# Infinity

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Mathematical concept

For the symbol, see [Infinity symbol](/source/Infinity_symbol). For other uses of "Infinity" and "Infinite", see [Infinity (disambiguation)](/source/Infinity_(disambiguation)). Not to be confused with [Infiniti](/source/Infiniti).

The [Sierpiński triangle](/source/Sierpi%C5%84ski_triangle) contains infinitely many (scaled-down) copies of itself.

**Infinity** is something which is boundless, limitless, or endless. It is denoted by ∞, called the [infinity symbol](/source/Infinity_symbol).

From the time of the [ancient Greeks](/source/Ancient_Greek_mathematics), the [philosophical nature of infinity](/source/Infinity_(philosophy)) has been the subject of debate. In the 17th century, with the introduction of the infinity symbol[1] and [infinitesimal calculus](/source/Infinitesimal_calculus), mathematicians began to work with [infinite series](/source/Infinite_series) and what some mathematicians (including [l'Hôpital](/source/Guillaume_de_l'H%C3%B4pital) and [Bernoulli](/source/Johann_Bernoulli))[2] regarded as infinitely small quantities, but infinity continued to be associated with endless processes. As mathematicians struggled with the foundation of [calculus](/source/Calculus), it remained unclear whether infinity could be considered as a number or [magnitude](/source/Magnitude_(mathematics)) and, if so, how this could be done.[1] At the end of the 19th century, [Georg Cantor](/source/Georg_Cantor) enlarged the mathematical study of infinity by studying [infinite sets](/source/Infinite_set) and [infinite numbers](/source/Transfinite_number), showing that they can be of various sizes.[1][3] For example, if a line is viewed as the set of all of its points, their infinite number (i.e., the [cardinality](/source/Cardinality) of the line) is larger than the number of [integers](/source/Integer).[4] In this usage, infinity is a mathematical concept, and infinite [mathematical objects](/source/Mathematical_object) 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](/source/Zermelo%E2%80%93Fraenkel_set_theory), on which most of modern mathematics can be developed, is the [axiom of infinity](/source/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](/source/Combinatorics) that may seem to have nothing to do with them.

In [physics](/source/Physics) and [cosmology](/source/Cosmology), it is an open question [whether the universe is spatially infinite or not](/source/Universe#Size_and_regions).

## History

Further information: [Infinity (philosophy)](/source/Infinity_(philosophy))

Ancient cultures had various ideas about the nature of infinity. The [ancient Indians](/source/Vedic_period) and the [Greeks](/source/Ancient_Greece) did not define infinity in precise formalism as does modern mathematics, and instead approached infinity as a philosophical concept.

### Early Greek

The earliest recorded idea of infinity in Greece may be that of [Anaximander](/source/Anaximander) (c. 610 – c. 546 BC) a [pre-Socratic](/source/Pre-Socratic_philosophy) Greek philosopher. He used the word *[apeiron](/source/Apeiron)*, which means "unbounded", "indefinite", and perhaps can be translated as "infinite".[1][5]

[Aristotle](/source/Aristotle) (350 BC) distinguished *potential infinity* from *[actual infinity](/source/Actual_infinity)*, which he regarded as impossible due to the various [paradoxes](/source/Paradoxes) it seemed to produce.[6] It has been argued that, in line with this view, the [Hellenistic](/source/Hellenistic) Greeks had a "horror of the infinite"[7][8] which would, for example, explain why [Euclid](/source/Euclid) (c. 300 BC) did not say that there are an infinity of primes but rather "Prime numbers are more than any assigned multitude of prime numbers."[9] It has also been maintained, that, in proving the [infinitude of the prime numbers](/source/Infinitude_of_the_prime_numbers), Euclid "was the first to overcome the horror of the infinite".[10] There is a similar controversy concerning Euclid's [parallel postulate](/source/Parallel_postulate), sometimes translated:

If a straight line falling across two [other] straight lines makes internal angles on the same side [of itself whose sum is] less than two right angles, then the two [other] straight lines, being produced to infinity, meet on that side [of the original straight line] that the [sum of the internal angles] is less than two right angles.[11]

Other translators, however, prefer the translation "the two straight lines, if produced indefinitely ...",[12] thus avoiding the implication that Euclid was comfortable with the notion of infinity. Finally, it has been maintained that a reflection on infinity, far from eliciting a "horror of the infinite", underlay all of early Greek philosophy and that Aristotle's "potential infinity" is an aberration from the general trend of this period.[13]

### Zeno: Achilles and the tortoise

Main article: [Zeno's paradoxes § Achilles and the tortoise](/source/Zeno's_paradoxes#Achilles_and_the_tortoise)

[Zeno of Elea](/source/Zeno_of_Elea) (c. 495 – c. 430 BC) did not advance any views concerning the infinite. Nevertheless, his paradoxes,[14] especially "Achilles and the Tortoise", were important contributions in that they made clear the inadequacy of popular conceptions. The paradoxes were described by [Bertrand Russell](/source/Bertrand_Russell) as "immeasurably subtle and profound".[15]

[Achilles](/source/Achilles) races a tortoise, giving the latter a head start.

- Step #1: Achilles runs to the tortoise's starting point while the tortoise walks forward.

- Step #2: Achilles advances to where the tortoise was at the end of Step #1 while the tortoise goes yet farther.

- Step #3: Achilles advances to where the tortoise was at the end of Step #2 while the tortoise goes yet farther.

- Step #4: Achilles advances to where the tortoise was at the end of Step #3 while the tortoise goes yet farther.

And this continues infinitely. Basically,

- Step #(n+1): Achilles advances to where the tortoise was at the end of Step #n while the tortoise goes yet farther.

Apparently, Achilles never overtakes the tortoise, since however many steps he completes, the tortoise remains ahead of him.

Zeno was not attempting to make a point about infinity. As a member of the [Eleatics](/source/Eleatic) school which regarded motion as an illusion, he saw it as a mistake to suppose that Achilles could run at all. Subsequent thinkers, finding this solution unacceptable, struggled for over two millennia to find other weaknesses in the argument.

Finally, in 1821, [Augustin-Louis Cauchy](/source/Augustin-Louis_Cauchy) provided both a satisfactory definition of a limit and a proof that, for 0 < *x* < 1,[16] a + a x + a x 2 + a x 3 + a x 4 + a x 5 + ⋯ = a 1 − x . {\displaystyle a+ax+ax^{2}+ax^{3}+ax^{4}+ax^{5}+\cdots ={\frac {a}{1-x}}.}

Suppose that Achilles is running at 10 meters per second, the tortoise is walking at 0.1 meters per second, and the latter has a 100-meter head start. The duration of the chase fits Cauchy's pattern with *a* = 10 seconds and *x* = 0.01. Achilles does overtake the tortoise; it takes him

- 10 + 0.1 + 0.001 + 0.00001 + ⋯ {\displaystyle 10+0.1+0.001+0.00001+\cdots } = 10 1 − 0.01 = 10 0.99 = 10.10101 … seconds . {\displaystyle ={\frac {10}{1-0.01}}={\frac {10}{0.99}}=10.10101\ldots {\text{ seconds}}.}

### Early Indian

The [Jain mathematical](/source/Indian_mathematics#Jain_mathematics_(400_BCE_–_200_CE)) text *[Surya Prajnapti](https://en.wikipedia.org/w/index.php?title=Surya_Prajnapti&action=edit&redlink=1)* (c. 4th–3rd century BCE) classifies all numbers into three sets: [enumerable](/source/Enumerable), innumerable, and infinite. Each of these was further subdivided into three orders:[17]

- Enumerable: lowest, intermediate, and highest

- Innumerable: nearly innumerable, truly innumerable, and innumerably innumerable

- Infinite: nearly infinite, truly infinite, infinitely infinite

### 17th century

In the 17th century, European mathematicians started using infinite numbers and infinite expressions in a systematic fashion. In 1655, [John Wallis](/source/John_Wallis) first used the notation ∞ {\displaystyle \infty } for such a number in his *De sectionibus conicis*,[18] and exploited it in area calculations by dividing the region into [infinitesimal](/source/Infinitesimal) strips of width on the order of 1 ∞ . {\displaystyle {\tfrac {1}{\infty }}.} [19] But in *Arithmetica infinitorum* (1656),[20] he indicates infinite series, infinite products and infinite continued fractions by writing down a few terms or factors and then appending "&c.", as in "1, 6, 12, 18, 24, &c."[21]

In 1699, [Isaac Newton](/source/Isaac_Newton) wrote about equations with an infinite number of terms in his work *[De analysi per aequationes numero terminorum infinitas](/source/De_analysi_per_aequationes_numero_terminorum_infinitas)*.[22]

## Symbol

Main article: [Infinity symbol](/source/Infinity_symbol)

