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).
Friedman, S. , Fuchino, S. , Sakai, H. (2018)., On the set-generic multiverse, in C. Antos, R. Honzik, C. Ternullo & S. D. Friedman (eds.), The hyperuniverse project and maximality, Dordrecht, Springer, pp. 109-124.
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