When looking at mathematical definitions, there are quite a few cases where we limit certain properties to countably infinite sets (e.g. $\sigma$-Algebras).
In some cases we set this limit as we'd lose our intuition of what would happen if we chose an even bigger infinity, in other cases there are hard facts at work.
Yet, the only property of countably infinite sets that comes to my mind is … well, that they are countable, and the higher cardinalities are not.
I'm sure though that there are many more characteristic properties of countably infinite sets that get lost when moving up to higher cardinalities – so what are they?
Best Answer
There are a lot of combinatorial properties you lose. For example:
$\mathbb{N}$ satisfies infinite Ramsey's theorem for pairs: if $c:[\mathbb{N}]^2\rightarrow \{0,1\}$ is a $2$-coloring of pairs of natural numbers, then there is a $c$-homogeneous set $H$ (that is, $c(\{x,y\})=c(\{x',y'\})$ for all $x\not=y, x'\not=y'$ in $H$) of the same size as $\mathbb{N}$. This fails in general for uncountable sets: if $X$ is uncountable, there may be a $2$-coloring of pairs of elements of $X$ with no size-$X$ homogeneous set. In fact, this fails for most uncountable cardinalities in a precise sense (see below), and provably fails of $\aleph_1$ (and indeed any uncountable $\kappa\le 2^{\aleph_0}$) in particular: fix an injection $i:\aleph_1\rightarrow\mathbb{R}$, and consider the coloring $$c:[\aleph_1]^2\rightarrow\{0,1\}: c(\{x,y\}=1\iff (x<_{\aleph_1}y\iff i(x)<_\mathbb{R}i(y)))$$ (with the two different orderings "decorated" for clarity). - it's not hard to show, using the separability (= countable dense subset) of $\mathbb{R}$, that $c$ cannot have an uncountable homogeneous set.
$\mathbb{N}$ satisfies Konig's lemma: any infinite-height, finitely-branching tree has an infinite path. The "$\kappa$-version" of this property, for an arbitrary infinite cardinal $\kappa$, should be: "Any $<\kappa$-branching tree of height $\kappa$ has a $\kappa$-length branch," but this too breaks down in general for uncountable sets, and in particular is again provably false for $\aleph_1$.
Combinatorial properties of $\mathbb{N}$ in general are very special, and often correspond to large cardinal properties when lifted to the uncountable setting. For example, it is consistent with ZFC that only $\aleph_0$ has the Ramsey property above.