Maximality and ontology: how axiom content varies across philosophical frameworks

Neil Barton, Sy-David Friedman
2017 Synthese  
Discussion of new axioms for set theory has often focused on conceptions of maximality, and how these might relate to the iterative conception of set. This paper provides critical appraisal of how certain maximality axioms behave on different conceptions of ontology concerning the iterative conception. In particular, we argue that forms of multiversism (the view that any universe of a certain kind can be extended) and actualism (the view that there are universes that cannot be extended in
more » ... ular ways) face complementary problems. The latter view is unable to use maximality axioms that make use of extensions, where the former has to contend with the existence of extensions violating maximality axioms. An analysis of two kinds of multiversism, a Zermelian form and Skolemite form, leads to the conclusion that the kind of maximality captured by an axiom differs substantially according to background ontology.
doi:10.1007/s11229-017-1336-9 pmid:32214510 pmcid:PMC7061924 fatcat:3zousqdz7bcw5b37hnoqx5ibbq