User blog comment:Edwin Shade/Enumerating the Countable Ordinals/@comment-25337554-20171208234807/@comment-30754445-20171218112038

"which by the reasoning that a point on the numberline chosen at random would be far beyond any current notation..."

The problem with this idea, is that the vast majority of real numbers have no finite description. Unlike (say) pi or e or the square root of two, most real numbers cannot be defined by any finite string of symbols.

Remember, there are only Aleph-Null strings of symbols, yet there are more than Aleph-Null real numbers. So if you could somehow pick a "random" real number, you won't even be able to give that number a concise definition, let alone do anything constructive with it.

It's a really strange state of affairs. You can define the entire set of real numbers easily enough, but the vast majority of the members of this set are completely unaccessible.

And of-course, if we limit ourselves only to those real numbers whose definitions we can actually write down, then we're back to countable-land. Might as well use ordinary integers for this purpose, since there are equally many of them.

BTW it is possible to create a surjection from the natural numbers to the set of ordinals less than X, for any arbitrary large countable ordinal X. There's no limit to the countable ordinals you can generate in this way, but the catch is that you have to set X in advance. It can be as large as you like, but once you set it to a specific ordinal, that will be the limit of your system.

(and another catch is that if your X is greater or equal to   ω  CK  1      , then your notation won't be computable)