infinite subset of an finite set?

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!