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 January 20th, 2018, 05:36 PM #1 Senior Member   Joined: May 2015 From: Arlington, VA Posts: 384 Thanks: 27 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, 05:59 PM #2 Senior Member     Joined: Sep 2015 From: USA Posts: 2,122 Thanks: 1102 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 06:33 PM.
January 20th, 2018, 07:24 PM   #3
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 Originally Posted by romsek 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.
They go way past the naturals. There are many countable ordinals past the naturals, and then they go into the uncountable ordinals.

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 ordinal-indexed sequences.

Quote:
 Originally Posted by Loren Is there a limit or discontinuity to the set of ordinal numbers, or do they correspond to the real numbers?
There are uncountable ordinals that are bijective with the real numbers. But no ordinal is order-equivalent to the reals in their usual order, because the reals in the usual order are not well-ordered.

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 07:32 PM.

 January 20th, 2018, 09:06 PM #4 Senior Member   Joined: May 2015 From: Arlington, VA Posts: 384 Thanks: 27 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: Banach–Tarski)
January 20th, 2018, 11:22 PM   #5
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 Originally Posted by Loren 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: Banach–Tarski)
If you accept the axiom of choice then there is a bijection between an initial segment of the ordinal numbers and the real numbers. This can then be used to give a cardinality to the real numbers that uses ordinal numbers.

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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