The infinity symbol ∞ {\displaystyle \infty } (sometimes called the [lemniscate](/source/Lemniscate)) is a mathematical symbol representing the concept of infinity. The symbol is encoded in [Unicode](/source/Unicode) at U+221E ∞ INFINITY (&infin;)[23] and in [LaTeX](/source/LaTeX) as \infty.[24]

It was introduced in 1655 by [John Wallis](/source/John_Wallis),[25][26] and since its introduction, it has also been used outside mathematics in modern mysticism[27] and literary [symbology](/source/Symbology).[28]

## Calculus

[Gottfried Leibniz](/source/Gottfried_Wilhelm_Leibniz), one of the co-inventors of [infinitesimal calculus](/source/Infinitesimal_calculus), speculated widely about infinite numbers and their use in mathematics. To Leibniz, both infinitesimals and infinite quantities were ideal entities, not of the same nature as appreciable quantities, but enjoying the same properties in accordance with the [Law of continuity](/source/Law_of_continuity).[29][2]

### Real analysis

In [real analysis](/source/Real_analysis), the symbol ∞ {\displaystyle \infty } , called "infinity", is used to denote an unbounded [limit](/source/Limit_of_a_function).[30] It is not a real number itself. The notation x → ∞ {\displaystyle x\rightarrow \infty } means that *x {\displaystyle x}* increases without bound, and x → − ∞ {\displaystyle x\to -\infty } means that *x {\displaystyle x}* decreases without bound. For example, if f ( t ) ≥ 0 {\displaystyle f(t)\geq 0} for every *t {\displaystyle t}*, then[31]

- ∫ a b f ( t ) d t = ∞ {\displaystyle \int _{a}^{b}f(t)\,dt=\infty } means that f ( t ) {\displaystyle f(t)} does not bound a finite area from a {\displaystyle a} to b . {\displaystyle b.}

- ∫ − ∞ ∞ f ( t ) d t = ∞ {\displaystyle \int _{-\infty }^{\infty }f(t)\,dt=\infty } means that the area under f ( t ) {\displaystyle f(t)} is infinite.

- ∫ − ∞ ∞ f ( t ) d t = a {\displaystyle \int _{-\infty }^{\infty }f(t)\,dt=a} means that the total area under f ( t ) {\displaystyle f(t)} is finite, and is equal to a . {\displaystyle a.}

Infinity can also be used to describe [infinite series](/source/Infinite_series), as follows:

- ∑ i = 0 ∞ f ( i ) = a {\displaystyle \sum _{i=0}^{\infty }f(i)=a} means that the sum of the infinite series [converges](/source/Convergent_series) to some real value a . {\displaystyle a.}

- ∑ i = 0 ∞ f ( i ) = ∞ {\displaystyle \sum _{i=0}^{\infty }f(i)=\infty } means that the sum of the infinite series properly [diverges](/source/Divergent_series) to infinity, in the sense that the partial sums increase without bound.[32]

In addition to defining a limit, infinity can be also used as a value in the extended real number system. Points labeled + ∞ {\displaystyle +\infty } and − ∞ {\displaystyle -\infty } can be added to the [topological space](/source/Topological_space) of the real numbers, producing the two-point [compactification](/source/Compactification_(mathematics)) of the real numbers. Adding algebraic properties to this gives us the [extended real numbers](/source/Extended_real_number).[33] We can also treat + ∞ {\displaystyle +\infty } and − ∞ {\displaystyle -\infty } as the same, leading to the [one-point compactification](/source/One-point_compactification) of the real numbers, which is the [real projective line](/source/Real_projective_line).[34] [Projective geometry](/source/Projective_geometry) also refers to a [line at infinity](/source/Line_at_infinity) in plane geometry, a [plane at infinity](/source/Plane_at_infinity) in three-dimensional space, and a [hyperplane at infinity](/source/Hyperplane_at_infinity) for general [dimensions](/source/Dimension_(mathematics_and_physics)), each consisting of [points at infinity](/source/Point_at_infinity).[35]

### Complex analysis

By [stereographic projection](/source/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](/source/Riemann_sphere).

In [complex analysis](/source/Complex_analysis) the symbol ∞ {\displaystyle \infty } , called "infinity", denotes an unsigned infinite [limit](/source/Limit_(mathematics)). The expression x → ∞ {\displaystyle x\rightarrow \infty } means that the magnitude | x | {\displaystyle |x|} of *x {\displaystyle x}* grows beyond any assigned value. A [point labeled ∞ {\displaystyle \infty }](/source/Point_at_infinity) can be added to the complex plane as a [topological space](/source/Topological_space) giving the [one-point compactification](/source/One-point_compactification) of the complex plane. When this is done, the resulting space is a one-dimensional [complex manifold](/source/Complex_manifold), or [Riemann surface](/source/Riemann_surface), called the extended complex plane or the [Riemann sphere](/source/Riemann_sphere).[36] Arithmetic operations similar to those given above for the extended real numbers can also be defined, though there is no distinction in the signs (which leads to the one exception that infinity cannot be added to itself). On the other hand, this kind of infinity enables [division by zero](/source/Division_by_zero), namely z / 0 = ∞ {\displaystyle z/0=\infty } for any nonzero [complex number](/source/Complex_number) *z {\displaystyle z}*. In this context, it is often useful to consider [meromorphic functions](/source/Meromorphic_function) as maps into the Riemann sphere taking the value of ∞ {\displaystyle \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](/source/M%C3%B6bius_transformation) (see [Möbius transformation § Overview](/source/M%C3%B6bius_transformation#Overview)).

### Nonstandard analysis

Infinitesimals (ε) and infinities (ω) on the hyperreal number line (1/ε = ω/1)

The original formulation of [infinitesimal calculus](/source/Infinitesimal_calculus) by [Isaac Newton](/source/Isaac_Newton) and Gottfried Leibniz used infinitesimal quantities. In the second half of the 20th century, it was shown that this treatment could be put on a rigorous footing through various [logical systems](/source/Logical_system), including [smooth infinitesimal analysis](/source/Smooth_infinitesimal_analysis) and [nonstandard analysis](/source/Nonstandard_analysis). In the latter, infinitesimals are invertible, and their inverses are infinite numbers. The infinities in this sense are part of a [hyperreal field](/source/Hyperreal_number); there is no equivalence between them as with the Cantorian [transfinites](/source/Transfinite_number). For example, if H is an infinite number in this sense, then H + H = 2H and H + 1 are distinct infinite numbers. This approach to [non-standard calculus](/source/Non-standard_calculus) is fully developed in [Keisler (1986)](#CITEREFKeisler1986).

## Set theory

Main articles: [Cardinality](/source/Cardinality) and [Ordinal number](/source/Ordinal_number)

One-to-one correspondence between an infinite set and its proper subset

A different form of "infinity" is the [ordinal](/source/Ordinal_number) and [cardinal](/source/Cardinal_number) infinities of set theory—a system of [transfinite numbers](/source/Transfinite_number) first developed by [Georg Cantor](/source/Georg_Cantor). In this system, the first transfinite cardinal is [aleph-null](/source/Aleph-null) (ℵ0), the cardinality of the set of [natural numbers](/source/Natural_number). This modern mathematical conception of the quantitative infinite developed in the late 19th century from works by Cantor, [Gottlob Frege](/source/Gottlob_Frege), [Richard Dedekind](/source/Richard_Dedekind) and others—using the idea of collections or sets.[1]

Dedekind's approach was essentially to adopt the idea of [one-to-one correspondence](/source/One-to-one_correspondence) as a standard for comparing the size of sets, and to reject the view of [Galileo](/source/Galileo_Galilei) (derived from [Euclid](/source/Euclid)) that the whole cannot be the same size as the part. (However, see [Galileo's paradox](/source/Galileo's_paradox) where Galileo concludes that positive integers cannot be compared to the subset of positive [square integers](/source/Square_number) since both are infinite sets.) An infinite set can simply be defined as one having the same size as at least one of its [proper subsets](/source/Proper_subset) (a strictly smaller subset); this notion of infinity is called [Dedekind infinite](/source/Dedekind_infinite). The diagram to the right gives an example: viewing lines as infinite sets of points, the left half of the lower blue line can be mapped in a one-to-one manner (green correspondences) to the higher blue line, and, in turn, to the whole lower blue line (red correspondences); therefore the whole lower blue line and its left half have the same cardinality, i.e. "size".[37]

Cantor defined two kinds of infinite numbers: [ordinal numbers](/source/Ordinal_number) and [cardinal numbers](/source/Cardinal_number). Ordinal numbers characterize [well-ordered](/source/Well-ordered) sets, or counting carried on to any stopping point, including points after an infinite number have already been counted. Generalizing finite and (ordinary) infinite [sequences](/source/Sequence) which are maps from the positive [integers](/source/Integers) leads to [mappings](/source/Function_(mathematics)) from ordinal numbers to transfinite sequences. Cardinal numbers define the size of sets, meaning how many members they contain, and can be standardized by choosing the first ordinal number of a certain size to represent the cardinal number of that size. The smallest ordinal infinity is that of the positive integers, and any set which has the cardinality of the integers is [countably infinite](/source/Countable_set).

