I *think* that this was answered by Andres Caicedo in a comment to an answer to this [question](https://mathoverflow.net/questions/59115/a-set-for-which-it-is-hard-to-determine-whether-or-not-it-is-countable/59181#59181).  

I quote:

> Given an infinite sequence of 1s and 2s, its initial segments are numbers (written in decimal notation, for example), so any such sequence corresponds to an infinite subset of ℕ, and any two of these sets have finite intersection.

This is basically the same answer as that of Todd Elsworth, perhaps phrased a bit more snappily.