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

TL;DR

This paper proves that in ZF set theory without the Axiom of Choice, the surjectively modified continuum function can be almost arbitrarily assigned to all infinite cardinals, contrasting with the constraints in ZFC.

Contribution

It establishes a version of Easton's Theorem for ZF, showing the continuum function's values can be freely specified without the Axiom of Choice.

Findings

The continuum function can be almost arbitrarily prescribed in ZF.

A class forcing construction achieves the desired continuum function values.

The model demonstrates the independence of continuum function behavior from 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 $\kappa$.