Cfrgtkky

In a universe where the continuum hypothesis ($CH$) fails we can ask about combinatorial cardinal characteristics of the continuum, but in a universe where $CH$ is true no such cardinals exist so this study becomes vacuous.

Does a similar phenomenon occur at the countable level in a universe without choice? Specifically, are there properties which are true for finite sets but false for $omega$ which are still true for the cardinal of an amorphous set, like divisibility as suggested here by François G. Dorais?

In a universe without choice we have the existence of amorphous sets and we can ask about their 'amorphous cardinals' which are incomparable with $omega$ (thank you Asaf for the correction) and may satisfy nice theorems, but in a universe with choice there are no infinite sets whose cardinality is incomparable with $omega$ so this study becomes vacuous in similar fashion to the uncountable case.

A possible candidate for characteristics smaller than $omega$ could come from theorems in finite group theory that become false for countable groups, since it is possible to have a group structure on an unbounded amorphous cardinal as constructed by Asaf Karagila here.

There is an article behind a paywall published in 2010 that appears to touch on these matters but I can't access it; if anyone is familiar with its contents and willing to give a brief exposition it would be greatly appreciated.

Any cardinal smaller than $aleph_0$ is finite. Amorphous sets are not "smaller", they are just incomparable with. They are very small, in some sense, for example we cannot even divide them into two infinite sets, but they are still infinite.

With respect to the article you linked, let me point out that amorphous sets cannot even be mapped onto $omega$, so they are definitely not the countable union of pairs.

Now, there are some combinatorial characteristics one can assign to general sets, which may be of interest in the case of amorphous sets. For example, if $A$ is amorphous, then any partition of $A$ is up to finitely many parts constant in size (i.e. all but finitely many parts are singletons, or pairs, or so on). We call this size the gauge of the partition, and we can ask what is the supremum of the gauges of possible partitions.

This can be $1$, or some finite $n$, or it can be "unbounded". We can prove, for example, that if $A$ is an amorphous set which can be made into a group, then it is unbounded. So it gives us some information.

But in general, this is not something too similar to cardinal characteristics in the traditional sense, and it is not something too helpful, since $omega$ is a very unique and a very concrete set, whereas amorphous sets can come in many different flavors, sizes, and support different structures.