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On the Set-Generic Multiverse

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The Hyperuniverse Project and Maximality

Abstract

The forcing method is a powerful tool to prove the consistency of set-theoretic assertions relative to the consistency of the axioms of set theory. Laver’s theorem and Bukovský’s theorem assert that set-generic extensions of a given ground model constitute a quite reasonable and sufficiently general class of standard models of set-theory.

In Sects. 2 and 3 of this note, we give a proof of Bukovsky’s theorem in a modern setting (for another proof of this theorem see Bukovský (Generic Extensions of Models of ZFC, a lecture note of a talk at the Novi Sad Conference in Set Theory and General Topology, 2014)). In Sect. 4 we check that the multiverse of set-generic extensions can be treated as a collection of countable transitive models in a conservative extension of ZFC. The last section then deals with the problem of the existence of infinitely-many independent buttons, which arose in the modal-theoretic approach to the set-generic multiverse by Hamkins and Loewe (Trans. Am. Math. Soc. 360(4):1793–1817, 2008).

Originally published in S. Friedman, S. Fuchino, H. Sakai, On the set-generic multiverse, in Sets and Computations. IMS Lecture Notes Series, vol. 33 (Institute of Mathematical Sciences, National University of Singapore, Singapore, 2017), pp. 25–44.

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Notes

  1. 1.

    In the terminology of [8], M is a ground of V .

  2. 2.

    Tadatoshi Miyamoto told us that James Baumgartner independently proved this theorem in an unpublished note using infinitary logic.

  3. 3.

    M[A] may be defined by M[A] =⋃ α ∈ On L(V αM ∪{A}). M[A] is a model of ZF: this can be seen easily e.g. by applying Theorem 13.9 in [13]. If M also satisfies AC then M[A] satisfies AC as well since, in this case, it is easy to see that a well-ordering of (V α )M ∪{A} belongs to M[A] for all α ∈ On.

  4. 4.

    More precisely, we assume that ZF proves the correctness of ⊢.

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Acknowledgements

The first author would like to thank the FWF (Austrian Science Fund) for its support through project number P 28420.

The second author is supported by Grant-in-Aid for Scientific Research (C) No. 21540150 and Grant-in-Aid for Exploratory Research No. 26610040 of the Ministry of Education, Culture, Sports, Science and Technology Japan (MEXT).

The third author is supported by Grant-in-Aid for Young Scientists (B) No. 23740076 of the Ministry of Education, Culture, Sports, Science and Technology Japan (MEXT).

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Correspondence to Sakaé Fuchino .

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Friedman, SD., Fuchino, S., Sakai, H. (2018). On the Set-Generic Multiverse. In: Antos, C., Friedman, SD., Honzik, R., Ternullo, C. (eds) The Hyperuniverse Project and Maximality. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-62935-3_5

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