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For other uses of "Infinity" and "Infinite", see Infinity (disambiguation).
The ∞ symbol in several typefaces



Main article: Infinity (philosophy)


Early Greek[edit]

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.[3] However, the earliest attestable accounts of mathematical infinity come from Zeno of Elea (c. 490 BCE? – c. 430 BCE?), a pre-Socratic Greek philosopher of southern Italy and member of the Eleatic School founded by Parmenides. Aristotle called him the inventor of the dialectic. He is best known for his paradoxes, described by Bertrand Russell as "immeasurably subtle and profound".


However, recent readings of the Archimedes Palimpsest have hinted that Archimedes at least had an intuition about actual infinite quantities.

Early Indian[edit]

The Indian mathematical text Surya Prajnapti (c. 3rd–4th century BCE) classifies all numbers into three sets: enumerable, innumerable, and infinite. Each of these was further subdivided into three orders:

  • Enumerable: lowest, intermediate, and highest
  • Innumerable: nearly innumerable, truly innumerable, and innumerably innumerable
  • Infinite: nearly infinite, truly infinite, infinitely infinite


17th century[edit]

European mathematicians started using infinite numbers in a systematic fashion in the 17th century. John Wallis first used the notation \infty for such a number, and exploited it in area calculations by dividing the region into infinitesimal strips of width on the order of \frac{1}{\infty}. Euler used the notation i for an infinite number, and exploited it by applying the binomial formula to the i'th power, and infinite products of i factors.


Infinity symbol[edit]

Main article: Infinity symbol


It was introduced in 1655 by John Wallis,[4][5] and, since its introduction, has also been used outside mathematics in modern mysticism[6] and literary symbology.[7]



Real analysis[edit]

In real analysis, the symbol \infty, called "infinity", is used to denote an unbounded limit.[10] x \rightarrow \infty means that x grows without bound, and x \to -\infty means the value of x is decreasing without bound. If f(t) ≥ 0 for every t, then[11]

  • \int_{a}^{b} \, f(t)\ dt \  = \infty means that f(t) does not bound a finite area from a to b
  • \int_{-\infty}^{\infty} \, f(t)\ dt \  = \infty means that the area under f(t) is infinite.
  • \int_{-\infty}^{\infty} \, f(t)\ dt \  = a means that the total area under f(t) is finite, and equals a

Infinity is also used to describe infinite series:

  • \sum_{i=0}^{\infty} \, f(i) = a means that the sum of the infinite series converges to some real value a.
  • \sum_{i=0}^{\infty}  \, f(i) = \infty means that the sum of the infinite series diverges in the specific sense that the partial sums grow without bound.

Infinity can be used not only to define a limit but as a value in the extended real number system. Points labeled +\infty and -\infty can be added to the topological space of the real numbers, producing the two-point compactification of the real numbers. Adding algebraic properties to this gives us the extended real numbers.[12] We can also treat +\infty and -\infty as the same, leading to the one-point compactification of the real numbers, which is the real projective line.[13] Projective geometry also refers to a line at infinity in plane geometry, a plane at infinity in three-dimensional space, and so forth for higher dimensions.

Complex analysis[edit]

By stereographic projection, the complex plane can be "wrapped" onto a sphere, with the top point of the sphere corresponding to infinity. This is the called the Riemann sphere.

In complex analysis the symbol \infty, called "infinity", denotes an unsigned infinite limit. x \rightarrow \infty means that the magnitude |x| of x grows beyond any assigned value. A point labeled \infty can be added to the complex plane as a topological space giving the one-point compactification of the complex plane. When this is done, the resulting space is a one-dimensional complex manifold, or Riemann surface, called the extended complex plane or the Riemann sphere. Arithmetic operations similar to those given above for the extended real numbers can also be defined, though there is no distinction in the signs (therefore one exception is that infinity cannot be added to itself). On the other hand, this kind of infinity enables division by zero, namely z/0 = \infty for any nonzero complex number z. In this context it is often useful to consider meromorphic functions as maps into the Riemann sphere taking the value of \infty at the poles. The domain of a complex-valued function may be extended to include the point at infinity as well. One important example of such functions is the group of Möbius transformations.

