A list of length eta, is a function from the ordinals < beta to a set of items. Mega-sequence will be used as a synonym for list.
How long does a list without duplicates of infinite binary series (IBS) have to be to force the list to have every IBS?
Let d be a list of length omega_1 of binary mega-sequences of length omega_1. Define the uncountable anti-diagonal as the mega-sequence of length omega_1 with b(eta) = 1 if d(eta)(eta) = 0 = 0 if d(eta)eta) = 1.
Clearly b isn't in the list d. Thusly no list of length omega_1 can list of all mega-sequences of length omega_1; there are more than omega_1 (aleph_1) of them. Since the set of binary mega-sequences of length omega_1 is equinumberous with the collections of subsets of ordinals < omega_1, this proves (the hard way) that there are more sets of countable ordinals than aleph_1.