April 7th, 2010, 06:31 PM  #1 
Newbie Joined: Nov 2008 Posts: 8 Thanks: 0  ordinal, uniqueness question
For any ordinal , there are unique , , such that . Hint: Choose largest such that , and largest such that . I do not see how to use the hint to prove this. Would the found in the hint be unique? Then I was struggling showing that the other two were unique. I would appreciate some hints on this. Thanks. 
April 7th, 2010, 08:26 PM  #2 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 933 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: ordinal, uniqueness question
Yes, all three are unique, provided alpha is infinite.

April 8th, 2010, 07:27 AM  #3 
Senior Member Joined: Oct 2007 From: Chicago Posts: 1,701 Thanks: 3  Re: ordinal, uniqueness question
Is N an arbitrary ordinal, or is it finite? If N isn't finite*, then there is no largest n... but I'm not even sure if the statement holds when N isn't finite. Otherwise, such an n (that the hint mentions) exists and it must be unique: the ordinals are wellordered! Same with x. From there, you end up with . Can you show that ? *(actually, if N isn't a successor) 
April 8th, 2010, 09:19 AM  #4  
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 933 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: ordinal, uniqueness question Quote:
 
April 8th, 2010, 12:35 PM  #5  
Senior Member Joined: Oct 2007 From: Chicago Posts: 1,701 Thanks: 3  Re: ordinal, uniqueness question Quote:
Quote:
I feel like it might work for limits as well, but I can't really focus on anything right now. Maybe I'll give an argument later today. I don't see any reason why it should stop at epsilon_0... (Ok, that's not true, but I'm not sure if the fixedpoint status is enough to "stop" this...) Ugh. I'm gonna need to sit down and figure this all out explicitly now. xianghu, does the idea for finite N make sense? Or should we explain it in a little more depth?  

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