Determinacy from strong compactness of ω1

In the absence of the Axiom of Choice, the “small” cardinal ω1 can exhibit properties more usually associated with large cardinals, such as strong compactness and supercompactness. For a local version of strong compactness, we say that ω1 is X-strongly compact (where X is any set) if there is a fine, countably complete measure on ℘ω1(X). Working in ZF+DC, we prove that the ℘(ω1)-strong compactness and ℘(R)-strong compactness of ω1 are equiconsistent with AD and ADR+DC respectively, where AD denotes the Axiom of Determinacy and ADR denotes the Axiom of Real Determinacy. The ℘(R)-supercompactness of ω1 is shown to be slightly stronger than ADR+DC, but its consistency strength is not computed precisely. An equiconsistency result at the level of ADR without DC is also obtained.