Abstract

We show that from a supercompact cardinal $\kappa$, there is a forcing extension $V[G]$ that has a symmetric inner model $N$ in which $\mathrm {ZF}+\lnot\mathrm {AC}$ 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 $\mathrm {ZF}+\lnot\mathrm {AC}_{\omega}$ in which either $\aleph_{1}$ and $\aleph_{2}$ are both singular and the continuum function at $\aleph_{1}$ can be precisely controlled, or $\aleph_{\omega}$ 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 $\mathrm {ZF}$. Some open questions concerning the continuum function in models of $\mathrm {ZF}$ with consecutive singular cardinals are posed