If a set is too large to be put in one-to-one correspondence with the positive integers, it is called *[uncountable](/source/Uncountable_set)*. [Cantor's theorem](/source/Cantor's_theorem) further proved that no largest cardinal exists, meaning that for every infinite set, there exists a larger set. Cantor proclaimed that the collection of all sets had an unincreasable [absolute infinity](/source/Absolute_infinite), which he identified with God.[38][39] Although [initially controversial](/source/Controversy_over_Cantor's_theory), Cantor's views prevailed and modern mathematics accepts actual infinity as part of a consistent and coherent theory.[40]

The theory of ordinal and cardinal numbers has been further developed since Cantor, with [large countable ordinals](/source/Large_countable_ordinal) and [large cardinals](/source/Large_cardinal) currently being studied in [mathematical logic](/source/Mathematical_logic).[41]

### Cardinality of the continuum

Main article: [Cardinality of the continuum](/source/Cardinality_of_the_continuum)

One of Cantor's most important results was that the cardinality of the continuum c {\displaystyle \mathbf {c} } is greater than that of the natural numbers ℵ 0 {\displaystyle {\aleph _{0}}} ; that is, there are more real numbers **R** than natural numbers **N**. Namely, Cantor showed that c = 2 ℵ 0 > ℵ 0 {\displaystyle \mathbf {c} =2^{\aleph _{0}}>{\aleph _{0}}} .[42]

Further information: [Cantor's diagonal argument](/source/Cantor's_diagonal_argument) and [Cantor's first set theory article](/source/Cantor's_first_set_theory_article)

The [continuum hypothesis](/source/Continuum_hypothesis) states that there is no [cardinal number](/source/Cardinal_number) between the cardinality of the reals and the cardinality of the natural numbers, that is, c = ℵ 1 = ℶ 1 {\displaystyle \mathbf {c} =\aleph _{1}=\beth _{1}} .

Further information: [Beth number § Beth one](/source/Beth_number#Beth_one)

This hypothesis cannot be proved or disproved within the widely accepted [Zermelo–Fraenkel set theory](/source/Zermelo%E2%80%93Fraenkel_set_theory), even assuming the [Axiom of Choice](/source/Axiom_of_Choice).[43]

[Cardinal arithmetic](/source/Cardinal_arithmetic) can be used to show not only that the number of points in a [real number line](/source/Real_number_line) is equal to the number of points in any [segment of that line](/source/Line_segment), but also that this is equal to the number of points on a plane and, indeed, in any [finite-dimensional](/source/Finite-dimensional) space.[44]

The first three steps of a fractal construction whose limit is a [space-filling curve](/source/Space-filling_curve), showing that there are as many points in a one-dimensional line as in a two-dimensional square

The first of these results is apparent by considering, for instance, the [tangent](/source/Tangent_(trigonometric_function)) function, which provides a [one-to-one correspondence](/source/One-to-one_correspondence) between the [interval](/source/Interval_(mathematics)) (−⁠π/2⁠, ⁠π/2⁠) and **R**.

See also: [Hilbert's paradox of the Grand Hotel](/source/Hilbert's_paradox_of_the_Grand_Hotel)

The second result was proved by Cantor in 1878, but only became intuitively apparent in 1890, when [Giuseppe Peano](/source/Giuseppe_Peano) introduced the [space-filling curves](/source/Space-filling_curve), curved lines that twist and turn enough to fill the whole of any square, or [cube](/source/Cube), or [hypercube](/source/Hypercube), or finite-dimensional space. These curves can be used to define a one-to-one correspondence between the points on one side of a square and the points in the square.[45]

## Geometry

Until the end of the 19th century, infinity was rarely discussed in [geometry](/source/Geometry), except in the context of processes that could be continued without any limit. For example, a [line](/source/Line_(geometry)) was what is now called a [line segment](/source/Line_segment), with the proviso that one can extend it as far as one wants; but extending it *infinitely* was out of the question. Similarly, a line was usually not considered to be composed of infinitely many points but was a location where a point may be placed. Even if there are infinitely many possible positions, only a finite number of points could be placed on a line. A witness of this is the expression "the [locus](/source/Locus_(mathematics)) of *a point* that satisfies some property" (singular), where modern mathematicians would generally say "the set of *the points* that have the property" (plural).

One of the rare exceptions of a mathematical concept involving [actual infinity](/source/Actual_infinity) was [projective geometry](/source/Projective_geometry), where [points at infinity](/source/Points_at_infinity) are added to the [Euclidean space](/source/Euclidean_space) for modeling the [perspective](/source/Perspective_(graphical)) effect that shows [parallel lines](/source/Parallel_lines) intersecting "at infinity". Mathematically, points at infinity have the advantage of allowing one to not consider some special cases. For example, in a [projective plane](/source/Projective_plane), two distinct [lines](/source/Line_(geometry)) intersect in exactly one point, whereas without points at infinity, there are no intersection points for parallel lines. So, parallel and non-parallel lines must be studied separately in classical geometry, while they need not be distinguished in projective geometry.

Before the use of [set theory](/source/Set_theory) for the [foundation of mathematics](/source/Foundation_of_mathematics), points and lines were viewed as distinct entities, and a point could be *located on a line*. With the universal use of set theory in mathematics, the point of view has dramatically changed: a line is now considered as *the set of its points*, and one says that a point *belongs to a line* instead of *is located on a line* (however, the latter phrase is still used).

In particular, in modern mathematics, lines are *infinite sets*.

### Infinite dimension

The [vector spaces](/source/Vector_space) that occur in classical [geometry](/source/Geometry) have always a finite [dimension](/source/Dimension_(vector_space)), generally two or three. However, this is not implied by the abstract definition of a vector space, and vector spaces of infinite dimension can be considered. This is typically the case in [functional analysis](/source/Functional_analysis) where [function spaces](/source/Function_space) are generally vector spaces of infinite dimension.

In topology, some constructions can generate [topological spaces](/source/Topological_space) of infinite dimension. In particular, this is the case of [iterated loop spaces](/source/Iterated_loop_space).

### Fractals

The structure of a [fractal](/source/Fractal) object is reiterated in its magnifications. Fractals can be magnified indefinitely without losing their structure and becoming "smooth"; they have infinite perimeters and can have infinite or finite areas. One such [fractal curve](/source/Fractal_curve) with an infinite perimeter and finite area is the [Koch snowflake](/source/Koch_snowflake).[46]

## Finitism

[Leopold Kronecker](/source/Leopold_Kronecker) was skeptical of the notion of infinity and how his fellow mathematicians were using it in the 1870s and 1880s. This skepticism was developed in the [philosophy of mathematics](/source/Philosophy_of_mathematics) called [finitism](/source/Finitism), an extreme form of mathematical philosophy in the general philosophical and mathematical schools of [constructivism](/source/Mathematical_constructivism) and [intuitionism](/source/Intuitionism).[47]

## Logic

In [logic](/source/Logic), an [infinite regress](/source/Infinite_regress) argument is "a distinctively philosophical kind of argument purporting to show that a thesis is defective because it generates an infinite series when either (form A) no such series exists or (form B) were it to exist, the thesis would lack the role (e.g., of justification) that it is supposed to play."[48]

In [first-order logic](/source/First-order_logic), both the [compactness theorem](/source/Compactness_theorem) and [Löwenheim–Skolem theorems](/source/L%C3%B6wenheim%E2%80%93Skolem_theorem) are used to construct [non-standard models](/source/Non-standard_model_of_arithmetic) with certain infinite properties.

