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?