Introduction
A set is uncountable if it contains so many elements that they cannot be put in onetoone correspondence with the set of natural numbers. In other words, there is no way that one can count off all elements in the set in such a way that, even though the counting will take forever, you will get to any particular element in a finite amount of time.
Uncountable is in contrast to countably infinite or countable.
The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal number is larger than that of the set of all natural numbers.
For example, the set of real numbers between 0 and 1 is an uncountable set because no matter what, you'll always have at least one number that is not included in the set. This set does not have a onetoone correspondence with the set of natural numbers. The proof of this involves creating an infinite list of numbers between 0 and 1 such as this.
\(0.12348…\)

\(0.34897…\)

\(0.98789…\)

\(0.43238…\)

\(0.55349...\)

No matter what kind of list you create, there will always be a number that is not in the list. This is found by using Cantor's diagonal argument, where you create a new number by taking the diagonal components of the list and adding 1 to each. So, you take the first place after the decimal in the first number and add one to it. You get \(1 + 1 = 2.\) Then you take the second place after the decimal in the second number and add 1 to it \((4 + 1 = 5).\) And so on to get your new number:
\(0.25840...\)
(When your number is a \(9,\) you get \(0\) when adding a \(1\))