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".
 A: Let me make a few remarks about the constructive aspects. The standard definition is the following: a set $X$ is finite if there is a natural number $n$ and a bijection between $X$ and $\{ i \in \mathbb{N} : i < n \}$. Some of the expected properties are true:


*

*The disjoint union of two finite sets is finite.

*The product of two finite sets is finite.

*The set of maps between two finite sets is finite.


On the other hand, there are some strange facts:


*

*Subsets of finite sets may not be finite.

*Quotients of finite sets may not be finite.


For example, given a proposition $\varphi$, $\{ i \in \mathbb{N} : \varphi \land i < 1  \}$ is finite if and only if $\varphi \lor \lnot \varphi$ holds. (This is because equality in $\mathbb{N}$ is decidable.) Thus one is tempted to look for weaker notions of finiteness.
Here is one alternative. The class of Kuratowski-finite sets is defined inductively as follows:


*

*The empty set is Kuratowski-finite.

*Every singleton set is Kuratowski-finite.

*The union of two Kuratowski-finite sets is Kuratowski-finite.


It is true that the quotient of a Kuratowski-finite set is automatically Kuratowski-finite. Indeed, every Kuratowski-finite set is in bijection with the quotient of some finite set – thus, one might call them finitely generated sets. In particular, Kuratowski-finiteness is strictly more general than finiteness. On the other hand, subsets of Kuratowski-finite sets may not be Kuratowski-finite.
A: 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 form 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.
A: 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.

A: At the elementary level of the sum of a typical infinite series such as $\sum_{n=1}^\infty \frac{1}{n^2}$ one can illustrate the idea of infinities smaller than the superscript $\infty$ in the sum by using the hyperreal framework.  Here a choice of a positive nonstandard hyperinteger $H$ gives a hyperfinite sum $\sum_{n=1}^H \frac{1}{n^2}$ which is infinitely close to the sum of the series but is not quite it.  Namely, one has $\sum_{n=1}^H \frac{1}{n^2}<\sum_{n=1}^\infty \frac{1}{n^2}$ (strict inequality) but $\sum_{n=1}^H \frac{1}{n^2}\approx\sum_{n=1}^\infty \frac{1}{n^2}$.  Another typical application is the hyperfinite sum $\sum_{n=1}^H \frac{1}{10^n}<1$.  Writing the lefthandside as zero, dot, followed by more than any finite number of $9$s is risky; see e.g., here.
