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.
Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
