A set is uncountable if it contains so many elements that they cannot be put in one-to-one 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 one-to-one 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.
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:
(When your number is a \(9,\) you get \(0\) when adding a \(1\))
Further Examples of Uncountable Sets
As we have already seen for countable sets, the concept of countability and cardinality will be explained through examples: