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March 4th, 2016, 04:11 PM   #1
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Cantors Diag Argument Proves Reals Countable

Let T be the set of all infinite binary series.

List all the elements of T and assume the list is uncountable.
Use Cantor's Diagonal Argument to show there is a member of T not in the list.

Contradiction. Therefore T is countable.
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March 4th, 2016, 04:54 PM   #2
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Quote:
Originally Posted by zylo View Post
List all the elements of T and assume the list is uncountable.
How is this possible? Or am I just reading into the wording the wrong way.
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March 4th, 2016, 05:01 PM   #3
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You aren't. Zylo doesn't understand that listing a set and counting a set are the same thing. He does have an irrational desperation to prove Cantor wrong though.
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March 4th, 2016, 05:51 PM   #4
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(sighs) Not again!

How about this. Make a "list" of T and assume it is countably infinite. Therefore there exists a bijection f from the integers to T.

Now use Cantor's diagonal argument to show there is an element of T that is not contained in the already listed elements of T. Therefore f is not onto and thus cannot be a bijection, a contradiction. Therefore T is not countably infinite.

Why is this so difficult for you?

-Dan

Last edited by topsquark; March 4th, 2016 at 05:59 PM.
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March 4th, 2016, 06:19 PM   #5
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I am starting to think that zylo is just posting these as a joke.
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March 5th, 2016, 12:44 AM   #6
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Quote:
Let T be the set of all infinite binary series.
Pause right here.

How do you prove that T exists?
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March 5th, 2016, 02:09 AM   #7
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Quote:
Originally Posted by zylo View Post
List all the elements of T and assume the list is uncountable. . . . Contradiction. Therefore T is countable.
You have assumed that T is uncountable and reached a contradiction, but you have also assumed that all the elements of T can be listed.

You've shown that at least one of your assumptions is untenable, but it may be your assumption that all the elements of T can be listed that is false, rather than your assumption that T is uncountable. That means that your argument is unsound (regardless of whether T is countable or uncountable).
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