What's between the finite and the infinite?
I'm wondering if there are any non-standard theories (built upon ZFC with some axioms weakened or replaced) that make formal sense of hypothetical set-like objects whose "cardinality" is "in between" the finite and the infinite. In a world like that non-finite may not necessarily mean infinite and there might be a "set" with countably infinite "power set".
There's a few things I can think of which might fit the bill:
We could work in a non-$\omega$ model of ZFC. In such a model, there are sets the model thinks are finite, but which are actually infinite; so there's a distinction between "internally infinite" and "externally infinite." (A similar thing goes on in non-standard analysis.)
Although their existence is ruled out by the axiom of choice, it is consistent with ZF that there are sets which are not finite but are Dedekind-finite: they don't have any non-trivial self-injections (that is, Hilbert's Hotel doesn't work for them). Such sets are similar to genuine finite sets in a number of ways: for instance, you can show that a Dedekind-finite set can be even (= partitionable into pairs) or odd (= partitionable into pairs and one singleton) or neither but not both. And in fact it is consistent with ZF that the Dedekind-finite cardinalities are linearly ordered, in which case they from a nonstandard model of true arithmetic; see https://mathoverflow.net/questions/172329/does-sageevs-result-need-an-inaccessible.
You could also work in non-classical logic - for instance, in a topos. I don't know much about this area, but lots of subtle distinctions between classically-equivalent notions crop up; I strongly suspect you'd find some cool stuff here.
Well, there are a few notions of "infinite" sets that aren't equivalent in $\mathsf{ZF}.$ One sort is called Dedekind-infinite ("D-infinite", for short) which is a set with a countably infinite subset, or equivalently, a set which has a proper subset of the same cardinality. So, a set is D-finite if and only if the Pigeonhole Principle holds on that set. The more common notion is Tarski-infinite (usually just called "infinite"), which describes sets for which there is no injection into any set of the form $\{0,1,2,...,n\}.$
It turns out, then, that the following are equivalent in $\mathsf{ZF}$:
- Every D-finite set is finite.
- D-finite unions of D-finite sets are D-finite.
- Images of D-finite sets are D-finite.
- Power sets of D-finite sets are D-finite.
Without a weak Choice principle (anything that implies $\aleph_0$ to be the smallest infinite cardinality, rather than simply a minimal infinite cardinality), the following may occur:
- There may be infinite, D-finite sets. In particular, there may be infinite sets whose cardinality is not comparable to $\aleph_0.$ Put another way, there may be infinite sets such that removing an element from such a set makes a set with strictly smaller cardinality.
- There may be a D-finite set of D-finite sets whose union is D-infinite.
- There may be a surjective function from a D-finite set to a D-infinite set.
- There may be a D-finite set whose power set is D-infinite.