Multiverse

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Universes

In model theory, one comes across many models of many set theories. However, if one were to believe in urelements other than $\emptyset$, then there are a couple views one can take on which model of set theory contains every urelement.

The multiverse

The existence of the multiverse states that there is no model of set theory that contains every set of every urelement. The multiverse itself is the collection of all universes of set theory.

The universe

The existence of the universe opposes that of the multiverse, claiming there is a model of set theory containing every set of urelements. The universe is this one model containing every set of urelements.

The hyperverse

The hyperverse is the collection of all countable transitive models of ZFC. In many ways, this is similar to the multiverse.