The set of real numbers is uncountable.

However, consider a reasonable human description of a number. Of course the description has to be finite for a human to begin to comprehend it. Anyway it seems pretty reasonable to say you could explain it in a combination of English and LaTeX. Which would imply, by base-257 encoding or whatever, that the set of real numbers that humans can describe specifically, even in principle, is countable.

Of course there seems to be a Russell's paradox lurking here... couldn't we specify "some undescribable number" with the axiom of choice? Well, we can certainly talk about "one undescribable number", but that phrase couldn't refer to any specific real number since... we can't describe it!

Or we might think of taking a well-ordering of the real numbers (whose existence follows from the axiom of choice) and the "first undescribable number" under it, but (I think) we cannot describe any specific well-ordering of the real numbers.

Of course this doesn't mean that "the set of describable numbers" is therefore mathematically well-defined as above; Gödel's incompleteness would probably come in and wreak havoc somewhere. But we can construct, and prove statements about, uncountably many more things than we can describe. In fact, by the same logic the number of "finitely describable" theorems is countable. And yet they can be results about uncountably many things.

Just something to think about.