On the set-generic multiverse

TL;DR

This paper explores the set-generic multiverse in set theory, providing modern proofs of key theorems, analyzing the structure of generic extensions, and addressing the existence of independent buttons within the multiverse framework.

Contribution

It offers a modern proof of Bukovský's theorem, examines the multiverse as a collection of models in ZFC, and investigates independent buttons in the set-generic multiverse.

Findings

Proof of Bukovský's theorem in a modern setting

Multiverse of set-generic extensions can be modeled within ZFC

Existence of infinitely-many independent buttons established

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 Bukovsk\'y'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 sections 2 and 3 of this note, we give a proof of Bukovsk\'y's theorem in a modern setting (for another proof of this theorem see Bukovsk\'y [4]). In section 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 J.Hamkins and B.Loewe [12].