Existence of semantically justified set axioms beyond ZF

Ascertain whether there exist set-theoretic axioms beyond the Zermelo–Fraenkel axioms that admit semantic justification grounded in an objective conception of the set-theoretic universe, and identify such axioms if they exist.

Background

The authors distinguish semantic and syntactic justifications of axioms, noting Gödel’s preference for semantic grounding via intuition of the set-theoretic universe. While ZF axioms have semantic justification in this framework, the paper explicitly states uncertainty about whether any additional set-theoretic axioms enjoy such semantic justification.

Resolving this question is central to the pursuit of a more complete, objectively grounded axiomatization of set theory, particularly regarding large cardinal axioms and consequences for undecidable statements.