## Applications

### Physics

In [physics](/source/Physics), approximations of [real numbers](/source/Real_number) are used for [continuous](/source/Continuum_(theory)) measurements and [natural numbers](/source/Natural_number) are used for [discrete](/source/Countable) measurements (i.e., counting). Concepts of infinite things such as an infinite [plane wave](/source/Plane_wave) exist, but there are no experimental means to generate them.[49]

#### Cosmology

The first published proposal that the universe is infinite came from Thomas Digges in 1576.[50] Eight years later, in 1584, the Italian philosopher and astronomer [Giordano Bruno](/source/Giordano_Bruno) proposed an unbounded universe in *On the Infinite Universe and Worlds*: "Innumerable suns exist; innumerable earths revolve around these suns in a manner similar to the way the seven planets revolve around our sun. Living beings inhabit these worlds."[51]

[Cosmologists](/source/Cosmology) have long sought to discover whether infinity exists in our physical [universe](/source/Universe): Are there an infinite number of stars? Does the universe have infinite volume? Does space "[go on forever](/source/Shape_of_the_universe)"? This is still an open question of [cosmology](/source/Physical_cosmology). 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 with respect to the Earth's curvature, one will eventually return to the exact spot one started from. The universe, at least in principle, might have a similar [topology](/source/Topology). If so, one might eventually return to one's starting point after travelling in a straight line through the universe for long enough.[52]

The curvature of the universe can be measured through [multipole moments](/source/Multipole_moments) in the spectrum of the [cosmic background radiation](/source/Cosmic_microwave_background_radiation). To date, analysis of the radiation patterns recorded by the [WMAP](/source/WMAP) spacecraft hints that the universe has a flat topology. This would be consistent with an infinite physical universe.[53][54][55]

However, the universe could be finite, even if its curvature is flat. An easy way to understand this is to consider two-dimensional examples, such as video games where items that leave one edge of the screen reappear on the other. The topology of such games is [toroidal](/source/Torus) and the geometry is flat. Many possible bounded, flat possibilities also exist for three-dimensional space.[56]

The concept of infinity also extends to the [multiverse](/source/Multiverse) hypothesis, which, when explained by astrophysicists such as [Michio Kaku](/source/Michio_Kaku), posits that there are an infinite number and variety of universes.[57] Also, [cyclic models](/source/Cyclic_model) posit an infinite amount of [Big Bangs](/source/Big_Bang), resulting in an infinite variety of universes after each Big Bang event in an infinite cycle.[58]

### Computing

The [IEEE floating-point](/source/IEEE_floating-point) standard (IEEE 754) specifies a positive and a negative infinity value (and also [indefinite](/source/NaN) values). These are defined as the result of [arithmetic overflow](/source/Arithmetic_overflow), [division by zero](/source/Division_by_zero), and other exceptional operations.[59]

Some [programming languages](/source/Programming_language), such as [Java](/source/Java_(programming_language))[60] and [J](/source/J_(programming_language)),[61] 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](/source/Greatest_element), as they compare (respectively) greater than or less than all other values. They have uses as [sentinel values](/source/Sentinel_value) in [algorithms](/source/Algorithm) involving [sorting](/source/Sorting), [searching](/source/Search_algorithm), or [windowing](/source/Window_function).[*[citation needed](https://en.wikipedia.org/wiki/Wikipedia:Citation_needed)*]

In languages that do not have greatest and least elements but do allow [overloading](/source/Operator_overloading) of [relational operators](/source/Relational_operator), 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](/source/Data_type), the infinity values may still be accessible and usable as the result of certain operations.[*[citation needed](https://en.wikipedia.org/wiki/Wikipedia:Citation_needed)*]

In programming, an [infinite loop](/source/Infinite_loop) is a [loop](/source/Loop_(computing)) whose exit condition is never satisfied, thus executing indefinitely.

## Arts, games, and cognitive sciences

[Perspective](/source/Perspective_(graphical)) artwork uses the concept of [vanishing points](/source/Vanishing_point), roughly corresponding to mathematical [points at infinity](/source/Point_at_infinity), located at an infinite distance from the observer. This allows artists to create paintings that realistically render space, distances, and forms.[62] Artist [M.C. Escher](/source/M.C._Escher) is specifically known for employing the concept of infinity in his work in this and other ways.[63]

Variations of [chess](/source/Chess) played on an unbounded board are called [infinite chess](/source/Infinite_chess).[64][65]

[Cognitive scientist](/source/Cognitive_science) [George Lakoff](/source/George_Lakoff) considers the concept of infinity in mathematics and the sciences as a metaphor. This perspective is based on the basic metaphor of infinity (BMI), defined as the ever-increasing sequence <1, 2, 3, …>.[66]

## See also

- [0.999...](/source/0.999...)

- [Absolute infinite](/source/Absolute_infinite)

- [Aleph number](/source/Aleph_number)

- [Ananta](/source/Ananta_(infinite))

- [Apeirophobia](/source/Apeirophobia)

- [Exponentiation](/source/Exponentiation)

- [Indeterminate form](/source/Indeterminate_form)

- [Infinite monkey theorem](/source/Infinite_monkey_theorem)

- [Names of large numbers](/source/Names_of_large_numbers)

- [Paradoxes of infinity](/source/Paradoxes_of_infinity)

- [Supertask](/source/Supertask)

- [Surreal number](/source/Surreal_number)

## References

1. ^ [***a***](#cite_ref-:1_1-0) [***b***](#cite_ref-:1_1-1) [***c***](#cite_ref-:1_1-2) [***d***](#cite_ref-:1_1-3) [***e***](#cite_ref-:1_1-4) [***f***](#cite_ref-:1_1-5) Allen, G Donald (2003). "3 The Emergence of Calculus". [*The History of Infinity*](https://web.archive.org/web/20200801202539/https://www.math.tamu.edu/~dallen/masters/infinity/infinity.pdf) (PDF). [Texas A&M University](/source/Texas_A%26M_University) Department of Mathematics. p. 7. Archived from [the original](https://www.math.tamu.edu/~dallen/masters/infinity/infinity.pdf) (PDF) on August 1, 2020. Retrieved Nov 15, 2019.

1. ^ [***a***](#cite_ref-Jesseph_2-0) [***b***](#cite_ref-Jesseph_2-1) Jesseph, Douglas Michael (1998-05-01). ["Leibniz on the Foundations of the Calculus: The Question of the Reality of Infinitesimal Magnitudes"](https://web.archive.org/web/20120111102635/http://muse.jhu.edu/journals/perspectives_on_science/v006/6.1jesseph.html). *[Perspectives on Science](/source/Perspectives_on_Science)*. **6** (1&2): 6–40. [doi](/source/Doi_(identifier)):[10.1162/posc_a_00543](https://doi.org/10.1162%2Fposc_a_00543). [ISSN](/source/ISSN_(identifier)) [1063-6145](https://search.worldcat.org/issn/1063-6145). [OCLC](/source/OCLC_(identifier)) [42413222](https://search.worldcat.org/oclc/42413222). [S2CID](/source/S2CID_(identifier)) [118227996](https://api.semanticscholar.org/CorpusID:118227996). Archived from [the original](http://muse.jhu.edu/journals/perspectives_on_science/v006/6.1jesseph.html) on 11 January 2012. Retrieved 1 November 2019 – via Project MUSE.

1. **[^](#cite_ref-3)** Gowers, Timothy; Barrow-Green, June (2008). *The Princeton companion to mathematics*. Imre Leader, Princeton University. Princeton: Princeton University Press. [ISBN](/source/ISBN_(identifier)) [978-1-4008-3039-8](https://en.wikipedia.org/wiki/Special:BookSources/978-1-4008-3039-8). [OCLC](/source/OCLC_(identifier)) [659590835](https://search.worldcat.org/oclc/659590835).

1. **[^](#cite_ref-4)** [Maddox 2002](#CITEREFMaddox2002), pp. 113–117

1. **[^](#cite_ref-5)** [Wallace 2004](#CITEREFWallace2004), p. 44

1. **[^](#cite_ref-6)** Aristotle. [*Physics*](http://classics.mit.edu/Aristotle/physics.3.iii.html). Translated by Hardie, R. P.; Gaye, R. K. The Internet Classics Archive. Book 3, Chapters 5–8.

1. **[^](#cite_ref-7)** Goodman, Nicolas D. (1981). "Reflections on Bishop's philosophy of mathematics". In Richman, F. (ed.). *Constructive Mathematics*. Lecture Notes in Mathematics. Vol. 873. Springer. pp. 135–145. [doi](/source/Doi_(identifier)):[10.1007/BFb0090732](https://doi.org/10.1007%2FBFb0090732). [ISBN](/source/ISBN_(identifier)) [978-3-540-10850-4](https://en.wikipedia.org/wiki/Special:BookSources/978-3-540-10850-4).

1. **[^](#cite_ref-8)** Maor, p. 3

1. **[^](#cite_ref-9)** Sarton, George (March 1928). ["*The Thirteen Books of Euclid's Elements*. Thomas L. Heath, Heiberg"](https://www.journals.uchicago.edu/doi/10.1086/346308). *Isis*. **10** (1): 60–62. [doi](/source/Doi_(identifier)):[10.1086/346308](https://doi.org/10.1086%2F346308). [ISSN](/source/ISSN_(identifier)) [0021-1753](https://search.worldcat.org/issn/0021-1753) – via The University of Chicago Press Journals.

1. **[^](#cite_ref-10)** Hutten, Ernest Hirschlaff (1962). [*The origins of science; an inquiry into the foundations of Western thought*](https://archive.org/details/originsofscience0000hutt_n9u7). Internet Archive. London, Allen and Unwin. pp. 1–241. [ISBN](/source/ISBN_(identifier)) [978-0-04-946007-2](https://en.wikipedia.org/wiki/Special:BookSources/978-0-04-946007-2). Retrieved 2020-01-09. {{[cite book](https://en.wikipedia.org/wiki/Template:Cite_book)}}: ISBN / Date incompatibility ([help](https://en.wikipedia.org/wiki/Help:CS1_errors#invalid_isbn_date))

1. **[^](#cite_ref-11)** Euclid (2008) [c. 300 BC]. [*Euclid's Elements of Geometry*](http://farside.ph.utexas.edu/Books/Euclid/Elements.pdf) (PDF). Translated by Fitzpatrick, Richard. Lulu.com. p. 6 (Book I, Postulate 5). [ISBN](/source/ISBN_(identifier)) [978-0-6151-7984-1](https://en.wikipedia.org/wiki/Special:BookSources/978-0-6151-7984-1).

