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".
[Math] What’s between the finite and the infinite
infinitylogicreference-requestsoft-question
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The proof is very intuitive (as you probably are feeling). But it can be written elaborately as follows, if you wish.
Your claim: For any finite set F, there exists an infinite subset I.
Try to prove: Let $F$ be a finite set defined as $F = \{f_1, f_2, \ldots , f_n\}$, where $n = 1, 2, \ldots$
Let $I$ be an infinite set defined as $I = \{i_1, i_2, \ldots, i_n, \ldots\}$, where n = 1, 2, ...
If I is a subset of F, then every element in I is also an element in F. If F contains finitely many elements, then only finitely many elements of I could belong to F.
However, I is infinite by definition, so clearly not all elements of I are contained in F.
Therefore, I is not a subset of F. This implies the claim is false.
Hence, for any finite set F, there does not exist an infinite subset I.
There is actually a proof you can probably find which does the same thing, just it takes a different angle: Prove that every subset of a finite set is finite. You can probably look this up somewhere!
I don't believe there are infinite planets in the universe. There are a large number, but it is not infinite. I don't believe anything in the universe is infinite, so there shouldn't be anything to reconcile here. Inverted World is sci-fi, so it's not even a theory. Just a nice tale!
Your thinking is correct. In fact, what you have argued is that there exists a bijection between the set of all convergent sequences of primes and the set $F\times P$. If this is your first example of doing this kind of task I suggest you try to actually write down this bijection. You have argued quite well that it must exist, but it's good exercise to actually do it at least once.
Noy, you also rightly say that $F\times P$ has a bijection to $\mathbb N^2$. As before, I suggest you actually write down this bijection.
Now, you simply need to see if $\mathbb N^2$ is countable, that is:
Does there exist a bijection between $\mathbb N$ and $\mathbb N^2$
To that end, I suggest you look into Cantor's pairing function.
Best Answer
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.