### The Set-Theoretic Multiverse

#### Posted by David Corfield

There’s an interesting paper out today on the ArXiv – Joel Hamkins’ The set-theoretic multiverse.

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.

So set theorists’ experience of dealing with various models of set theory, and the small modifications needed to generate models either satisfying or not satisfying the continuum hypothesis, tells them that settling its truth or falsity by devising an evident axiom from which either it or its negation follows is not a live option. Gödel’s platonism was wrong then, yet Hamkins retains a form of realism:

The multiverse view is one of higher-order realism – Platonism about universes – and I defend it as a realist position asserting actual existence of the alternative set theoretic universes into which our mathematical tools have allowed us to glimpse. The multiverse view, therefore, does not reduce via proof to a brand of formalism. In particular, we may prefer some of the universes in the multiverse to others, and there is no obligation to consider them all as somehow equal.

Naturally, I looked to see the part Hamkins sees category theory playing in this multiverse. While not extensively considered in the paper, he writes

Set theory appears to have discovered an entire cosmos of set-theoretic universes, revealing a category-theoretic nature for the subject, in which the universes are connected by the forcing relation or by large cardinal embeddings in complex commutative diagrams, like constellations filling a dark night sky. (p. 3)

and

In what appears to be an interesting case of convergent evolution in the foundations of mathematics, this latter universe concept coincides almost completely with the concept of Grothendieck universe, now pervasively used in category theory [Krö01]. The only difference is that the category theorists also view $V_{\omega}$ as a Grothendieck universe, which amounts to considering $\aleph_0$ as an incipient inaccessible cardinal. Surely the rise of Grothedieck universes in category theory shares strong affinities with the multiverse view in set theory, although most set theorists find Grothendieck universes clumsy in comparison with the more flexible concept of a (transitive) model of set theory; nevertheless, the category theorists will point to the multiverse concepts present in the theory of toposes as more general still (see [Bla84]). (p. 13)

Bla84 is Andreas Blass, The interaction between category theory and set theory.

We’ve had our own expression of multiverse thinking in this thread.

## Re: The Set-Theoretic Multiverse

Is “multiverse” really a good term to use? Surely this will unnecessarily attract more physics/quantum mechanics crackpots than you want…