Consecutive singular cardinals and the continuum function

TL;DR

This paper demonstrates how to construct models of ZF set theory with singular cardinals and precisely controlled continuum functions, using forcing from a supercompact cardinal, and explores related models with multiple singular cardinals.

Contribution

It introduces methods to control the continuum function at singular cardinals in models of ZF without the Axiom of Choice, extending previous results and posing open questions.

Findings

Existence of models with both  and  singular and controlled continuum functions

Construction of models with multiple singular consecutive cardinals

Separation of lengths of sequences of subsets of singular cardinals

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 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.