1. **[^](#cite_ref-12)** [Heath, Sir Thomas Little](/source/Thomas_Heath_(classicist)); Heiberg, Johan Ludvig (1908). [*The Thirteen Books of Euclid's Elements*](https://books.google.com/books?id=dkk6AQAAMAAJ&q=right+angles+infinite&pg=PR8). Vol. v. 1. The University Press. p. 212.

1. **[^](#cite_ref-13)** Drozdek, Adam (2008). In the Beginning Was the*Apeiron*: Infinity in Greek Philosophy. Stuttgart, Germany: Franz Steiner Verlag. [ISBN](/source/ISBN_(identifier)) [978-3-515-09258-6](https://en.wikipedia.org/wiki/Special:BookSources/978-3-515-09258-6).

1. **[^](#cite_ref-Zeno's_paradoxes_14-0)** ["Zeno's Paradoxes"](https://plato.stanford.edu/entries/paradox-zeno/). *Stanford University*. October 15, 2010. Retrieved April 3, 2017.

1. **[^](#cite_ref-15)** [Russell 1996](#CITEREFRussell1996), p. 347

1. **[^](#cite_ref-16)** [Cauchy, Augustin-Louis](/source/Augustin-Louis_Cauchy) (1821). [*Cours d'Analyse de l'École Royale Polytechnique*](https://books.google.com/books?id=UrT0KsbDmDwC&pg=PA1). Libraires du Roi & de la Bibliothèque du Roi. p. 124. Retrieved October 12, 2019.

1. **[^](#cite_ref-17)** Ian Stewart (2017). [*Infinity: a Very Short Introduction*](https://books.google.com/books?id=iewwDgAAQBAJ&pg=PA117). Oxford University Press. p. 117. [ISBN](/source/ISBN_(identifier)) [978-0-19-875523-4](https://en.wikipedia.org/wiki/Special:BookSources/978-0-19-875523-4). [Archived](https://web.archive.org/web/20170403200429/https://books.google.com/books?id=iewwDgAAQBAJ&pg=PA117) from the original on April 3, 2017.

1. **[^](#cite_ref-18)** Cajori, Florian (2007). [*A History of Mathematical Notations*](https://books.google.com/books?id=OQZxHpG2y3UC&q=infinity). Vol. 1. Cosimo, Inc. p. 214. [ISBN](/source/ISBN_(identifier)) [9781602066854](https://en.wikipedia.org/wiki/Special:BookSources/9781602066854).

1. **[^](#cite_ref-19)** [Cajori 1993](#CITEREFCajori1993), Sec. 421, Vol. II, p. 44

1. **[^](#cite_ref-20)** ["Arithmetica Infinitorum"](https://archive.org/details/ArithmeticaInfinitorum/page/n5/mode/2up).

1. **[^](#cite_ref-21)** [Cajori 1993](#CITEREFCajori1993), Sec. 435, Vol. II, p. 58

1. **[^](#cite_ref-22)** Grattan-Guinness, Ivor (2005). [*Landmark Writings in Western Mathematics 1640-1940*](https://books.google.com/books?id=UdGBy8iLpocC). Elsevier. p. 62. [ISBN](/source/ISBN_(identifier)) [978-0-08-045744-4](https://en.wikipedia.org/wiki/Special:BookSources/978-0-08-045744-4). [Archived](https://web.archive.org/web/20160603085825/https://books.google.com/books?id=UdGBy8iLpocC) from the original on 2016-06-03. [Extract of p. 62](https://books.google.com/books?id=UdGBy8iLpocC&pg=PA62)

1. **[^](#cite_ref-23)** AG, Compart. ["Unicode Character "∞" (U+221E)"](https://www.compart.com/en/unicode/U+221E). *Compart.com*. Retrieved 2019-11-15.

1. **[^](#cite_ref-24)** ["List of LaTeX mathematical symbols - OeisWiki"](https://oeis.org/wiki/List_of_LaTeX_mathematical_symbols). *oeis.org*. Retrieved 2019-11-15.

1. **[^](#cite_ref-25)** Scott, Joseph Frederick (1981), [*The mathematical work of John Wallis, D.D., F.R.S., (1616–1703)*](https://books.google.com/books?id=XX9PKytw8g8C&pg=PA24) (2 ed.), [American Mathematical Society](/source/American_Mathematical_Society), p. 24, [ISBN](/source/ISBN_(identifier)) [978-0-8284-0314-6](https://en.wikipedia.org/wiki/Special:BookSources/978-0-8284-0314-6), [archived](https://web.archive.org/web/20160509151853/https://books.google.com/books?id=XX9PKytw8g8C&pg=PA24) from the original on 2016-05-09

1. **[^](#cite_ref-26)** [Martin-Löf, Per](/source/Per_Martin-L%C3%B6f) (1990), "Mathematics of infinity", *COLOG-88 (Tallinn, 1988)*, Lecture Notes in Computer Science, vol. 417, Berlin: Springer, pp. 146–197, [doi](/source/Doi_(identifier)):[10.1007/3-540-52335-9_54](https://doi.org/10.1007%2F3-540-52335-9_54), [ISBN](/source/ISBN_(identifier)) [978-3-540-52335-2](https://en.wikipedia.org/wiki/Special:BookSources/978-3-540-52335-2), [MR](/source/MR_(identifier)) [1064143](https://mathscinet.ams.org/mathscinet-getitem?mr=1064143)

1. **[^](#cite_ref-27)** O'Flaherty, Wendy Doniger (1986), [*Dreams, Illusion, and Other Realities*](https://books.google.com/books?id=vhNNrX3bmo4C&pg=PA243), University of Chicago Press, p. 243, [ISBN](/source/ISBN_(identifier)) [978-0-226-61855-5](https://en.wikipedia.org/wiki/Special:BookSources/978-0-226-61855-5), [archived](https://web.archive.org/web/20160629143323/https://books.google.com/books?id=vhNNrX3bmo4C&pg=PA243) from the original on 2016-06-29

1. **[^](#cite_ref-28)** Toker, Leona (1989), [*Nabokov: The Mystery of Literary Structures*](https://books.google.com/books?id=Jud1q_NrqpcC&pg=PA159), Cornell University Press, p. 159, [ISBN](/source/ISBN_(identifier)) [978-0-8014-2211-9](https://en.wikipedia.org/wiki/Special:BookSources/978-0-8014-2211-9), [archived](https://web.archive.org/web/20160509095701/https://books.google.com/books?id=Jud1q_NrqpcC&pg=PA159) from the original on 2016-05-09

1. **[^](#cite_ref-29)** [Bell, John Lane](/source/John_Lane_Bell). ["Continuity and Infinitesimals"](https://plato.stanford.edu/entries/continuity/). In [Zalta, Edward N.](/source/Edward_N._Zalta) (ed.). *[Stanford Encyclopedia of Philosophy](/source/Stanford_Encyclopedia_of_Philosophy)*. [ISSN](/source/ISSN_(identifier)) [1095-5054](https://search.worldcat.org/issn/1095-5054). [OCLC](/source/OCLC_(identifier)) [429049174](https://search.worldcat.org/oclc/429049174).

1. **[^](#cite_ref-30)** [Taylor 1955](#CITEREFTaylor1955), p. 63

1. **[^](#cite_ref-31)** These uses of infinity for integrals and series can be found in any standard calculus text, such as, [Swokowski 1983](#CITEREFSwokowski1983), pp. 468–510

1. **[^](#cite_ref-32)** ["Properly Divergent Sequences - Mathonline"](http://mathonline.wikidot.com/properly-divergent-sequences). *mathonline.wikidot.com*. Retrieved 2019-11-15.

