Two sets are of the same cardinality if a bijection exists between the two sets.
Two sets represent the same ordinal if an order preserving bijection exists between them. The ordinals produce distinctions between sets that general bijections did not. So introducing ordering allows us to refine the idea of size (no one shoot me for that statement).
Are there other structures that do this? More specifically I'm looking for an extra structure (other than 'ordering') that can be imposed on a set, such that:
- two sets (plus structures) are defined as equivalent if there is a structure-preserving bijection between them
- these equivalence classes can be ordered in a natural structure that 'plays nice with' (e.g. refines, but I'm open to other ideas if that cuts out something interesting) the cardinals.
The structures I'm aware of don't seem to work. As far as I can see there isn't a canonical way to talk about topological structures or group structures on a set as imposing some sort of weak ordering on the class of sets. Are there non-trivial and/or interesting examples of this? Or do they all in some way reduce to things that are identical to the above two cases?
There are two parts to this answer, one approaching the problem, and one giving an example for a solution in the absence of choice.
Part I
You first need to define what are the expected properties from this notion of cardinality.
Since we are going to deal with "structure" let us think of sets as structures of the language of membership (or equality, if you prefer), so they are really just lacking meaningful structure.
This is a somewhat philosophical question, because it requires you to come up with a notion. Let me write what I find as a reasonable notion of "size":
To see that the usual definition of cardinality has these properties note that all are trivial apart for the fourth one, but it too is true because we treat sets as structureless objects, so any additional structure is preserved by bijections (i.e. transport of structure) and does not interfere with the no-structure on sets.
Now we want to direct our attention to $\leq$, whose definition we will also generalize from the usual order: $$|M|\leq|N|\iff\exists f\: M\xrightarrow{f} N\text{ monomorphism}\\|M|=|N|\iff\exists f: M\xrightarrow{f} N\text{ isomorphism}$$
Again, this is obvious that the usual cardinals have these properties (mono. are just injections, and iso. are just bijections).
As this seem to be going in the direction of categories, I should mention that the fourth condition should be rephrased using functors, what I have in mind is something like "Whenever $F$ is a forgetful functor, and $F(x)=M$, there is a functor $G$, naturally isomorphic to $F$ such that $G(x)=N$, or something like that.
Now you have some idea about what sort of categories can generalize cardinalities. Or at least some idea about how to approach the problem.
Part II
Now let us discuss for a moment what happens with the cardinals when the axiom of choice fails. I will also deviate slightly from my previous characterization of "good measures for cardinality".
The axiom of choice is equivalent to saying that $\leq$ is a linear ordering on the cardinals. When we assume its negation then this ordering is not linear anymore. So we cannot represent every cardinal as an initial ordinal, and we are resorted to other tricks (e.g. Scott's trick). It might be the case that there is a function which chooses a representative set for each cardinality, but it is also consistent that this proposition fails.
This is one example for how we extend the notion of cardinals from $\sf ZFC$.
Another example is to consider $\leq^*$ which is defined using surjections rather than injections. This order does not always satisfy the Cantor-Bernstein property (i.e. in some models this relation is not anti-symmetric modulo equivalence). But this is an interesting order nonetheless.
If one allows a weaker notion of cardinality to hold, i.e. do not require the CB property, then this order makes a very natural example.