Stronger Forms of Determinacy
Several set-theoretic principles about determinacy stronger than Borel determinacy are studied in descriptive set theory. They are closely related to large cardinal axioms.
The axiom of projective determinacy states that all projective subsets of a Polish space are determined. It is known to be unprovable in ZFC but relatively consistent with it and implied by certain large cardinal axioms. The existence of a measurable cardinal is enough to imply over ZFC that all analytic subsets of Polish spaces are determined.
The axiom of determinacy states that all subsets of all Polish spaces are determined. It is inconsistent with ZFC but equiconsistent with certain large cardinal axioms.
Read more about this topic: Borel Determinacy Theorem
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