1. **[^](#cite_ref-33)** Aliprantis, Charalambos D.; Burkinshaw, Owen (1998), [*Principles of Real Analysis*](https://books.google.com/books?id=m40ivUwAonUC&pg=PA29) (3rd ed.), San Diego, CA: Academic Press, Inc., p. 29, [ISBN](/source/ISBN_(identifier)) [978-0-12-050257-8](https://en.wikipedia.org/wiki/Special:BookSources/978-0-12-050257-8), [MR](/source/MR_(identifier)) [1669668](https://mathscinet.ams.org/mathscinet-getitem?mr=1669668), [archived](https://web.archive.org/web/20150515120230/https://books.google.com/books?id=m40ivUwAonUC&pg=PA29) from the original on 2015-05-15

1. **[^](#cite_ref-34)** [Gemignani 1990](#CITEREFGemignani1990), p. 177

1. **[^](#cite_ref-35)** Beutelspacher, Albrecht; Rosenbaum, Ute (1998), *Projective Geometry / from foundations to applications*, Cambridge University Press, p. 27, [ISBN](/source/ISBN_(identifier)) [978-0-521-48364-3](https://en.wikipedia.org/wiki/Special:BookSources/978-0-521-48364-3)

1. **[^](#cite_ref-36)** Rao, Murali; Stetkær, Henrik (1991). [*Complex Analysis: An Invitation : a Concise Introduction to Complex Function Theory*](https://books.google.com/books?id=wdTntZ_N0tYC&pg=PA113). World Scientific. p. 113. [ISBN](/source/ISBN_(identifier)) [9789810203757](https://en.wikipedia.org/wiki/Special:BookSources/9789810203757).

1. **[^](#cite_ref-37)** Eric Schechter (2005). [*Classical and Nonclassical Logics: An Introduction to the Mathematics of Propositions*](https://books.google.com/books?id=70W4Q-kzdicC) (illustrated ed.). Princeton University Press. p. 118. [ISBN](/source/ISBN_(identifier)) [978-0-691-12279-3](https://en.wikipedia.org/wiki/Special:BookSources/978-0-691-12279-3). [Extract of page 118](https://books.google.com/books?id=70W4Q-kzdicC&pg=PA118)

1. **[^](#cite_ref-Dasgupta2013_38-0)** Abhijit Dasgupta (2013). *Set Theory: With an Introduction to Real Point Sets*. [Springer Science & Business Media](/source/Springer_Science_%26_Business_Media). pp. 362–363. [ISBN](/source/ISBN_(identifier)) [978-1-4614-8854-5](https://en.wikipedia.org/wiki/Special:BookSources/978-1-4614-8854-5).

1. **[^](#cite_ref-39)** Ignacio Jané (May 1995). "The role of the absolute infinite in Cantor's conception of set". *Erkenntnis*. **42** (3): 375–402. [doi](/source/Doi_(identifier)):[10.1007/BF01129011](https://doi.org/10.1007%2FBF01129011). [JSTOR](/source/JSTOR_(identifier)) [20012628](https://www.jstor.org/stable/20012628). [S2CID](/source/S2CID_(identifier)) [122487235](https://api.semanticscholar.org/CorpusID:122487235). Cantor (1) took the absolute to be a manifestation of God [...] When the absolute is first introduced in Grundlagen, it is linked to God. "the true infinite or absolute, which is in God, admits no kind of determination" (Cantor 1883b, p. 175) This is not an incidental remark, for Cantor is very explicit and insistent about the relation between the absolute and God.

1. **[^](#cite_ref-40)** A.W. Moore (2012). [*The Infinite*](https://books.google.com/books?id=z-UJhZmQnhAC) (2nd, revised ed.). Routledge. p. xiv. [ISBN](/source/ISBN_(identifier)) [978-1-134-91213-1](https://en.wikipedia.org/wiki/Special:BookSources/978-1-134-91213-1). [Extract of page xiv](https://books.google.com/books?id=z-UJhZmQnhAC&pg=PR14)

1. **[^](#cite_ref-41)** [Hamkins, Joel David](/source/Joel_David_Hamkins) (2012-01-04). ["Climb into Cantor's attic"](https://jdh.hamkins.org/climb-into-cantors-attic/). *Joel David Hamkins*. Retrieved 2026-01-20.

1. **[^](#cite_ref-42)** [Dauben, Joseph](/source/Joseph_Dauben) (1993). ["Georg Cantor and the Battle for Transfinite Set Theory"](http://acmsonline.org/home2/wp-content/uploads/2016/05/Dauben-Cantor.pdf) (PDF). *9th ACMS Conference Proceedings*: 4.

1. **[^](#cite_ref-43)** [Cohen 1963](#CITEREFCohen1963), p. 1143

1. **[^](#cite_ref-44)** Felix Hausdorff (2021). [*Set Theory*](https://books.google.com/books?id=TFA_EAAAQBAJ). American Mathematical Soc. p. 44. [ISBN](/source/ISBN_(identifier)) [978-1-4704-6494-3](https://en.wikipedia.org/wiki/Special:BookSources/978-1-4704-6494-3). [Extract of page 44](https://books.google.com/books?id=TFA_EAAAQBAJ&pg=PA44)

1. **[^](#cite_ref-45)** [Sagan 1994](#CITEREFSagan1994), pp. 10–12

1. **[^](#cite_ref-46)** Michael Frame; Benoit Mandelbrot (2002). [*Fractals, Graphics, and Mathematics Education*](https://books.google.com/books?id=Wz7iCaiB2C0C) (illustrated ed.). Cambridge University Press. p. 36. [ISBN](/source/ISBN_(identifier)) [978-0-88385-169-2](https://en.wikipedia.org/wiki/Special:BookSources/978-0-88385-169-2). [Extract of page 36](https://books.google.com/books?id=Wz7iCaiB2C0C&pg=PA36)

1. **[^](#cite_ref-47)** [Kline 1972](#CITEREFKline1972), pp. 1197–1198

1. **[^](#cite_ref-48)** *Cambridge Dictionary of Philosophy*, Second Edition, p. 429

1. **[^](#cite_ref-49)** [Doric Lenses](http://www.doriclenses.com/administrer/upload/pdf/NOT_AXI_ENG_070212_doricl97_doricle_kvgwQP.pdf) [Archived](https://web.archive.org/web/20130124011604/http://www.doriclenses.com/administrer/upload/pdf/NOT_AXI_ENG_070212_doricl97_doricle_kvgwQP.pdf) 2013-01-24 at the [Wayback Machine](/source/Wayback_Machine) – Application Note – Axicons – 2. Intensity Distribution. Retrieved 7 April 2014.

1. **[^](#cite_ref-50)** John Gribbin (2009), *In Search of the Multiverse: Parallel Worlds, Hidden Dimensions, and the Ultimate Quest for the Frontiers of Reality*, [ISBN](/source/ISBN_(identifier)) [978-0-470-61352-8](https://en.wikipedia.org/wiki/Special:BookSources/978-0-470-61352-8). p. 88

1. **[^](#cite_ref-51)** Brake, Mark (2013). [*Alien Life Imagined: Communicating the Science and Culture of Astrobiology*](https://books.google.com/books?id=sWGqzfL0snEC&pg=PA63) (illustrated ed.). Cambridge University Press. p. 63. [ISBN](/source/ISBN_(identifier)) [978-0-521-49129-7](https://en.wikipedia.org/wiki/Special:BookSources/978-0-521-49129-7).

1. **[^](#cite_ref-52)** Koupelis, Theo; Kuhn, Karl F. (2007). [*In Quest of the Universe*](https://books.google.com/books?id=6rTttN4ZdyoC) (illustrated ed.). Jones & Bartlett Learning. p. 553. [ISBN](/source/ISBN_(identifier)) [978-0-7637-4387-1](https://en.wikipedia.org/wiki/Special:BookSources/978-0-7637-4387-1). [Extract of p. 553](https://books.google.com/books?id=6rTttN4ZdyoC&pg=PA553)

1. **[^](#cite_ref-NASA_Shape_53-0)** ["Will the Universe expand forever?"](https://map.gsfc.nasa.gov/universe/uni_shape.html). NASA. 24 January 2014. [Archived](https://web.archive.org/web/20120601032707/http://map.gsfc.nasa.gov/universe/uni_shape.html) from the original on 1 June 2012. Retrieved 16 March 2015.

1. **[^](#cite_ref-Fermi_Flat_54-0)** ["Our universe is Flat"](http://www.symmetrymagazine.org/article/april-2015/our-flat-universe?email_issue=725). FermiLab/SLAC. 7 April 2015. [Archived](https://web.archive.org/web/20150410200411/http://www.symmetrymagazine.org/article/april-2015/our-flat-universe?email_issue=725) from the original on 10 April 2015.

1. **[^](#cite_ref-55)** Marcus Y. Yoo (2011). "Unexpected connections". *Engineering & Science*. LXXIV1: 30.

1. **[^](#cite_ref-56)** Weeks, Jeffrey (2001). [*The Shape of Space*](https://archive.org/details/shapeofspace0000week). CRC Press. [ISBN](/source/ISBN_(identifier)) [978-0-8247-0709-5](https://en.wikipedia.org/wiki/Special:BookSources/978-0-8247-0709-5).

1. **[^](#cite_ref-57)** Kaku, M. (2006). Parallel worlds. Knopf Doubleday Publishing Group.

1. **[^](#cite_ref-Nautilus2014_58-0)** McKee, Maggie (25 September 2014). ["Ingenious: Paul J. Steinhardt – The Princeton physicist on what's wrong with inflation theory and his view of the Big Bang"](http://nautil.us/issue/17/big-bangs/ingenious-paul-j-steinhardt). *Nautilus*. No. 17. NautilusThink Inc. Retrieved 31 March 2017.

