TITLE: Do any proofs/properties rely on the distinction between some uncountable size and a larger uncountable size, in order for the proof/property to hold?
QUESTION [12 upvotes]: (Sorry if my terminology is a bit imprecise; I'm trying to describe a rather vague notion in my head about when something "relies on" the distinction between finite/countable/uncountable, but I am finding it difficult to put into words precisely.)
There are proofs or properties that only hold when something is finite, but breaks down when it becomes infinite. For example, the finite intersection of open sets is open, but not arbitrary infinite intersections. As another example, $ a_n>0 \ \forall n \in \mathbb{N} $ does not imply $ \lim_{n \rightarrow \infty}{a_n}>0 $.
Similarly, there are also proofs or properties that only hold when something is countable. For example, normal induction can only be used if the variable takes values from a countable set. 
I was wondering, are there any common proofs or properties that rely on a similar distinction between some uncountable size and another larger uncountable size? For example, property X is true iff some set S has size $\leq \aleph_n$ where $n>0$ ?
(Not sure what to tag; please edit if necessary :) )

REPLY [4 votes]: It is not a trivial theorem, but $(\ell^\infty)^*$ is the $\rm ba$ space, whose cardinality is $2^{2^{\aleph_0}}$. The reason is that we can identify this space with finitely additive measures, and every ultrafilter on $\Bbb N$ induces such measure, and by a fairly straightforward argument, there are $2^{2^{\aleph_0}}$ such ultrafilters. The upper bound can be obtained by noting that the algebraic dual, which is strictly larger, has cardinality $2^{2^{\aleph_0}}$, since the dimension of $\ell^\infty$, as a linear space, is $2^{\aleph_0}$.
 
Now, since $\ell^1$ is a separable Banach space, its cardinality is only $2^{\aleph_0}$. This gives a "quick" proof as to why $(\ell^\infty)^*\ncong\ell^1$.