I'm faced with a set of strictly increasing functions $\Bbb N\to \Bbb N$, i.e. positive integer valued sequences. The only thing I know about them is that they are pairwise eventually disjoint, by which I mean that given $U$, $V$ in this set, $U(\Bbb N)\cap V(\Bbb N)$ is finite.
Is this family countable? Uncountable? Does it depend?
Obviously the set of all positive integer sequences is uncountable, so this feels like a bit of a long shot, but I'm hoping the pairwise eventually disjoint condition might put a cap on the cardinality.
One thing I've been able to do, using a sort of diagonal argument, is to show that given such a set, if it's countable, I can find a new sequence that's eventually pairwise disjoint to every element in the set, and yet has no number assigned to it. But this new sequence won't actually be in the set, so this doesn't produce a contradiction like a diagonal argument is supposed to.
If the set is countable, then I can "decapitate" each sequence by a finite amount to obtain a set of pairwise disjoint sequences, which I'd really like to be able to do.