Nonstandard analysis[edit]


Set theory[edit]

Main articles: Cardinality and Ordinal number
One-to-one correspondence between infinite set and proper subset




Cardinality of the continuum[edit]

One of Cantor's most important results was that the cardinality of the continuum \mathbf c is greater than that of the natural numbers {\aleph_0}; that is, there are more real numbers R than natural numbers N. Namely, Cantor showed that \mathbf{c} = 2^{\aleph_0} > {\aleph_0} (see Cantor's diagonal argument or Cantor's first uncountability proof).

The continuum hypothesis states that there is no cardinal number between the cardinality of the reals and the cardinality of the natural numbers, that is, \mathbf{c} = \aleph_1 = \beth_1 (see Beth one). However, this hypothesis can neither be proved nor disproved within the widely accepted Zermelo–Fraenkel set theory, even assuming the Axiom of Choice.




Geometry and topology[edit]

Infinite-dimensional spaces are widely used in geometry and topology, particularly as classifying spaces, notably Eilenberg−MacLane spaces. Common examples are the infinite-dimensional complex projective space K(Z,2) and the infinite-dimensional real projective space K(Z/2Z,1).



Mathematics without infinity[edit]




Theoretical applications of physical infinity[edit]






Cosmologists have long sought to discover whether infinity exists in our physical universe: Are there an infinite number of stars? Does the universe have infinite volume? Does space "go on forever"? This is an open question of cosmology. Note that the question of being infinite is logically separate from the question of having boundaries. The two-dimensional surface of the Earth, for example, is finite, yet has no edge. By travelling in a straight line one will eventually return to the exact spot one started from. The universe, at least in principle, might have a similar topology. If so, one might eventually return to one's starting point after travelling in a straight line through the universe for long enough.

If, on the other hand, the universe were not curved like a sphere but had a flat topology, it could be both unbounded and infinite. The curvature of the universe can be measured through multipole moments in the spectrum of the cosmic background radiation. As to date, analysis of the radiation patterns recorded by the WMAP spacecraft hints that the universe has a flat topology. This would be consistent with an infinite physical universe.[citation needed]







Some programming languages, such as Java[20] and J,[21] allow the programmer an explicit access to the positive and negative infinity values as language constants. These can be used as greatest and least elements, as they compare (respectively) greater than or less than all other values. They are useful as sentinel values in algorithms involving sorting, searching, or windowing.

In languages that do not have greatest and least elements, but do allow overloading of relational operators, it is possible for a programmer to create the greatest and least elements. In languages that do not provide explicit access to such values from the initial state of the program, but do implement the floating point data type, the infinity values might still be accessible and usable as the result of certain operations.

Arts and cognitive sciences[edit]

Perspective artwork utilizes the concept of imaginary vanishing points, or points at infinity, located at an infinite distance from the observer. This allows artists to create paintings that realistically render space, distances, and forms.[22] Artist M. C. Escher is specifically known for employing the concept of infinity in his work in this and other ways.



See also[edit]


  2. ^ Maddox 2002, pp. 113 –117
  3. ^ Wallace 2004, pg. 44
  5. ^ Martin-Löf, Per (1990), "Mathematics of infinity", COLOG-88 (Tallinn, 1988), Lecture Notes in Computer Science 417, Berlin: Springer, pp. 146–197, doi:10.1007/3-540-52335-9_54, MR 1064143 .
  8. ^ Continuity and Infinitesimals entry by John Lane Bell in the Stanford Encyclopedia of Philosophy
  9. ^ Jesseph, Douglas Michael (1998). "Leibniz on the Foundations of the Calculus: The Question of the Reality of Infinitesimal Magnitudes". Perspectives on Science 6 (1&2): 6–40. ISSN 1063-6145. OCLC 42413222. Archived from the original on 16 February 2010. Retrieved 16 February 2010. 
  10. ^ Taylor 1955, p. 63
  13. ^ Gemignani 1990, p. 177
  19. ^ Cambridge Dictionary of Philosophy, Second Edition, p. 429


Further reading[edit]

  • L. C. Jain (1982). Exact Sciences from Jaina Sources. 
  • L. C. Jain (1973). "Set theory in the Jaina school of mathematics", Indian Journal of History of Science.
  • Navjyoti Singh (1988). Jaina Theory of Actual Infinity and Transfinite Numbers. Journal of Asiatic Society 30. 

External links[edit]