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April 7th, 2010, 06:31 PM   #1
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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.
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April 7th, 2010, 08:26 PM   #2
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Re: ordinal, uniqueness question

Yes, all three are unique, provided alpha is infinite.
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April 8th, 2010, 07:27 AM   #3
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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 . Can you show that ?

*(actually, if N isn't a successor)
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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.
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April 8th, 2010, 12:35 PM   #5
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Re: ordinal, uniqueness question

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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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