Large Countable Ordinals
When you get to uncountable ordinals, you run into the problem that any language has only finitely many symbols, so you can only write out countably many ordinals explicitly, so you run out of symbols when you try to express all the larger ordinals in some kind of a generalization of Cantor Normal Form. Indeed you run into this problem long before the first uncountable ordinal, it is already an issue with large countable ordinals.
Read more about this topic: Ordinal Arithmetic
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