January 20th, 2018, 06:36 PM  #1 
Senior Member Joined: May 2015 From: Arlington, VA Posts: 299 Thanks: 25 Math Focus: Number theory  Limit of ordinality
Is there a limit or discontinuity to the set of ordinal numbers, or do they correspond to the real numbers?

January 20th, 2018, 06:59 PM  #2 
Senior Member Joined: Sep 2015 From: USA Posts: 1,755 Thanks: 900 
They correspond to the natural numbers. They have no limit. I'm not sure what you mean by discontinuity. They certainly aren't continuous over the reals. Last edited by greg1313; January 20th, 2018 at 07:33 PM. 
January 20th, 2018, 08:24 PM  #3  
Senior Member Joined: Aug 2012 Posts: 1,702 Thanks: 449  Quote:
As far as continuity, you can put a topology on the ordinals based on their order. This is the order topology. Under this topology you can define open and closed sets, and limits of sequences or ordinalindexed sequences. Quote:
An uncountable ordinal does have discontinuities if you think of it that way, because every ordinal has a successor. There's no third ordinal between an ordinal and its successor. On the other hand there are limit ordinals like $\omega$, which is the first transfinite ordinal. It goes at the end of the natural numbers: $0, 1, 2, 3, 4, \dots, \omega$. A limit ordinal is one that has no immediate predecessor, so $\omega$ is a limit ordinal. The topology of any ordinal is much different than the usual topology on the reals. Last edited by Maschke; January 20th, 2018 at 08:32 PM.  
January 20th, 2018, 10:06 PM  #4 
Senior Member Joined: May 2015 From: Arlington, VA Posts: 299 Thanks: 25 Math Focus: Number theory 
Maschke, Thank you for bringing up the role of topology with sets. Are you saying that a bijection between ordinals and reals may apply to any ordinal sequence? Do the real numbers have a cardinality? How does one describe the transition from naturals (countable sets) to, say, the power set (uncountables)? Keep keeping an open mind (Re: Banachâ€“Tarski) 
January 21st, 2018, 12:22 AM  #5  
Senior Member Joined: Oct 2009 Posts: 229 Thanks: 81  Quote:
There are of course also definitions of cardinality that do not use the axiom of choice, but then a lot of weird things happen.  

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