The term “infinity” does not itself have a settled mathematical definition, but instead refers to an intuitive notion of being boundless. The word, Latin in origin (infinitas, “without finish”), corresponded originally to the Greek word \(\alpha\pi\varepsilon\iota\rho\omicron\nu\) (apeiron), meaning without form or limit.

The Greek philosopher Aristotle argued that nothing that actually exists could be limitless, because to exist means to have a particular form, and form implies limits. On his authority mathematicians refused to treat infinity as meaningful until late in the 19th century, when the invention of set theory by Georg Cantor introduced infinite collections with well-defined properties.

In modern mathematics the notion corresponds to infinite sets, including countably infinite and uncountable sets. In certain non-standard conceptions of the real number line there is a notion of infinity as a reciprocal of an infinitessimal, a quantity smaller than any definable real number. It may also be used less formally to indicate a procedure that may be carried out indefinitely, as in the “infinite divisibility” of a continuous line segment into smaller segments. In projective geometry the terms “point at infinity,” “line at infinity,” etc., are used to refer to certain formally defined objects. Finally, in formal number theory and mathematical logic there is a well-defined inference rule called “transfinite (infinite) induction.”

Citation Info

  • [MLA] “infinity.” Platonic Realms Interactive Mathematics Encyclopedia. Platonic Realms, 10 Apr 2014. Web. 28 Feb 2017. <>
  • [APA] infinity (10 Apr 2014). Retrieved 28 Feb 2017 from the Platonic Realms Interactive Mathematics Encyclopedia:


Get the ultimate math study-guide Math & Me: Embracing Successproduct thumbnail image Available in the Math Store
detail from Escher pic Belvedere

Are you a mathematical artist?

Platonic Realms is preparing an online gallery space to showcase and market the works of painters, sculptors, and other artists working in a tangible medium.

If your work celebrates mathematical themes we want to hear from you!

Please let us know about yourself using the contact page.