Why is the set of all real numbers uncountable? I understand Cantor's diagonal argument, but it just doesn't seem to jive for some reason.
Lets say I assign the following numbers ad infinitum...


*

*$1\to 0.1$

*$2\to 0.2$

*$3\to 0.3$
...

*$10\to 0.10$

*$11\to 0.11$


and so on... How come there's supposedly at least one more real number than you can map to a member of $\mathbb{N}$?

How come there's supposedly at least one more real number than you can map to a member of $\mathbb N$?
Well, suppose there isn't - that Cantor's conclusion, his theorem, is wrong, because our enumeration covers all real numbers.  Wonderful, but let us see what happens when we take our enumeration and apply Cantor's diagonal technique to obtain a real number that can't be in this sequence.  But that contradicts our supposition!  Hence our supposition - that we can have an enumeration of all reals - is false.  That the argument can be applied to any enumeration is what it takes for Cantor's theorem to be true.
Wilfred Hodges wrote an excellent survey of wrong refutations of Catonr' argument, his An Editor Recalls Some Hopeless Papers (Postscript).  Section 7, dealing with how the counterfactual assumption confuses, might be of particular interest.
