Maximality and ontology: how axiom content varies across philosophical frameworks

Synthese 197 (2):623-649 (2017)
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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 particular 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.

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Citations of this work

Universism and extensions of V.Carolin Antos, Neil Barton & Sy-David Friedman - 2021 - Review of Symbolic Logic 14 (1):112-154.
Are Large Cardinal Axioms Restrictive?Neil Barton - forthcoming - Philosophia Mathematica.

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References found in this work

The iterative conception of set.George Boolos - 1971 - Journal of Philosophy 68 (8):215-231.
The potential hierarchy of sets.Øystein Linnebo - 2013 - Review of Symbolic Logic 6 (2):205-228.

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