A note on surjective cardinals

TL;DR

This paper investigates the structure of surjective cardinals within set theory, showing limitations of certain axioms and proposing a modified algebraic framework that simplifies proofs of key properties.

Contribution

It demonstrates that surjective cardinals do not form a traditional cardinal algebra under ZF+DC_kappa, and introduces a 'surjective cardinal algebra' with a finite refinement postulate.

Findings

Surjective cardinals do not form a cardinal algebra under ZF+DC_kappa.

A new 'surjective cardinal algebra' framework is proposed.

The cancellation law for surjective cardinals is proved more simply.

Abstract

For cardinals $\mathfrak{a}$ and $\mathfrak{b}$, we write $\mathfrak{a}=^\ast\mathfrak{b}$ if there are sets $A$ and $B$ of cardinalities $\mathfrak{a}$ and $\mathfrak{b}$, respectively, such that there are partial surjections from $A$ onto $B$ and from $B$ onto $A$. $=^\ast$-equivalence classes are called surjective cardinals. In this article, we show that $\mathsf{ZF}+\mathsf{DC}_\kappa$, where $\kappa$ is a fixed aleph, cannot prove that surjective cardinals form a cardinal algebra, which gives a negative solution to a question proposed by Truss [J. Truss, Ann. Pure Appl. Logic 27, 165--207 (1984)]. Nevertheless, we show that surjective cardinals form a ``surjective cardinal algebra'', whose postulates are almost the same as those of a cardinal algebra, except that the refinement postulate is replaced by the finite refinement postulate. This yields a smoother proof of the cancellation law for surjective cardinals, which states that $m\cdot\mathfrak{a}=^\ast m\cdot\mathfrak{b}$ implies $\mathfrak{a}=^\ast\mathfrak{b}$ for all cardinals $\mathfrak{a},\mathfrak{b}$ and all nonzero natural numbers $m$.