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I have some notes about curious facts about ordinal numbers, for example that their addition is not commutative, multiplication is not distributive from the right hand side and that the exponent rule doesn't always hold. Also that some things that are undefined in analysis like $0^0, \infty^0, 1^\infty$ are actually defined for ordinal numbers. I know that there's been some investigations devoted to pursuing ordinal arithmetic along the lines of classical results in number theory for example.

Do you happen to know other curious facts about ordinal numbers (compared to facts in analysis or other)?

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    Nice list, but the notation is so small (at least in my browser!)2012-11-21
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    I don't think there are many fun facts on ordinals, or maybe I just don't know what you want to hear...2012-11-21
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    John Baez (http://math.ucr.edu/home/baez/) has been writing a series of posts on Google+ (https://plus.google.com/117663015413546257905/posts) on countable ordinals, which you may find interesting. He is a very engaging expositor. The latest one is on the Hercules-Hydra game: https://plus.google.com/117663015413546257905/posts/HSS2PMVENe3 (and see also https://plus.google.com/117663015413546257905/posts/PjyerUy3AeH )2012-11-21
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    Well for example other fun facts I know of is that Fermat's Last Theorem and Goldbach's Hypothesis are known to be false in ordinal number theory (Sierpinski [1950]).2012-11-21
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    But that is really trivial once you know how ordinal arithmetics work. Are you looking for trivial facts, or do you seek something which has some rather deep intuition as an example?2012-11-21
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    As a sanity check, by "$\infty^0$" and "$1^\infty$", do you really mean for $\infty$ as a variable whose values range over the infinite ordinals? As opposed, e.g. to suggesting there is an ordinal named $\infty$ or that it is the only infinite ordinal?2012-11-21
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    Theorem: Every ordinal is interesting. Proof: Assume there is an uninteresting ordinal. Then there is a least uninteresting ordinal. But this makes the ordinal interesting. ><2014-03-08

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