> But there is a countable list that DOES include every possible real - you just can't always figure out whether things on that list are reals! A (hopefully) h

> But there is a countable list that DOES include every possible real - you just can't always figure out whether things on that list are reals! A (hopefully) helpful way to look at it: Cantor's diagonalization argument doesn't work if every function from natural numbers to reals, that includes all reals, is partial. Which, per the halting problem[0], is exactly the behaviour you get if your correspondance function involves doing arbitrary computations decoded from the input natural. 0: The class of propositions "Turing machine #N halts.", in addition to true and false propositions, also contains infinitely many propositions that are neither true nor false[1], so compacting out the non-halting naturals doesn't help. 1: aka infinitely many counterexamples to the axiom of excluded middle