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 April 7th, 2010, 06:31 PM #1 Newbie   Joined: Nov 2008 Posts: 8 Thanks: 0 ordinal, uniqueness question For any ordinal $\alpha < \omega^N$, there are unique $n, $x< \omega$, $\beta < \omega^n$ such that $\alpha=\omega^n \cdot x + \beta$. Hint: Choose $n$ largest such that $\omega^n \leq \alpha$, and $x$ largest such that $\omega^n \cdot x \leq \alpha$. I do not see how to use the hint to prove this. Would the $n$ 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: 937 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 well-ordered! Same with x. From there, you end up with $\omega^n.x + \beta$. Can you show that $\beta<\omega^n$? *(actually, if N isn't a successor)
April 8th, 2010, 09:19 AM   #4
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Re: ordinal, uniqueness question

Quote:
 Originally Posted by cknapp 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.
Right, sorry. I'm accustomed to unquantified Latin letters referring to natural numbers in this context (Greek and Hebrew for possibly-infinite ordinals). I suppose it might even hold slightly higher, though... to omega or even possibly below epsilon-naught.

April 8th, 2010, 12:35 PM   #5
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Re: ordinal, uniqueness question

Quote:
 Originally Posted by CRGreathouse Right, sorry. I'm accustomed to unquantified Latin letters referring to natural numbers in this context (Greek and Hebrew for possibly-infinite ordinals).
Yeah, I'm used to that as well, but since it's not explicit, it doesn't hurt to ask.

Quote:
 I suppose it might even hold slightly higher, though... to omega or even possibly below epsilon-naught.
Morally, I feel it should hold for all successor ordinals....
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 fixed-point 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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