Dedekind-finite cardinals having countable partitions

TL;DR

This paper investigates the structure of certain infinite sets that lack countable subsets but are not surjectively Dedekind finite, exploring their properties under different notions of surjectivity.

Contribution

It introduces and analyzes the properties of sets with no countable subset that are not surjectively Dedekind finite in two distinct senses.

Findings

Characterization of sets with no countable subset that are not surjectively Dedekind finite.

Comparison between two notions of surjectivity for such sets.

Insights into the structure of these special infinite sets.

Abstract

We study the possible structures which can be carried by sets which have no countable subset, but which fail to be `surjectively Dedekind finite', in two possible senses, that there is a surjection to $\omega$, or alternatively, that there is no surjection to a proper superset.