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 March 6th, 2016, 08:58 PM #1 Banned Camp   Joined: Mar 2015 From: New Jersey Posts: 1,720 Thanks: 126 Cantors Diagonal Argument and Epimenides Let S be the set of all infinite binary sequences. By Cantor's Diagonal Argument: S countable -> S uncountable -> S countable -> ...... , which is circular, which is Epimenides Liar Paradox: I am a liar T -> I am a liar F -> I am a liar T ......... Last edited by skipjack; March 6th, 2016 at 09:07 PM.
 March 6th, 2016, 09:07 PM #2 Global Moderator   Joined: Dec 2006 Posts: 20,968 Thanks: 2217 S uncountable -> S countable isn't shown by Cantor's (or any other) argument.
 March 6th, 2016, 09:37 PM #3 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,683 Thanks: 2661 Math Focus: Mainly analysis and algebra Zylo, what is your motivation behind all of these threads? Are you channelling Leopold Kronecker? You have the appearance of somebody who is desperate to achieve something, but it's not clear if that something has a clear definition or not. What is clear is that you don't have enough understanding to be making any claims. If you don't understand a particular result, you should admit that and then work to understand it, rather than simply claiming it to be false. I would go so far as to suggest that if you disagree with any result, you should do the same. In this case, you have yet again ignored the responses to your previous thread. In that thread you claimed that the diagonal argument could be used, with an assumption of the uncountability of a set, to prove the countability of that set. Now you use the same result, despite every response to the first thread pointing out how your argument was mistaken. Why do you refuse to listen to the arguments of others? Last edited by skipjack; March 6th, 2016 at 11:17 PM.
 March 8th, 2016, 05:37 AM #4 Math Team   Joined: Jan 2015 From: Alabama Posts: 3,264 Thanks: 902 What Cantor's proof says is that "if S is the set of all infinite binary sequences then S--> there exist an infinite binary sequence that is not in S", a contradiction.
March 8th, 2016, 06:08 AM   #5
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Quote:
 Originally Posted by Country Boy What Cantor's proof says is that "if S is the set of all infinite binary sequences then S--> there exist an infinite binary sequence that is not in S", a contradiction.
No, Cantor's proof says that, given any countable set S of infinite binary sequences, there exists an infinite binary sequence that is not in S. This result then provides a contradiction if we assume that the set of infinite binary sequences is countable.

If we assume that the set of infinite binary sequences is uncountable, Cantor's argument doesn't tell us anything interesting. In fact it doesn't apply, because the argument requires that the set be listed and no uncountable set can be listed.

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