An Easton-like Theorem for Zermelo-Fraenkel Set Theory Without Choice (Preliminary Report)

TL;DR

This paper extends Easton's theorem to Zermelo-Fraenkel set theory without the Axiom of Choice, demonstrating how to control the surjective exponential function on all infinite cardinals under certain conditions.

Contribution

It introduces a choiceless version of Easton's theorem, allowing arbitrary cardinal values for the surjective exponential function without assuming choice.

Findings

The surjective exponential function can be forced to take arbitrary cardinal values.

Monotonicity and Cantor's theorem are sufficient conditions in the choiceless context.

The result holds irrespective of cofinalities.

Abstract

By Easton's theorem one can force the exponential function on regular cardinals to take rather arbitrary cardinal values provided monotonicity and Koenig's lemma are respected. In models without choice we employ a "surjective" version of the exponential function. We then prove a choiceless Easton's theorem: one can force the surjective exponential function on all infinite cardinals to take arbitrary cardinal values, provided monotonicity and Cantor's theorem are satisfied, irrespective of cofinalities.