Consecutive Singular Cardinals and the Continuum Function

Abstract

We show that from a supercompact cardinal $\kappa$, there is a forcing extension $V\left[G\right]$ that has a symmetric inner model $N$ in which $\mathrm{ZF}+\neg \mathrm{AC}$ 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 $\mathrm{ZF}+\neg {\mathrm{AC}}_{\omega }$ 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 $\mathrm{ZF}$. Some open questions concerning the continuum function in models of $\mathrm{ZF}$ with consecutive singular cardinals are posed.