Something I worked on a bit last year, and continue to work on, is how *what an axiom says* is dependent on the ontological framework in which it is evaluated. In particular, when an axiom of set theory uses *extensions* of universes in order to evaluate its content (as is the case with Friedman’s Inner Model Hypothesis [which states that any parameter-free sentence true in an extension is true in the universe]), the axiom can have quite a radically different meaning based on philosophical perspective. This is because while axioms might not have their `intended’ content (in the case of the Inner Model Hypothesis for example, you might not believe that there are extensions of the set-theoretic universe), they can quite often be coded. You then get an interesting phenomenon where what an axiom says varies according to whether it states a property of a universe by asserting a relationship between actually existing bona fide universes, or alternatively whether it states that a universe behaves a certain way with respect to kinds of coding mechanism (dependent upon one’s philosophical views). In fact, interpretation of certain complicated axioms in set theory turn out to have very similar features to standard debates between anti-realists and realists in the philosophy of mathematics.

You can find more details here: Journal, Pre-print.

Abstract:

Discussion of new axioms for set theory has often focussed 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.