>The proof of uncountability of real numbers starts with a process of enumerating this infinite list of infinitely long binary numbers. Then says that we constr

>The proof of uncountability of real numbers starts with a process of enumerating this infinite list of infinitely long binary numbers. Then says that we construct a number that can't be on them by flipping bits the down the diagonal. I don't know what you mean by "process". Any list of numerals like this will be missing (at least) a numerals is what he says. Maybe let's take a step back. It needs a proof by contraposition. The "normal" position is "S -> T" (so if S then T), which also fixes the contraposition "(not T) -> (not S)". Cantor's idea is to have a list of (possibly infinite length) binary literals representing real numbers between 0 and 1 (and if those are uncountable, so are ALL the real numbers :P). The important part is that first consider the set of all such literals, whether countable or not. Let's call those T (for "total"). But then consider the subset of those that can be enumerated. Let's call those S (for "subset"). The countability means that you have a function that literally enumerates those (i.e. a function "natural number -> literal", 1:1) Enumerate the literal and enumerate the digit at some position in the literal. Let a_n be the nth literal. The a_n need to be unique (in order for those to be an enumeration). Let d(n, m) be the nth literal's digit at position n. Now construct a new literal q that differs from the first string in the first position, from the second string in the second position and so on. This q is NOT IN S. (it differs from any of the elements of S in at least one position each) Thereby, we proved that if S is countable, then it's not all of T. (because we found some literal that's not on the list; the list of all needs to be bigger!) So the "position" proof is: "Assume S is countable. Then show it's not all of T." But that, by contraposition, means, if it's all of T then it's not countable. Which is what we wanted.