1. **[^](#cite_ref-59)** ["Infinity and NaN (The GNU C Library)"](https://www.gnu.org/software/libc/manual/html_node/Infinity-and-NaN.html). *www.gnu.org*. Retrieved 2021-03-15.

1. **[^](#cite_ref-60)** Gosling, James; et al. (27 July 2012). ["4.2.3."](http://docs.oracle.com/javase/specs/jls/se7/html/jls-4.html#jls-4.2.3). *The Java Language Specification* (Java SE 7 ed.). California: Oracle America, Inc. [Archived](https://web.archive.org/web/20120609071157/http://docs.oracle.com/javase/specs/jls/se7/html/jls-4.html#jls-4.2.3) from the original on 9 June 2012. Retrieved 6 September 2012.

1. **[^](#cite_ref-61)** Stokes, Roger (July 2012). ["19.2.1"](https://web.archive.org/web/20120325064205/http://www.rogerstokes.free-online.co.uk/19.htm#10). *Learning J*. Archived from [the original](http://www.rogerstokes.free-online.co.uk/19.htm#10) on 25 March 2012. Retrieved 6 September 2012.

1. **[^](#cite_ref-62)** Morris Kline (1985). [*Mathematics for the Nonmathematician*](https://books.google.com/books?id=gXSMukf1aFIC) (illustrated, unabridged, reprinted ed.). Courier Corporation. p. 229. [ISBN](/source/ISBN_(identifier)) [978-0-486-24823-3](https://en.wikipedia.org/wiki/Special:BookSources/978-0-486-24823-3). [Extract of page 229, Section 10-7](https://books.google.com/books?id=gXSMukf1aFIC&pg=PA229)

1. **[^](#cite_ref-63)** Schattschneider, Doris (2010). ["The Mathematical Side of M. C. Escher"](https://www.ams.org/notices/201006/rtx100600706p.pdf) (PDF). *Notices of the AMS*. **57** (6): 706–718.

1. **[^](#cite_ref-64)** [Infinite chess at the Chess Variant Pages](http://www.chessvariants.com/boardrules.dir/infinite.html) [Archived](https://web.archive.org/web/20170402082426/http://www.chessvariants.com/boardrules.dir/infinite.html) 2017-04-02 at the [Wayback Machine](/source/Wayback_Machine) An infinite chess scheme.

1. **[^](#cite_ref-65)** ["Infinite Chess, PBS Infinite Series"](https://www.youtube.com/watch?v=PN-I6u-AxMg) [Archived](https://web.archive.org/web/20170407211614/https://www.youtube.com/watch?v=PN-I6u-AxMg) 2017-04-07 at the [Wayback Machine](/source/Wayback_Machine) PBS Infinite Series, with academic sources by J. Hamkins (infinite chess: Evans, C.D.A; Joel David Hamkins (2013). "Transfinite game values in infinite chess". [arXiv](/source/ArXiv_(identifier)):[1302.4377](https://arxiv.org/abs/1302.4377) [[math.LO](https://arxiv.org/archive/math.LO)]. and Evans, C.D.A; Joel David Hamkins; Norman Lewis Perlmutter (2015). "A position in infinite chess with game value $ω^4$". [arXiv](/source/ArXiv_(identifier)):[1510.08155](https://arxiv.org/abs/1510.08155) [[math.LO](https://arxiv.org/archive/math.LO)].).

1. **[^](#cite_ref-66)** Elglaly, Yasmine Nader; Quek, Francis. ["Review of "Where Mathematics comes from: How the Embodied Mind Brings Mathematics Into Being" By George Lakoff and Rafael E. Nunez"](https://web.archive.org/web/20200226004335/http://www.se.rit.edu/~yasmine/assets/papers/Embodied%20math.pdf) (PDF). *CHI 2009*. Archived from [the original](http://www.se.rit.edu/~yasmine/assets/papers/Embodied%20math.pdf) (PDF) on 2020-02-26. Retrieved 2021-03-25.

### Bibliography

- Cajori, Florian (1993) [1928 & 1929], [*A History of Mathematical Notations (Two Volumes Bound as One)*](https://archive.org/details/historyofmathema00cajo_0), Dover, [ISBN](/source/ISBN_(identifier)) [978-0-486-67766-8](https://en.wikipedia.org/wiki/Special:BookSources/978-0-486-67766-8)

- Gemignani, Michael C. (1990), *Elementary Topology* (2nd ed.), Dover, [ISBN](/source/ISBN_(identifier)) [978-0-486-66522-1](https://en.wikipedia.org/wiki/Special:BookSources/978-0-486-66522-1)

- [Keisler, H. Jerome](/source/Howard_Jerome_Keisler) (1986), [*Elementary Calculus: An Approach Using Infinitesimals*](http://www.math.wisc.edu/~keisler/calc.html) (2nd ed.)

- Maddox, Randall B. (2002), *Mathematical Thinking and Writing: A Transition to Abstract Mathematics*, Academic Press, [ISBN](/source/ISBN_(identifier)) [978-0-12-464976-7](https://en.wikipedia.org/wiki/Special:BookSources/978-0-12-464976-7)

- [Kline, Morris](/source/Morris_Kline) (1972), *Mathematical Thought from Ancient to Modern Times*, New York: Oxford University Press, pp. 1197–1198, [ISBN](/source/ISBN_(identifier)) [978-0-19-506135-2](https://en.wikipedia.org/wiki/Special:BookSources/978-0-19-506135-2)

- [Russell, Bertrand](/source/Bertrand_Russell) (1996) [1903], *The Principles of Mathematics*, New York: Norton, [ISBN](/source/ISBN_(identifier)) [978-0-393-31404-5](https://en.wikipedia.org/wiki/Special:BookSources/978-0-393-31404-5), [OCLC](/source/OCLC_(identifier)) [247299160](https://search.worldcat.org/oclc/247299160)

- Sagan, Hans (1994), *Space-Filling Curves*, Springer, [ISBN](/source/ISBN_(identifier)) [978-1-4612-0871-6](https://en.wikipedia.org/wiki/Special:BookSources/978-1-4612-0871-6)

- Swokowski, Earl W. (1983), [*Calculus with Analytic Geometry*](https://archive.org/details/calculuswithanal00swok) (Alternate ed.), Prindle, Weber & Schmidt, [ISBN](/source/ISBN_(identifier)) [978-0-87150-341-1](https://en.wikipedia.org/wiki/Special:BookSources/978-0-87150-341-1)

- Taylor, Angus E. (1955), *Advanced Calculus*, Blaisdell Publishing Company

- [Wallace, David Foster](/source/David_Foster_Wallace) (2004), *Everything and More: A Compact History of Infinity*, Norton, W.W. & Company, Inc., [ISBN](/source/ISBN_(identifier)) [978-0-393-32629-1](https://en.wikipedia.org/wiki/Special:BookSources/978-0-393-32629-1)

### Sources

- Aczel, Amir D. (2001). *The Mystery of the Aleph: Mathematics, the Kabbalah, and the Search for Infinity*. New York: Pocket Books. [ISBN](/source/ISBN_(identifier)) [978-0-7434-2299-4](https://en.wikipedia.org/wiki/Special:BookSources/978-0-7434-2299-4).

- [D.P. Agrawal](/source/D.P._Agrawal) (2000). *[Ancient Jaina Mathematics: an Introduction](http://www.infinityfoundation.com/mandala/t_es/t_es_agraw_jaina.htm)*, [Infinity Foundation](http://infinityfoundation.com).

- Bell, J.L.: Continuity and infinitesimals. Stanford Encyclopedia of philosophy. Revised 2009.

- Cohen, Paul (1963), "The Independence of the Continuum Hypothesis", *[Proceedings of the National Academy of Sciences of the United States of America](/source/Proceedings_of_the_National_Academy_of_Sciences_of_the_United_States_of_America)*, **50** (6): 1143–1148, [Bibcode](/source/Bibcode_(identifier)):[1963PNAS...50.1143C](https://ui.adsabs.harvard.edu/abs/1963PNAS...50.1143C), [doi](/source/Doi_(identifier)):[10.1073/pnas.50.6.1143](https://doi.org/10.1073%2Fpnas.50.6.1143), [PMC](/source/PMC_(identifier)) [221287](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC221287), [PMID](/source/PMID_(identifier)) [16578557](https://pubmed.ncbi.nlm.nih.gov/16578557).

- Jain, L.C. (1982). *Exact Sciences from Jaina Sources*.

- Jain, L.C. (1973). "Set theory in the Jaina school of mathematics", *Indian Journal of History of Science*.

