The idea of infinity is powerful. The concepts of the infinity and the eternal have been a source of debate among philosophers, artists, intellectuals, scientists, and individuals from all walks of life throughout history.

The idea of infinity is very crucial in mathematics. Dealing with infinitely huge sets—collections of numbers that never end, such as the natural numbers: 1, 2, 3, 4, 5, and so on—leads to the practically immediate appearance of infinity.

However, not all infinite sets are created equal. In reality, there are various sizes or levels of infinity; some infinite sets are enormously larger than others.

Georg Cantor, a brilliant mathematician, developed the theory of infinite sets in the late 19th century. Many of Cantor's theories and propositions form the cornerstone of contemporary mathematics. One of Cantor's most interesting ideas was a method for comparing the dimensions of infinite sets and using this concept to demonstrate the existence of numerous infinities.

Some surprising facts that arise from this concept, is how the set of natural numbers, 1,2,3,4, and so on, is actually the same cardinality (or size) as the set of even numbers only, or odd numbers only. This happens as a result of forming a bijection (one-to-one correspondence) between the two sets. But then the set of all real numbers between zero and one, is a bigger size of infinity than the set of all natural numbers!

This can be proved by contradiction (a powerful technique in mathematics), and trying the diagonal method (a method to show that there is no one-to-one correspondence between the two sets, which means they can’t be the same “size”). No matter what alignment we attempt between the real numbers and the natural numbers, we are still able to create a number that does not appear in the correspondence.

The size of these two sets is different. The natural numbers and the real numbers are both infinite sets, but the real numbers constitute a set that is substantially larger than the naturals; they reflect some "higher level" of infinity. This leads to the deep and perhaps unsettling insight that there must be numerous levels of infinity. According to Cantor's theorem, there is always a set with a bigger cardinality for a given set.