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.
In dependent type theory, the set-theoretic multiverse is given by the existence of multiple inequivalent models of set theory as types with well-founded relations , which are made into Tarski universes via the dependent sum type:
In addition, this extends to any other notion of set theory, such as the categorical models of set theory as well-pointed categories , which are made into Tarski universes by the hom-set
where is the terminal separator of the category .
This all coexists with the usual type theoretic notion of universes of sets as Tarski universes with universal type family in which for every type in the universe, satisfies UIP.
Joel David Hamkins: The set-theoretic multiverse, Review of Symbolic Logic 5 3 (2012) 416-449 [arXiv:1108.4223, doi:10.1017/S1755020311000359]
Claudio Ternullo, Maddy on the Multiverse, in: Reflections on the Foundations of Mathematics, Synthese Library 407 Springer (2019) [doi:10.1007/978-3-030-15655-8_3, pdf]
Last revised on July 27, 2024 at 17:45:27. See the history of this page for a list of all contributions to it.