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.