infinity

Introduction & Top Questions
Mathematical infinities
Physical infinities
Metaphysical infinities

References & Edit History Related Topics

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Understand the use of integers to count infinity

At a Glance

infinity summary

Related Questions

How are derivatives related to the slope of a curve at one point?
How can a limit replace infinitesimals when defining a derivative?
How can all fractions be arranged in a list with the natural numbers?
How can an infinite set have the same size as one of its proper subsets?
How can the counting numbers be paired one-to-one with the even numbers?
How can the natural numbers be paired one-to-one with the even numbers?
How can two circles with different circumferences be matched point for point?
How can two circles with different circumferences have the same number of points?
How did Abraham Robinson show that infinitesimals could be used consistently?
How did Archimedes use the method of exhaustion to calculate areas?
How did Archimedes use the method of exhaustion without assuming an actual infinity?
How did Aristotle distinguish potential infinity from actual infinity?
How did Cantor’s ideas contribute to the development of modern set theory?
How did Eudoxus use the method of exhaustion to calculate areas?
How did Eudoxus use the method of exhaustion to work with infinite processes?
How did Galileo Galilei pair counting numbers with their squares?
How did Galileo's paradox compare counting numbers with their squares?
How did Georg Cantor demonstrate that some infinities are larger than others?
How did Georg Cantor develop set theory?
How did Georg Cantor's diagonal argument show that real numbers cannot be listed?
How did Georg Cantor's theory change mathematicians' understanding of infinity?
How did Georg Cantor’s work on trigonometric series lead him to study sets?
How did Gottfried Wilhelm Leibniz use infinitesimals in calculus?
How did Karl Weierstrass give a rigorous definition of a limit?
How did Paul Cohen contribute to the study of the continuum hypothesis?
How did Richard Dedekind define an infinite set?
How did Richard Dedekind define an infinite set using one-to-one correspondence?
How did Russell’s paradox reveal problems with unrestricted set formation?
How did the ancient Greeks use the word apeiron when discussing infinity?
How do Zeno’s paradoxes connect endless division with the problem of motion?
How do Zeno's paradoxes involve infinitely many divisions of distance?
How does Cantor’s diagonal argument show that real numbers cannot be completely listed?
How does Cantor's theorem compare a set with its power set?
How does Cantor's theorem show that there are infinitely many different sizes of infinity?
How does a one-to-one correspondence compare the sizes of infinite sets?
How does cardinal arithmetic differ from ordinary arithmetic for infinite numbers?
How does one-to-one correspondence compare the sizes of two sets?
How does the cardinality of the real numbers compare with aleph-null?
How does the continuum hypothesis relate the real numbers to aleph-one?
How does the diagonal of a square with side length 1 lead to the irrational number √2?
How is aleph-one related to aleph-null?
How is infinity used in mathematics?
How many subsets does a finite set with n elements have?
What are transfinite cardinal numbers?
What contradiction is produced by the idea of a set containing every set?
What did Aristotle mean by potential infinity?
What does ZFC stand for in set theory?
What does aleph-null represent in set theory?
What does aleph-null ℵ0 represent in set theory?
What does the continuum hypothesis claim about the real numbers?
What does the continuum represent in Cantor's theory of infinite sets?
Who invented the infinity symbol ∞?
Why are the rational numbers considered countable?
Why are the real numbers uncountable?
Why are transcendental numbers more numerous than integers even though both collections are infinite?
Why did Leopold Kronecker oppose Georg Cantor's ideas about infinite numbers?
Why did Leopold Kronecker oppose Georg Cantor’s theories about infinite numbers?
Why did Zeno argue that a moving object must first reach infinitely many halfway points?
Why did mathematicians develop Zermelo-Fraenkel set theory with the axiom of choice?
Why did the ratio between a square’s diagonal and side challenge the followers of Pythagoras?
Why do the natural numbers and even numbers both have cardinality aleph-null?
Why do the natural numbers and integers have the same cardinality?
Why does Galileo’s pairing of counting numbers with even numbers seem surprising?
Why does a set with n elements have a power set containing 2^n subsets?
Why does aleph-null multiplied by aleph-null equal aleph-null?
Why does aleph-null plus aleph-null equal aleph-null?
Why does every power set have more elements than its original set?
Why does the decimal expansion of √2 never end or repeat?
Why does the sequence 1, 1/2, 1/3, 1/4 approach zero?
Why does transfinite cardinal arithmetic differ from ordinary arithmetic?
Why does ℵ0 plus ℵ0 equal ℵ0?
Why does ℵ0 times ℵ0 equal ℵ0?
Why is the cardinality of the real numbers larger than aleph-null?
Why is the diagonal of a unit square equal to the irrational number √2?
Why is the power set of an infinite set larger than the original set?
Why were early mathematicians critical of infinitesimals?
What is the value of pi?
What are the uses of pi?

Science Mathematics

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External Websites

Stanford Encyclopedia of Philosophy - Infinity
European Mathematical Information Service - Infinity of Numbers: How Students understand it (PDF)
Mathematics LibreTexts - How Big is Infinity? Or Is it Â“InfinitiesÂ”?
Internet Encyclopedia of Philosophy - The Infinite
University of Montana - ScholarWorks - The Constructs of PhD Students about Infinity: An Application of Repertory Grids (PDF)

infinity

mathematics

Written by

Rudy Rucker

Professor of Mathematics and Computer Science, San Jose State University, California (1986–2004). Author of Geometry, Relativity and the Fourth Dimension; The Fourth Dimension; Mind Tools; Freeware;...

Rudy Rucker

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Britannica Editors

Encyclopaedia Britannica's editors oversee subject areas in which they have extensive knowledge, whether from years of experience gained by working on that content or via study for an advanced degree....

Britannica Editors

Last updated

Nov. 8, 2026 •History

infinity, the concept of something that is unlimited, endless, without bound. The common symbol for infinity, ∞, was invented by the English mathematician John Wallis in 1655. Three main types of infinity may be distinguished: the mathematical, the physical, and the metaphysical. Mathematical infinities occur, for instance, as the number of points on a continuous line or as the size of the endless sequence of counting numbers: 1, 2, 3,…. Spatial and temporal concepts of...

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