Types of Infinity

Infinity, cannot be treated as a number, but rather a concept in the sense that it is without any limit. Georg Cantor a famous Mathematician used sets to show how some infinities are larger than others. He did this by showing how a bijection cannot be shown therefore the cardinality (size of the set) must be different between them.

Given two sets A and B, a bijection is said to be true if:

1)There is an Injection between set A and set B (One to one mapping).
2)There is a Surjection between set A and set B (All elements of set A are mapped to an element of set B).

Therefore by proving a bijection between two sets we can conclude that they have the same cardinality.

For example:

Set A (1,2,3,....) and set B (2,4,6,....) can be proved to have the same cardinality. This is because using the function (f(x)=2x), we can create a bijection between set A and set B.

There is a one to one mapping because 2x only has one value.
All elements of set A are mapped to an element of set B because the elements of B are double that of A!

Therefore we have shown that the size of the set 1,2,3,4,5,..... is the same size as the set of 2,4,6,8,10,.....

These are known as countable infinities, because hypothetically speaking, the only thing that stops you from not being able to count all of the numbers 1,2,3,4,... is the time restraint. It is countable because if you counted forever, you could not have missed any numbers between 1 and n, which makes it listable/countable.

However you cannot list the uncountable infinity, because there is no order in which numbers can be arranged, that once you had counted forever, you would have missed no numbers. And for that reason this type of infinity is of different size to the cardinality of the set of natural numbers.