In more rigorous language: " V: a vector space having an uncountable base S: The set of subspaces of V that have countable dimension. Can we construct explicitly a chain in the poset S (ordered by inclusion), such that this chain has NO upper bound in S? "

Apparently, this chain must have uncountable terms. Also,because S doesn't satisfy Zorn's lemma, we know such chain must exist in S.

But how do we construct it?