Sets of Infinity, deeply explored by the mathematician, George Cantor. Georg Cantor (1845-1918) is considered the pioneer of sets of infinities: ###### Set theory Cantor created modern set theory and established the importance of one-to-one correspondence between sets. **Infinite sets** Cantor showed that there are different sizes of infinite sets, and that some are larger than others. **Real numbers** Cantor proved that the set of real numbers is larger than the set of natural numbers. He concluded that the real numbers are uncountably infinite. ###### Continuum hypothesis Cantor conjectured that there is no intermediate size of infinity between the countable natural numbers and the uncountable real numbers. This conjecture is now known as the continuum hypothesis. Cantor's work transformed mathematicians' ideas of infinity. His theory of infinite sets was later formalized into Zermelo–Fraenkel set theory, which is now commonly accepted as a foundation of mathematics. **Infinity Sets** In set theory, an infinite set is a set that is not finite. There are multiple types of infinite sets, including countable and uncountable sets: **Countably infinite** A set that has the same size as the set of natural numbers. For example, the set of even numbers and the set of rational numbers are countably infinite. **Uncountably infinite** A set that is larger than the set of natural numbers. For example, the set of real numbers is uncountably infinite. The size of a set is called its cardinality. For finite sets, the cardinality is calculated by counting the elements. For infinite sets, the size can be compared using a one-to-one correspondence. The axiom of infinity in Zermelo–Fraenkel set theory states that infinite sets exist.