Paper detail

Consecutive singular cardinals and the continuum function

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.