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Infinity (∞) ○◂｜Definition｜1st｜20251119205401-00-⌔
Infinity
Infinity is something which is boundless, limitless, or endless. It is denoted by ∞, called the infinity symbol.
From the time of the ancient Greeks, the philosophical nature of infinity has been the subject of debate. In the 17th century, with the introduction of the infinity symbol1 and infinitesimal calculus, mathematicians began to work with infinite series and what some mathematicians (including l’Hôpital and Bernoulli)2 regarded as infinitely small quantities, but infinity continued to be associated with endless processes. As mathematicians struggled with the foundation of calculus, it remained unclear whether infinity could be considered as a number or magnitude and, if so, how this could be done.1 At the end of the 19th century, Georg Cantor enlarged the mathematical study of infinity by studying infinite sets and infinite numbers, showing that they can be of various sizes.13 For example, if a line is viewed as the set of all of its points, their infinite number (i.e., the cardinality of the line) is larger than the number of integers.4 In this usage, infinity is a mathematical concept, and infinite mathematical objects can be studied, manipulated, and used just like any other mathematical object.
The mathematical concept of infinity refines and extends the old philosophical concept, in particular by introducing infinitely many different sizes of infinite sets. Among the axioms of Zermelo–Fraenkel set theory, on which most of modern mathematics can be developed, is the axiom of infinity, which guarantees the existence of infinite sets.1 The mathematical concept of infinity and the manipulation of infinite sets are widely used in mathematics, even in areas such as combinatorics that may seem to have nothing to do with them.
In physics and cosmology, it is an open question whether the universe is spatially infinite or not.
Printed 2026-06-28.
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Link to original Footnotes
Allen, G Donald (2003). “3 The Emergence of Calculus”. The History of Infinity (PDF). Texas A&M University Department of Mathematics. p. 7. Archived from the original (PDF) on August 1, 2020. Retrieved Nov 15, 2019. ↩ ↩2 ↩3 ↩4
Jesseph, Douglas Michael (1998-05-01). “Leibniz on the Foundations of the Calculus: The Question of the Reality of Infinitesimal Magnitudes”. Perspectives on Science. 6 (1&2): 6–40. doi:10.1162/posc_a_00543. ISSN 1063-6145. OCLC 42413222. S2CID 118227996. Archived from the original on 11 January 2012. Retrieved 1 November 2019 – via Project MUSE. ↩
Gowers, Timothy; Barrow-Green, June (2008). The Princeton companion to mathematics. Imre Leader, Princeton University. Princeton: Princeton University Press. ISBN 978-1-4008-3039-8. OCLC 659590835. ↩
Maddox 2002, pp. 113–117 ↩
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