.), Handbook of set theory (pp. 885–1147). New York: Springer.
Gitman, V., & Hamkins, J. D. (2011). A natural model of the multiverse axioms. Preprint. arXiv:1104.4450 [math.LO].
Gödel, K. (1947). What is Cantor’s continuum problem? In K. Gödel, Collected works (Vol. II, pp. 176–187). Oxford: Oxford University Press.
Hamkins, J. D. (2012a). Pluralism in set theory: Does every mathematical statement have a definite truth value? Held at Philosophy Colloquium: Cuny Graduate Center.
Hamkins, J. D. (2012b). The set-theoretic multiverse. The Review of Symbolic Logic, 5(3), 416–449.
Hamkins, J. D., & Loewe, B. (2008). The modal logic of forcing. Transactions of the American Mathematical Society, 360(4), 1793–1817.
Hamkins, J. D., & Seabold, D. E. (2012). Well-founded Boolean ultrapowers as large cardinal embeddings. arXiv:1206.6075 [math.LO].
Isaacson, D. (2011). The reality of mathematics and the case of set theory. In Z. Noviak & A. Simonyi (Eds.), Truth, reference, and realism (pp. 1–76). Hungary: Central European University Press.
Jech, T. (2002). Set theory. New York: Springer.
Koellner, P. (2013). Hamkins on the multiverse. Held at Exploring the frontiers of incompleteness.
Kunen, K. (1980). Set theory: An introduction to independence proofs. Amsterdam: Elsevier.
Maddy, P. (1988). Believing the axioms. I. The Journal of Symbolic Logic, 53(2), 481–511.
Maddy, P. (2011). Defending the axioms. Oxford: Oxford University Press.
Martin, D. (2001). Multiple universes of sets and indeterminate truth values. Topoi, 20(1), 5–16.
McGee, V. (1997). How we learn mathematical language. The Philosophical Review, 106(1), 35–68.
Meadows, T. (2013). What can a categoricity theorem tell us? The Review of Symbolic Logic, 6, 524–544.
Potter, M. (2004). Set theory and its philosophy: A critical introduction. Oxford: Oxford University Press.
Shapiro, S. (1991). Foundations without foundationalism: A case for second-order logic. Oxford: Oxford University Press.
Steel, J. (2014). Gödel’s program. In J. Kennedy (Ed.), Interpreting Gödel (pp. 153–179). Cambridge: Cambridge University Press.
Zermelo, E. (1930). On boundary numbers and domains of sets. In W. B. Ewald (Eds.), From Kant to Hilbert: A source book in mathematics (Vol. 2, pp. 1208–1233). Oxford: Oxford University Press.
Acknowledgments
The author is very grateful to Ben Fairbairn, Joel Hamkins, Peter Koellner, Ian Rumfitt, and audiences in London and Milan for insightful and useful feedback on the issues discussed. Special mention must be made of Victoria Gitman, Alex Kocurek, Chris Scambler, and two anonymous referees whose detailed comments improved the paper immensely. The author also wishes to thank the Arts and Humanities Research Council for their support during the preparation of the paper.
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Barton, N. (2016). Multiversism and Concepts of Set: How Much Relativism Is Acceptable?. In: Boccuni, F., Sereni, A. (eds) Objectivity, Realism, and Proof . Boston Studies in the Philosophy and History of Science, vol 318. Springer, Cham. https://doi.org/10.1007/978-3-319-31644-4_11
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