- Joseph, George G. (2000). *The Crest of the Peacock: Non-European Roots of Mathematics* (2nd ed.). [Penguin Books](/source/Penguin_Books). [ISBN](/source/ISBN_(identifier)) [978-0-14-027778-4](https://en.wikipedia.org/wiki/Special:BookSources/978-0-14-027778-4).

- H. Jerome Keisler: Elementary Calculus: An Approach Using Infinitesimals. First edition 1976; 2nd edition 1986. This book is now out of print. The publisher has reverted the copyright to the author, who has made available the 2nd edition in .pdf format available for downloading at [http://www.math.wisc.edu/~keisler/calc.html](http://www.math.wisc.edu/~keisler/calc.html)

- [Eli Maor](/source/Eli_Maor) (1991). [*To Infinity and Beyond*](https://books.google.com/books?id=lXjF7JnHQoIC&q=To+Infinity+and+beyond). Princeton University Press. [ISBN](/source/ISBN_(identifier)) [978-0-691-02511-7](https://en.wikipedia.org/wiki/Special:BookSources/978-0-691-02511-7).

- O'Connor, John J. and Edmund F. Robertson (1998). ['Georg Ferdinand Ludwig Philipp Cantor'](http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Cantor.html) [Archived](https://web.archive.org/web/20060916095918/http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Cantor.html) 2006-09-16 at the [Wayback Machine](/source/Wayback_Machine), *[MacTutor History of Mathematics archive](/source/MacTutor_History_of_Mathematics_archive)*.

- O'Connor, John J. and Edmund F. Robertson (2000). ['Jaina mathematics'](http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Jaina_mathematics.html) [Archived](https://web.archive.org/web/20081220145242/http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Jaina_mathematics.html) 2008-12-20 at the [Wayback Machine](/source/Wayback_Machine), *MacTutor History of Mathematics archive*.

- Pearce, Ian. (2002). ['Jainism'](http://www-history.mcs.st-andrews.ac.uk/history/Projects/Pearce/Chapters/Ch5.html), *MacTutor History of Mathematics archive*.

- [Rucker, Rudy](/source/Rudy_Rucker) (1995). *Infinity and the Mind: The Science and Philosophy of the Infinite*. Princeton University Press. [ISBN](/source/ISBN_(identifier)) [978-0-691-00172-2](https://en.wikipedia.org/wiki/Special:BookSources/978-0-691-00172-2).

- Singh, Navjyoti (1988). "Jaina Theory of Actual Infinity and Transfinite Numbers". *Journal of the Asiatic Society*. **30**.

## External links

**Infinity**  at Wikipedia's [sister projects](https://en.wikipedia.org/wiki/Wikipedia:Wikimedia_sister_projects)

- [Media](https://commons.wikimedia.org/wiki/Category:Infinity) from Commons
- [Quotations](https://en.wikiquote.org/wiki/Infinity) from Wikiquote

Look up ***[infinity](https://en.wiktionary.org/wiki/Special:Search/infinity)*** in Wiktionary, the free dictionary.

Wikibooks has a book on the topic of: ***[Infinity is not a number](https://en.wikibooks.org/wiki/Infinity_is_not_a_number)***

- Fieser, James; Dowden, Bradley (eds.). ["The Infinite"](https://iep.utm.edu/infinite). *[Internet Encyclopedia of Philosophy](/source/Internet_Encyclopedia_of_Philosophy)*. [ISSN](/source/ISSN_(identifier)) [2161-0002](https://search.worldcat.org/issn/2161-0002). [OCLC](/source/OCLC_(identifier)) [37741658](https://search.worldcat.org/oclc/37741658).

- [Infinity](https://www.bbc.co.uk/programmes/p0054927) on [*In Our Time*](/source/In_Our_Time_(radio_series)) at the [BBC](/source/BBC)

- *[A Crash Course in the Mathematics of Infinite Sets](http://www.earlham.edu/~peters/writing/infapp.htm) [Archived](https://web.archive.org/web/20100227033849/http://www.earlham.edu/~peters/writing/infapp.htm) 2010-02-27 at the [Wayback Machine](/source/Wayback_Machine)*, by Peter Suber. From the St. John's Review, XLIV, 2 (1998) 1–59. The stand-alone appendix to *Infinite Reflections*, below. A concise introduction to Cantor's mathematics of infinite sets.

- *[Infinite Reflections](http://www.earlham.edu/~peters/writing/infinity.htm) [Archived](https://web.archive.org/web/20091105182928/http://www.earlham.edu/~peters/writing/infinity.htm) 2009-11-05 at the [Wayback Machine](/source/Wayback_Machine)*, by Peter Suber. How Cantor's mathematics of the infinite solves a handful of ancient philosophical problems of the infinite. From the St. John's Review, XLIV, 2 (1998) 1–59.

- Grime, James. ["Infinity is bigger than you think"](https://web.archive.org/web/20171022173525/http://www.numberphile.com/videos/countable_infinity.html). *Numberphile*. [Brady Haran](/source/Brady_Haran). Archived from [the original](http://www.numberphile.com/videos/countable_infinity.html) on 2017-10-22. Retrieved 2013-04-06.

- [Hotel Infinity](https://web.archive.org/web/20040910082530/http://www.c3.lanl.gov/mega-math/workbk/infinity/infinity.html)

- John J. O'Connor and Edmund F. Robertson (1998). ['Georg Ferdinand Ludwig Philipp Cantor'](http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Cantor.html) [Archived](https://web.archive.org/web/20060916095918/http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Cantor.html) 2006-09-16 at the [Wayback Machine](/source/Wayback_Machine), *[MacTutor History of Mathematics archive](/source/MacTutor_History_of_Mathematics_archive)*.

- John J. O'Connor and Edmund F. Robertson (2000). ['Jaina mathematics'](http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Jaina_mathematics.html) [Archived](https://web.archive.org/web/20081220145242/http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Jaina_mathematics.html) 2008-12-20 at the [Wayback Machine](/source/Wayback_Machine), *MacTutor History of Mathematics archive*.

- Ian Pearce (2002). ['Jainism'](http://www-history.mcs.st-andrews.ac.uk/history/Projects/Pearce/Chapters/Ch5.html), *MacTutor History of Mathematics archive*.

- [The Mystery Of The Aleph: Mathematics, the Kabbalah, and the Search for Infinity](https://www.washingtonpost.com/wp-srv/style/longterm/books/chap1/mysteryaleph.htm)

- [Dictionary of the Infinite](http://dictionary.of-the-infinite.com) (compilation of articles about infinity in physics, mathematics, and philosophy)

v t e Infinity (∞) History Ananta (infinite) Apeiron Controversy over Cantor's theory Galileo's paradox Hilbert's paradox of the Grand Hotel Infinity (philosophy) Paradoxes of infinity Paradoxes of set theory Branches of mathematics Complex analysis Internal set theory Nonstandard analysis Set theory Synthetic differential geometry Formalizations of infinity 0.999... Absolute infinite Actual infinity Aleph number Beth number Cardinal numbers Cardinality of the continuum Dedekind-infinite set Division by zero (Complex infinity) Epsilon number Gimel function Hilbert space Hyperreal numbers Infinite set Infinitesimal Ordinal numbers Point at infinity Large cardinal Sphere at infinity (Kleinian group) Supertask Surreal numbers Transfinite numbers Geometries Differential geometry of surfaces Möbius plane Möbius transformation Riemannian manifold Mathematicians Georg Cantor David Hilbert Gottfried Wilhelm Leibniz August Ferdinand Möbius Bernhard Riemann Abraham Robinson

v t e Large numbers Examples in numerical order Hundred Thousand Ten thousand Hundred thousand Million Billion Trillion Quadrillion Quintillion Sextillion Septillion Octillion Nonillion Decillion Eddington number Googol Shannon number Googolplex Skewes's number Moser's number Graham's number TREE(3) SSCG(3) BH(3) Rayo's number Expression methods Notations Scientific notation Knuth's up-arrow notation Conway chained arrow notation Steinhaus–Moser notation Operators Hyperoperation Tetration Ackermann function Grzegorczyk hierarchy Fast-growing hierarchy Slow-growing hierarchy Hardy hierarchy Veblen function Related articles (alphabetical order) Busy beaver Extended real number line Indefinite and fictitious numbers Infinitesimal Largest known prime number List of numbers Long and short scales Number systems Number names Orders of magnitude Power of two Power of three Power of 10 Sagan Unit Names History

v t e Major topics in mathematical analysis Calculus: Integration Differentiation Differential equations ordinary partial stochastic Fundamental theorem of calculus Calculus of variations Vector calculus Tensor calculus Matrix calculus Lists of integrals Table of derivatives Real analysis Complex analysis Hypercomplex analysis (quaternionic analysis) Functional analysis Fourier analysis Least-squares spectral analysis Harmonic analysis P-adic analysis (P-adic numbers) Algebraic analysis Tropical analysis Measure theory Microlocal analysis Representation theory Functions Continuous function Special functions Limit Series Infinity Mathematics portal

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