Nam Trang




Monday, April 3, 2017 - 4:00pm to 5:30pm




We investigate various aspects of compactness of \omega_1 under ZF+ DC (the Axiom of Dependent Choice). We say that \omega_1 is X-supercompact if there is a normal, fine, countably complete nonprincipal measure on \powerset_{\omega_1}(X) (in the sense of Solovay). We say \omega_1 is X-strongly compact if there is a fine, countably complete nonprincipal measure on \powerset_{\omega_1}(X). A long-standing open question in set theory asks whether (under ZFC) "supercompactness" can be equiconsistent with "strong compactness. We ask the same question under ZF+DC. More specifically, we discuss whether the theories "\omega_1 is X-supercompact" and "\omega_1 is X-strongly compact" can be equiconsistent for various X. The global question is still open but we show that the local version of the question is false for various X. We also discuss various results in constructing and analyzing canonical models of AD^+ + \omega_1 is X-supercompact.