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Topic: Uncountable Diagonal Problem
Replies: 52   Last Post: Jan 6, 2013 2:43 PM

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William Elliot

Posts: 1,434
Registered: 1/8/12
Uncountable Diagonal Problem
Posted: Dec 30, 2012 3:47 AM
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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.



Date Subject Author
12/30/12
Read Uncountable Diagonal Problem
William Elliot
12/30/12
Read Re: Uncountable Diagonal Problem
Butch Malahide
12/30/12
Read Re: Uncountable Diagonal Problem
Shmuel (Seymour J.) Metz
12/30/12
Read Re: Uncountable Diagonal Problem
ross.finlayson@gmail.com
12/30/12
Read Re: Uncountable Diagonal Problem
Virgil
12/30/12
Read Re: Uncountable Diagonal Problem
ross.finlayson@gmail.com
12/30/12
Read Re: Uncountable Diagonal Problem
Virgil
12/30/12
Read Re: Uncountable Diagonal Problem
ross.finlayson@gmail.com
12/30/12
Read Re: Uncountable Diagonal Problem
Virgil
12/30/12
Read Re: Uncountable Diagonal Problem
ross.finlayson@gmail.com
12/31/12
Read Re: Uncountable Diagonal Problem
Virgil
12/31/12
Read Re: Uncountable Diagonal Problem
ross.finlayson@gmail.com
12/31/12
Read Re: Uncountable Diagonal Problem
Virgil
12/31/12
Read Re: Uncountable Diagonal Problem
ross.finlayson@gmail.com
1/1/13
Read Re: Uncountable Diagonal Problem
Virgil
1/1/13
Read Re: Uncountable Diagonal Problem
Virgil
1/1/13
Read Re: Uncountable Diagonal Problem
ross.finlayson@gmail.com
1/1/13
Read Re: Uncountable Diagonal Problem
Virgil
1/1/13
Read Re: Uncountable Diagonal Problem
Virgil
1/1/13
Read Re: Uncountable Diagonal Problem
ross.finlayson@gmail.com
1/1/13
Read Re: Uncountable Diagonal Problem
Virgil
1/1/13
Read Re: Uncountable Diagonal Problem
ross.finlayson@gmail.com
1/1/13
Read Re: Uncountable Diagonal Problem
Virgil
1/1/13
Read Re: Uncountable Diagonal Problem
ross.finlayson@gmail.com
1/1/13
Read Re: Uncountable Diagonal Problem
Virgil
1/2/13
Read Re: Uncountable Diagonal Problem
ross.finlayson@gmail.com
1/2/13
Read Re: Uncountable Diagonal Problem
Virgil
1/2/13
Read Re: Uncountable Diagonal Problem
ross.finlayson@gmail.com
1/2/13
Read Re: Uncountable Diagonal Problem
Virgil
1/3/13
Read Re: Uncountable Diagonal Problem
ross.finlayson@gmail.com
1/3/13
Read Re: Uncountable Diagonal Problem
Virgil
1/3/13
Read Re: Uncountable Diagonal Problem
Virgil
1/3/13
Read Re: Uncountable Diagonal Problem
ross.finlayson@gmail.com
1/3/13
Read Re: Uncountable Diagonal Problem: Ross is WRONG, again!
Virgil
1/3/13
Read Re: Uncountably Nested Intervals
ross.finlayson@gmail.com
1/4/13
Read Re: Uncountably Nested Intervals
Virgil
1/4/13
Read Re: Uncountably Nested Intervals
ross.finlayson@gmail.com
1/4/13
Read Re: Uncountably Nested Intervals
fom
1/4/13
Read Re: Uncountably Nested Intervals
ross.finlayson@gmail.com
1/5/13
Read Re: Uncountably Nested Intervals
fom
1/5/13
Read Re: Uncountably Nested Intervals
ross.finlayson@gmail.com
1/5/13
Read Re: Uncountably Nested Intervals
fom
1/6/13
Read Re: Uncountably Nested Intervals
ross.finlayson@gmail.com
1/4/13
Read Re: Uncountably Nested Intervals
Virgil
12/30/12
Read Re: Uncountable Diagonal Problem
forbisgaryg@gmail.com
12/30/12
Read Re: Uncountable Diagonal Problem
ross.finlayson@gmail.com
12/31/12
Read Re: Uncountable Diagonal Problem
Virgil
12/31/12
Read Re: Uncountable Diagonal Problem
ross.finlayson@gmail.com
12/31/12
Read Re: Uncountable Diagonal Problem
Virgil
1/1/13
Read Re: Uncountable Diagonal Problem
Graham Cooper
1/1/13
Read Re: Uncountable Diagonal Problem
Virgil
1/1/13
Read Re: Uncountable Diagonal Problem
camgirls@hush.com
12/31/12
Read Re: Uncountable Diagonal Problem
Shmuel (Seymour J.) Metz

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