The usual philosophical position in set theory is the universe view, according to which there is an absolute concept of set in an absolute universe in which every set-theoretic assertion has a definite truth value. The consequence is that formally undecidable statements like the continuum hypothesis have definitive answers.
In contrast, @2012hamkinsSettheoretic argues for the multiverse view:
There are diverse distinct concepts of set, each instantiated in a corresponding set-theoretic universe, which exhibit diverse set-theoretic truths. Each such universe exists independently in the same Platonic sense that proponents of the universe view regard their universe to exist. … By adopting a particular concept of set, we in effect adopt that universe as our current mathematical universe; we jump inside and explore the nature of set theory offered by that universe. (2)
The multiverse view is a realist position about the actual existence of alternative set-theoretic universes: “Platonism about universes … into which our mathematical tools have allowed us to glimpse.”