Ahh, I'm starting to get it. This is the same argument as below. This set is linearly independent, but, as a Hamel basis, it fails to span the entire space. Ergo it is not an algebraic basis.  Notice that algebraic basis is a red link, and yet its definition is central to the argument. I presume that there must be a proof that no algebraic basis for for such a space can ever be countable; does such a theorem have some famous name?    