A directed set is a pair $(A,leq)$ where $leq$ is a reflexive, transitive relation such that for any $x,yin A$ we have some $z$ such that $x,yleq z$. (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?