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What is the largest known ordinal number $\alpha$ such that a uniform notation scheme has been developed for all ordinals up to $\alpha$ (there should be no "gaps" in what ordinals are representable), with an algorithm allowing to effectively compare any two ordinals written in that notation?

(I understand that every such scheme can in principle be extended by adding ad hoc symbol for $\alpha$ itself, but I am interested in notations that have been actually described.)

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    What is a uniform notation scheme?2011-11-17
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    I mean a finite set of rules which identify some symbolic expressions with all ordinals up to a certain point - similar to Cantor normal form for ordinals < \epsilon_0.2011-11-17
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    If I could answer this question and give you such an ordinal $\alpha$, wouldn't I have just written down a symbolic expression that describes $\alpha$?2011-11-17
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    I understand that in principle every notation can be extended in this way. My question is more about what notations have been actually described, with an algorithm allowing comparison of ordinals written in that notation.2011-11-18
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    @nikov: It seems that the only reasonable answer is "letters" let $\alpha,\beta$ be two ordinals. Either $\alpha\in\beta$, $\alpha=\beta$ or $\beta\in\alpha$. This is a nice algorithm. Also note that if you replace $\omega$ by something larger in the CNF then you just get to a "larger" epsilon-like number.2011-11-18
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    @QiaochuYuan: To make it more clear, imagine a game where you and your opponent each is given a sheet of paper and 10 min to describe a notation and a corresponding comparison algorithm. One whose notation is able to represent all elements in a bigger ordinal wins. What approach would you choose?2011-11-18
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    @nikov: this strikes me as a very different question from the body question. Do you want to edit your question?2011-11-18
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    @VladimirReshetnikov Hm, that sounds like a fun game for a googologist.2017-04-26

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