My Math Forum Question

 Number Theory Number Theory Math Forum

 June 28th, 2013, 10:37 AM #1 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
Global Moderator

Joined: Nov 2006
From: UTC -5

Posts: 12,826
Thanks: 84

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 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 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 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 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 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
Global Moderator

Joined: Nov 2006
From: UTC -5

Posts: 12,826
Thanks: 84

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 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.
June 28th, 2013, 11:15 AM   #11
Global Moderator

Joined: Nov 2006
From: UTC -5

Posts: 12,826
Thanks: 84

Re: Question

Quote:
 Originally Posted by krausebj0 If you wish to discuss my previous paper here
Not allowed. But so far we've managed to keep this thread separate and I have high hopes that it can continue.

Quote:
 Originally Posted by krausebj0 Can anyone prove that E does not equal D?
You haven't defined them.

 June 28th, 2013, 11:16 AM #12 Joined: Nov 2011 Posts: 599 Thanks: 0 Re: Question Both E and D are defined. I am using only induction, if you must get really technical.
 June 28th, 2013, 11:23 AM #13 Joined: Nov 2011 Posts: 599 Thanks: 0 Re: Question Sorry CRG. I just do not cough up edited posts every time. Let D = { 0.1011..., 0.110111..., 0.1110111..., ... } Let E = { 0.111..., 0.1011..., 0.110111..., 0.1110111..., ... } I put the ... in each set. Is that better?
 June 28th, 2013, 11:24 AM #14 Global Moderator     Joined: Nov 2006 From: UTC -5 Posts: 12,826 Thanks: 84 Re: Question Giving examples separated by ellipses is not the same as a definition.
June 28th, 2013, 11:27 AM   #15

Joined: Nov 2011

Posts: 599
Thanks: 0

Re: Question

Quote:
 Originally Posted by CRGreathouse Giving examples separated by ellipses is not the same as a definition.
Then for crying out loud. What is your definition of E and D?

 Tags question

 Thread Tools Display Modes Linear Mode

 Contact - Home - Top