> consider the set of numbers that cannot be described using any finite collection of symbols. Such numbers must exist because there are only a countably infin

> consider the set of numbers that cannot be described using any finite collection of symbols. Such numbers must exist because there are only a countably infinite number of numbers that can be described using a finite collection of symbols, but there are an uncountably infinite number of real numbers. So not only are there numbers that cannot be described using a finite number of symbols, there are vastly more of these than numbers that can be so described. Maybe you already know, but this argument is wrong, see Hamkins "Pointwise definable models of set theory". It's a triple whammy: 1) You can't state the argument formally, because the property of describability is undefinable (Tarski's undefinability of truth). 2) Whatever your theory claims about uncountability, from the outside there's always a countable model of it (Lowenheim-Skolem). 3) And if you have a countable model, there's nothing stopping you from having a finite formula for every member of it (making it a "pointwise definable model"). All the details are worked out in the paper: yes Virginia, there are models of ZFC whose every member - every set, every function, every real number - is definable by a finite formula. Also a lot of nice counterexamples in this reddit thread: https://www.reddit.com/r/math/comments/d9sp9h/the_math_tea_argument_there_must_exist_real/ https://www.reddit.com/r/math/comments/d9sp9h/the_math_tea_a... For example: > "Take your favorite real in some model M. We can take a forcing extension to a model M[G] in which the real is coded in the continuum function below aleph_omega. Now it's definable." Or: > "There is a computable real that ZFC doesn't prove is a computable real."