User blog comment:Edwin Shade/Enumerating the Countable Ordinals/@comment-30754445-20171206034326

Nice.

There's no need for multiplication ("2"), though. You can represent any ordinal below e0 with the other 6 symbols.

As for the notion of ennumerating all countable ordinals: Them being uncountable is not your problem. You can give every ordinal a real number (there are certainly enough real numbers to do the job). The problem is that these ordinal are supposed to well-ordered, and there's no way to actually construct a well-ordered notation for an uncountable set.

Actually, the problem is even worse. Consider the following question: "does the set of real numbers have the same size as the set of countable ordinals"?

The surprising answer is that there's no way to tell. It has been proven that this question cannot be answered with either a definite "yes" or a definite "no". This is getting into a really deep rabbit hole regarding the foundation of mathematics, but suffice to say that there's no way to use either assumptions (yes or no) in an actual construction.

So even if we obtained, miraculously, a mapping from real numbers to the ordinals, we would be unable to answer the most basic question of "how high does this notation go?". If you could show that such a mapping gets exactly up to ω₁, then you've also shown that there are ω₁ real numbers - which, as I've already stated, is an impossible task.