| > And the main way we prove that the Reals are uncountable is to use a proof by contradiction. It would take too long to spell it out, but they aren't really just contradicting "Reals are Countable". It's "All that other stuff we think is true AND Reals are Countable" that gets contradicted. Btw that's not true. Cantor's diagonal argument isn't a proof by contradiction, it's a purely constructive argument. The way he made it in his original paper is a bit more technical than this but this is the "modern" version that's a bit easier to put into layman's terms. Say you say you can construct a (countably finite) list containing all the real numbers. I say I don't care how you made your list I can give you a number that's not on it, and in fact cut your list down to just numbers between 0 and 1. If you have all the real numbers you must have all the numbers between 0 and 1, but even if you make a countably infinite list of numbers between 0 and 1 I'll give you a procedure that will construct a number that's not on your list no matter how you made it. 1) Read the first number on your list. If it has 1 in the first decimal place, make the first decimal place of my number a two otherwise make it a 1.
2) Read the second number on your list. If it has 1 in the second decimal place, make the second decimal place of my number a two otherwise make it a 1.
.... Proceed in that manner. At the n-th step I read the n-th number on your list. If it has 1 in the n-th decimal place make the n-th decimal place of my number a 2 otherwise make it a 1. Now: My number is clearly nowhere on your list as it differs at in least one decimal place from every number on your list. Therefore it is not possible to construct a countably infinite list of real numbers. |