Consecutive singular cardinals and the continuum function
arXiv:1112.1890 · doi:10.1215/00294527-1960434
Abstract
We show that from a supercompact cardinal κ, there is a forcing extension V[G] that has a symmetric inner model N in which ZF + not AC holds, κ and κ^+ are both singular, and the continuum function at κ can be precisely controlled, in the sense that the final model contains a sequence of distinct subsets of κ of length equal to any predetermined ordinal. We also show that the above situation can be collapsed to obtain a model of ZF + not AC_ω in which either (1) aleph_1 and aleph_2 are both singular and the continuum function at aleph_1 can be precisely controlled, or (2) aleph_ω and aleph_{ω+1} are both singular and the continuum function at aleph_ω can be precisely controlled. Additionally, we discuss a result in which we separate the lengths of sequences of distinct subsets of consecutive singular cardinals κ and κ^+ in a model of ZF. Some open questions concerning the continuum function in models of ZF with consecutive singular cardinals are posed.
to appear in the Notre Dame Journal of Formal Logic, issue 54:3, June 2013