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 $F:Card\to Card$ 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 $\kappa$.

Cited by in corpus (1)

• An Easton-like Theorem for Zermelo-Fraenkel Set Theory with the Axiom of Dependent Choice