Georg Cantor formalized many ideas related to infinity and infinite sets during the late 19th and early 20th centuries. Infinity cultures had various ideas about the nature of infinity.

The ancient Indians and Greeks did not define infinity in precise formalism as does modern mathematics, and instead approached infinity as a philosophical concept. The earliest recorded idea of infinity comes from Anaximander, a pre-Socratic Greek philosopher who lived in Miletus. He used the word apeiron which means infinite or limitless. Socratic Greek philosopher of southern Italy and member of the Eleatic School founded by Parmenides. Aristotle called him the inventor of the dialectic.

However, recent readings of the Archimedes Palimpsest have found that Archimedes had an understanding about actual infinite quantities. According to Nonlinear Dynamic Systems and Controls, Archimedes has been found to be “the first to rigorously address the science of infinity with infinitely large sets using precise mathematical proofs. In this work, two basic types of infinite numbers are distinguished. European mathematicians started using infinite numbers and expressions in a systematic fashion in the 17th century. In 1699 Isaac Newton wrote about equations with an infinite number of terms in his work De analysi per aequationes numero terminorum infinitas. Mathematics is the science of the infinite. It was introduced in 1655 by John Wallis, and, since its introduction, has also been used outside mathematics in modern mysticism and literary symbology.

Leibniz, one of the co-inventors of infinitesimal calculus, speculated widely about infinite numbers and their use in mathematics. Infinity can be used not only to define a limit but as a value in the extended real number system. Adding algebraic properties to this gives us the extended real numbers. By stereographic projection, the complex plane can be “wrapped” onto a sphere, with the top point of the sphere corresponding to infinity. This is called the Riemann sphere.