# nLab set-theoretic multiverse

The set-theoretic multiverse

foundations

# The set-theoretic multiverse

## Idea

The set-theoretic multiverse is a philosophical perspective on set theory, advocated by Joel David Hamkins, according to which

there are many distinct concepts of set, each instantiated in a corresponding set-theoretic universe.

This is in contrast to the “universe view”, which

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 set-theoretic multiverse is at least informally analogous to such categorical notions as Topos, the 2-category of toposes, with each topos regarded as a universe of (“variable”) sets. See at topos theory and at categorical logic for more on this.

## In dependent type theory

In dependent type theory, the set-theoretic multiverse is given by the existence of multiple inequivalent models of set theory as types $V$ with well-founded relations $\in$, which are made into Tarski universes via the dependent sum type:

$x:V \vdash \mathrm{El}(x) \coloneqq \sum_{y:V} y \in x$

In addition, this extends to any other notion of set theory, such as the categorical models of set theory as well-pointed categories $\mathcal{E}$, which are made into Tarski universes by the hom-set

$x:V \vdash \mathrm{El}(x) \coloneqq \mathrm{hom}(1, x)$

where $1:\mathcal{E}$ is the terminal separator of the category $\mathcal{E}$.

This all coexists with the usual type theoretic notion of universes of sets as Tarski universes $U$ with universal type family $T$ in which for every type $A:U$ in the universe, $T(A)$ satisfies UIP.

## References

Last revised on July 27, 2024 at 17:45:27. See the history of this page for a list of all contributions to it.