The set-theoretic multiverse

TL;DR

The paper advocates for a multiverse perspective in set theory, proposing that multiple set concepts exist across various universes, which better explains set-theoretic diversity and the unresolved status of the continuum hypothesis.

Contribution

It introduces and defends the multiverse view as an alternative to the universe view, explaining set-theoretic phenomena and the nature of the continuum hypothesis.

Findings

Multiverse view accounts for diverse set-theoretic possibilities.

The continuum hypothesis is settled within the multiverse perspective.

The universe view is challenged by the multiverse explanation.

Abstract

The multiverse view in set theory, introduced and argued for in this article, is the view that there are many distinct concepts of set, each instantiated in a corresponding set-theoretic universe. The universe view, in contrast, asserts that there is an absolute background set concept, with a corresponding absolute set-theoretic universe in which every set-theoretic question has a definite answer. The multiverse position, I argue, explains our experience with the enormous diversity of set-theoretic possibilities, a phenomenon that challenges the universe view. In particular, I argue that the continuum hypothesis is settled on the multiverse view by our extensive knowledge about how it behaves in the multiverse, and as a result it can no longer be settled in the manner formerly hoped for.