User blog comment:Edwin Shade/Enumerating the Countable Ordinals/@comment-25337554-20171208234807/@comment-5529393-20171209024304

There do exists well-orderings of uncountable sets; for example, all uncountable ordinals are well-ordered by inclusion. Furthermore, in ZFC all sets are well-orderable. (In ZF the statement "all sets are well-orderable" is equivalent to the axiom of choice.)  Even more furthermore, if we assume the axiom V=L, which means that all sets in the universe are constructible, then we can define an explicit well-ordering of the universe. This will in turn define an explicit well-ordering on each set in the universe.

On the other hand, if the axiom of choice is false, then there exist non-well-orderable sets. And even in ZFC, you can't give a formula that provably defines a well-ordering of the reals; it is consistent that such a formula exists, as with V=L above, but it is also consistent that no such formula exists.

Concerning your previous paragraph, as Wojowu said it is indeed possible to assign a countable ordinal to each real number such that every countable ordinal is assigned a real number. The problem is, we can't really pick a random real number; you can say, "if we randomly pick a real number using this particular distribution, then we may get a very large countable ordinal" but to pick an explicit countable ordinal using this method, we have to specify an explicit real number, meaning we would have to define it explicitly. Which doesn't seem to be much different from defining a countable ordinal explicitly, so I don't think this is really what you want.

Here's a conundrum for you: Say we were to define a probability distribution on the countable ordinals, such that any countable subset has probability 0. Say we pick to ordinals, a and b. Which is bigger? Well, there are only countably many ordinals less than or equal to a, so with probability 1 b is bigger. On the other hand, there are only countably many ordinals less than or equal to b, so with probability 1 a is bigger. Paradox! (This is the basic paradox behind Freiling's argument against the continuum hypothesis.)