does it ever strike you as weird that there are uncountably many real numbers, but only countably many can be named or precisely described in the english language in a finite amount of time?
obviously all rational numbers can be described in a finite amount of time
("53623562112541251251345146*10^2452/426257567257" is something you can say in a finite amount of time)
so can anything that can be built out of roots of integer polynomials ("algebraic numbers")
("square root of 7 plus 38 times the cube root of 17, divided by the seventh root of 2011")
but also there are lots of non-algebraic numbers that can be described in the english language in a finite amount of time
"the circumference of a circle divided by its diameter"
"euler's number"
"the limit of the series \sum 1/10^(n^3+n^n)" (probably, i don't actually know for sure that this one is non-algebraic, but it's definitely irrational)
"the fine structure constant" (i guess it's unknown whether this is rational or algebraic or anything)
but even with all this, we can name and describe so many numbers, not just all the (infinitely many) whole numbers, rational numbers, algebraic numbers, but so many more than that, and yet there's still way more that could never be named
like we can name all the stars, we can name every atom in the universe and infinitely more universes than that, but we can't name all the numbers between 0 and 1?
