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 $\mathbb{N}$?

Cantor's diagonal is a trick to show that given any list of reals, a real can be found that is not in the list. 
First a few properties:


*

*You know that two numbers differ if just one digit differs. 

*If a number shares the previous property with every number in a set, it is not part of the set.


Cantor's diagonal is a clever solution to finding a number which satisfies these properties. The number which is the diagonal is transformed s.t. it doesn't share the first digit of the first number nor the second digit with the second and so on. Thus the number is unique to the list. 
This is why Cantor's diagonal as a method proves the result that the reals are uncountable.
