Four Cardinals and Their Relations in ZF
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 $\operatorname{seq}^{11}(M)$ 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 $[M]^2$, $M^2$, $\operatorname{fin}(M)$, $\operatorname{seq}^{11}(M)$ 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.
 Publication:

arXiv eprints
 Pub Date:
 September 2021
 DOI:
 10.48550/arXiv.2109.11315
 arXiv:
 arXiv:2109.11315
 Bibcode:
 2021arXiv210911315H
 Keywords:

 Mathematics  Logic;
 {\bf 03E35};
 03E10;
 03E25
 EPrint:
 19 pages, 1 figure