basic constructions:
strong axioms
further
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.
Joel David Hamkins, The set-theoretic multiverse, Review of Symbolic Logic 5:416-449 (2012), arxiv