Cantor used the notion of an "injection" to formalize the size of two sets: A is "smaller" than B if A injects into B.
Simply put, the question is - how does this situation change if we use surjections instead of injections in our notion of size? And if we use "surjections both ways" to define equivalence classes rather than "injections both ways?"
If we are working in ZFC, these should be equivalent. Without AC, I don't think they will be. Basically, I am wondering if surjections are "better behaved" than injections if there is no AC. For example, without AC, the reals don't even inject into the ordinals, but do the ordinals surject onto the reals?
To be really clear, I think there is some added subtlety in that without AC you don't necessarily get the "dual CBS theorem" - that surjections from sets $A \to B$ and $B \to A$ give you a bijection $A \leftrightarrow B$. In other words, without AC, the existence of a bijection can be stronger than the existence of two surjections, so that the latter should yield coarser equivalence classes than the former - which is what I am asking about.
So summarizing, let's define the "surjective cardinals" as equivalence classes of sets that mutually surject onto one another (or the least-rank representative of each such class). Then I am curious about basic questions like (all assuming no $AC$):
- What is the general structure of these "surjective cardinals" without AC?
- Do the surjective cardinals have any nice ordering properties wrt surjections (are they linearly ordered, or well-ordered, or ...)?
- Are the ordinals "surjectively larger" than the reals, without AC?
- If so, is there some least ordinal that is surjectively larger than the reals? (Regardless of if we know which one it is)
- Is there a corresponding notion of CH for surjective cardinals?
Are these well-studied? Does anyone have any references?