Date: Dec 30, 2012 3:47 AM
Author: William Elliot
Subject: Uncountable Diagonal Problem
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.