A directed set is a pair $(A,\leq)$ where $\leq$ is a reflexive, transitive relation such that for any $x,y\in A$ we have some $z$ such that $x,y\leq z$. (This comes up when dealing with categorical limits and topological nets).
In particular $(\mathbb{N},\leq)$ and $(\mathbb{R},\leq)$ are directed sets.
To help get comfortable with them, I imposed a "smallness" criteria: Let's say a "finite-type" directed set is a directed set where every element has finitely many predecessors (smaller elements).
My Guess: Finite-type directed sets are always countable.
As before $(\mathbb{N},\leq)$ is an example, but now $(\mathbb{R},\leq)$ is too big and is a non-example. Another example is $(\mathbb{N}^2,\leq)$ where $(a,b)\leq (c,d)$ iff $(c,d)-(a,b)\in \mathbb{N}^2$ and it's higher dimensional analogues. However, I've personally been unable to equip $\mathbb{N}^\mathbb{N}$ with an appropriate finite-type directed set structure.
Is there a clean proof or counterexample regarding my guess? Or does this somehow end up touching upon foundational things such as the axiom of choice?
Let $X$ be any set, and let $A$ be the collection of finite subsets of $X$; $A$ is directed by $\subseteq$, and each member of $A$ has only finitely many predecessors in that order. However, if $X$ is infinite, then $|A|=|X|$, so the cardinality of $A$ can be as large as you like.