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 June 28th, 2013, 10:37 AM #1 Senior Member   Joined: Nov 2011 Posts: 599 Thanks: 0 Question Let D = { 0.1011..., 0.110111..., 0.1110111..., ... } Let E = { 0.111..., 0.1011..., 0.110111..., 0.1110111..., ... } Both sets have a cardinality equal to aleph null. If E is well ordered, then does it have a last element? Does well ordering E change its cardinality?
June 28th, 2013, 10:43 AM   #2
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Re: Question

Quote:
 Originally Posted by krausebj0 Let D = { 0.1011..., 0.110111..., 0.1110111..., } Let E = { 0.111..., 0.1011..., 0.110111..., 0.1110111..., } Both sets have a cardinality equal to aleph null. If E is well ordered, then does it have a last element?
What is a "last element"?

Quote:
 Originally Posted by krausebj0 Does well ordering E change its cardinality?
No.

 June 28th, 2013, 10:44 AM #3 Senior Member   Joined: Nov 2011 Posts: 599 Thanks: 0 Re: Question 0.111... would be greater than any element of D. If E is well ordered, then D is a subset of E (D is a subset of E regardless). 0.111... would be the last element of E, right?
 June 28th, 2013, 10:55 AM #4 Senior Member   Joined: Nov 2011 Posts: 599 Thanks: 0 Re: Question Here is what you are not realizing. D = E
 June 28th, 2013, 11:09 AM #6 Senior Member   Joined: Nov 2011 Posts: 599 Thanks: 0 Re: Question Ai is bounded. If Ai is not bounded, then Ai = 0.10111..., 0.110111..., 0.1110111..., is an infinite sequence. If Ai is bounded, then the above is not an infinite sequence. Ai is only infinitely large if A = D.
 June 28th, 2013, 11:10 AM #7 Senior Member   Joined: Nov 2011 Posts: 599 Thanks: 0 Re: Question If you wish to discuss my previous paper here, I can make a surjection from D onto RI just by making sure A = D. Ai is bounded.
 June 28th, 2013, 11:11 AM #8 Senior Member   Joined: Nov 2011 Posts: 599 Thanks: 0 Re: Question Back to the point. Can anyone prove that E does not equal D?
June 28th, 2013, 11:13 AM   #9
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Re: Question

Quote:
 Originally Posted by krausebj0 0.111... would be greater than any element of D. If E is well ordered, then D is a subset of E (D is a subset of E regardless). 0.111... would be the last element of E, right?
You haven't defined "last element", nor explained its connection to your other question.

 June 28th, 2013, 11:14 AM #10 Senior Member   Joined: Nov 2011 Posts: 599 Thanks: 0 Re: Question If nobody can prove that E does not equal D, then I assert E equals D. Once I assert that, you guys better stop locking my threads.

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