| [edit: for clarity i'm not going to use the term "name"] the proof crucially relies upon each real number being represented as an infinitely long (perhaps ending in an infinite sequence of repeated digits for less interesting numbers) sequence of digits. i.e. the representation that Cantor uses for reals is infinite sequences -- effectively representing each real number infinitely long (in the countable sense) strings. Cantor represents each real number as an infinite sequence of digits. apart from technical details`+` you can think of this as the base 10 or base 2 expansion of the number. for simplicity, you need only consider counting the real numbers between 0 and 1. there are plenty enough of those. each such number can be addressed in base 2 as some infinite sequence of 1s and 0s after the binary point. Cantor's argument shows that if you try to enumerate all such numbers (i.e. count them) by their series expansion, then you can generate a new number that doesnt apppear in the enumerated list, which is a proof by contradiction, refuting the assumption that you could enumerate them all in the first place. `+` : technical details include that some numbers have non-unique representations as infinite sequences of digits. For example, in base 10, 1 can be represented as 1.000... (where the 0s keep going) or 0.999... (where the 9s keep going). if this irritates you, let z = 0.999... then 10 * z = 9.999...
then 10 * z - z = 9
then 9 * z = 9
so z = 1 |