An Easton-like theorem for Zermelo-Fraenkel Set Theory without Choice

Abstract

We show that in Zermelo-Fraenkel Set Theory without the Axiom of Choice a surjectively modified continuum function $\theta(\kappa)$ can take almost arbitrary values for all infinite cardinals. This choiceless version of Easton's Theorem is in sharp contrast to the situation in ZFC, where for singular cardinals $\kappa$, the value of $2^\kappa$ is strongly influenced by the behaviour of the continuum function below. Our construction can roughly be described as follows: In a ground model $V \models ZFC + GCH$ with a "reasonable" function $F: Card \rightarrow Card$ on the infinite cardinals, a class forcing ${\mathbb P}$ is introduced, which blows up the power sets of all cardinals according to $F$ . The eventual model $N \models ZF$ is a symmetric extension by ${\mathbb P}$ such that $\theta^N(\kappa) = F(\kappa)$ holds for all $\kappa$.