Abstract
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.
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