We show that from a supercompact cardinal \kappa, there is a forcing
extension V[G] that has a symmetric inner model N in which ZF + not AC holds,
\kappa\ and \kappa^+ are both singular, and the continuum function at \kappa\
can be precisely controlled, in the sense that the final model contains a
sequence of distinct subsets of \kappa\ 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_\omega\ 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_\omega\ and aleph_{\omega+1} are both singular and the continuum
function at aleph_\omega\ can be precisely controlled. Additionally, we discuss
a result in which we separate the lengths of sequences of distinct subsets of
consecutive singular cardinals \kappa\ and \kappa^+ in a model of ZF. Some open
questions concerning the continuum function in models of ZF with consecutive
singular cardinals are posed.Comment: to appear in the Notre Dame Journal of Formal Logic, issue 54:3, June
201
