Four Cardinals and Their Relations in ZF

TL;DR

This paper investigates the possible orderings of four specific set cardinalities in ZF set theory, establishing that at least five of six potential orderings are consistent without the Axiom of Choice.

Contribution

It demonstrates the consistency of multiple linear orderings among four key set cardinalities within ZF, expanding understanding of their relationships without choice.

Findings

At least five of six possible orderings are consistent in ZF.

The four cardinalities can be pairwise distinct and comparable.

The results do not rely on the Axiom of Choice.

Abstract

For a set $M$, $\operatorname{fin}(M)$ denotes the set of all finite subsets of $M$, $M^2$ denotes the Cartesian product $M\times M$, $[M]^2$ denotes the set of all $2$-element subsets of $M$, and $\operatorname{seq}^{1-1}(M)$ denotes the set of all finite sequences without repetition which can be formed with elements of $M$. Furthermore, for a set $S$, let $|S|$ denote the cardinality of $S$. Under the assumption that the four cardinalities $|[M]^2|$, $|M^2|$, $|\operatorname{fin}(M)|$, $|\operatorname{seq}^{1-1}(M)|$ are pairwise distinct and pairwise comparable in ZF, there are six possible linear orderings between these four cardinalities. We show that at least five of the six possible linear orderings are consistent with ZF.