Choiceless Chain Conditions

TL;DR

This paper examines alternative conditions equivalent to the countable chain condition in forcing theory, especially in contexts lacking the axiom of choice, to better understand their utility and implications.

Contribution

It identifies and analyzes conditions equivalent to the countable chain condition without relying on the axiom of choice.

Findings

Several conditions are equivalent to the countable chain condition in ZFC.

Antichains are less useful without the axiom of choice.

The study clarifies the role of chain conditions in choice-less set theory.

Abstract

Chain conditions are one of the major tools used in the theory of forcing. We say that a partial order has the countable chain condition if every antichain (in the sense of forcing) is countable. Without the axiom of choice antichains tend to be of little use, for various reasons, and in this short note we study a number of conditions which in ZFC are equivalent to the countable chain condition.