Does Boolos’s stage theory resolve the justification of the Axiom of Choice?

Determine whether George Boolos’s stage theory of the iterative formation of sets provides an adequate resolution of the justification of the Axiom of Choice within the iterative conception of sets, addressing Kurt Gödel’s concern about justifying the Axiom of Choice in the construction of the set-theoretic universe.

Background

The paper discusses Gödel’s iterative conception of sets and the related issue of justifying the Axiom of Choice (AC). George Boolos’s stage theory offers a formal account of stages in the iterative construction of sets, validating all ZF axioms but indicating limitations regarding AC. The authors note that Gödel’s lecture manuscript emphasizes concern with justifying AC, and they explicitly state that it is unclear whether Boolos’s construction resolves this concern.

Clarifying whether Boolos’s stage theory provides a semantic or methodological resolution to AC’s justification would bear directly on foundational views that prioritize objective justification of set-theoretic axioms.

References

The manuscript of the lecture in which G¨odel presents his simple type theory reveals his concern for the justification of the Axiom of Choice. It is not clear that Boolos’ construction resolves it (see [1]).

— Kurt Gödel and the Logic of Concepts (2406.05442 - Kostić et al., 8 Jun 2024) in Section 3.1