{"history":[{"id":"d42f635a-9111-4841-ac77-1847b1edbed6","wordChanged":"4","newWord":":::summary\n:::\n\n# \n**4** is something which is boundless, endless, or larger than any [natural number](https://bliptext.com/articles/natural-number).\nIt is often denoted by the [infinity symbol](https://bliptext.com/articles/infinity-symbol) **INFINITY**\nFrom the time of the [ancient Greeks](https://bliptext.com/articles/greek-mathematics), the [philosophical nature of infinity](https://bliptext.com/articles/infinity-philosophy) has been the subject of many discussions among philosophers.\nIn the 17th century, with the introduction of the infinity symbol and the [infinitesimal calculus](https://bliptext.com/articles/infinitesimal-calculus), mathematicians began to work with [infinite series](https://bliptext.com/articles/infinite-series) and what some mathematicians (including [l'Hôpital](https://bliptext.com/articles/guillaume-de-l-h-pital) and [Bernoulli](https://bliptext.com/articles/johann-bernoulli)) regarded as infinitely small quantities, but infinity continued to be associated with endless processes.\nAs mathematicians struggled with the foundation of [calculus](https://bliptext.com/articles/calculus), it remained unclear whether infinity could be considered as a number or [magnitude](https://bliptext.com/articles/magnitude-mathematics) and, if so, how this could be done.\nAt the end of the 19th century, [Georg Cantor](https://bliptext.com/articles/georg-cantor) enlarged the mathematical study of infinity by studying [infinite sets](https://bliptext.com/articles/infinite-set) and [infinite numbers](https://bliptext.com/articles/transfinite-number), showing that they can be of various sizes.\nFor example, if a line is viewed as the set of all of its points, their infinite number (i.e., the [cardinality](https://bliptext.com/articles/cardinality) of the line) is larger than the number of [integers](https://bliptext.com/articles/integer).\nIn this usage, infinity is a mathematical concept, and infinite [mathematical objects](https://bliptext.com/articles/mathematical-object) can be studied, manipulated, and used just like any other mathematical object.\nThe mathematical concept of infinity refines and extends the old philosophical concept, in particular by introducing infinitely many different sizes of infinite sets.\nAmong the axioms of [Zermelo–Fraenkel set theory](https://bliptext.com/articles/zermelo-fraenkel-set-theory), on which most of modern mathematics can be developed, is the [axiom of infinity](https://bliptext.com/articles/axiom-of-infinity), which guarantees the existence of infinite sets.\nThe mathematical concept of infinity and the manipulation of infinite sets are widely used in mathematics, even in areas such as [combinatorics](https://bliptext.com/articles/combinatorics) that may seem to have nothing to do with them.\nFor example, [Wiles's proof of Fermat's Last Theorem](https://bliptext.com/articles/wiles-s-proof-of-fermat-s-last-theorem) implicitly relies on the existence of [Grothendieck universes](https://bliptext.com/articles/grothendieck-universe), very large infinite sets, for solving a long-standing problem that is stated in terms of [elementary arithmetic](https://bliptext.com/articles/elementary-arithmetic).\nIn [physics](https://bliptext.com/articles/physics) and [cosmology](https://bliptext.com/articles/cosmology), [whether the universe is spatially infinite or not](https://bliptext.com/articles/universe), is an open question.\n# [The Sierpiński Triangle contains an infinite amount of itself because of its recursive pattern. | upload.wikimedia.org/wikipedia/commons/c/c5/Sierpi%C5%84ski///Triangle.png]\n# [Due to the constant light reflection between opposing mirrors, it seems that there is a boundless amount of space and repetition inside of them. | upload.wikimedia.org/wikipedia/commons/9/9e/Reflections///1090029.jpg]\n\n# History\nAncient cultures had various ideas about the nature of infinity.\nThe [ancient Indians](https://bliptext.com/articles/vedic-period) and the [Greeks](https://bliptext.com/articles/ancient-greece) did not define infinity in precise formalism as does modern mathematics, and instead approached infinity as a philosophical concept.\n\n## Early Greek\nThe earliest recorded idea of infinity in Greece may be that of [Anaximander](https://bliptext.com/articles/anaximander) (c. 610 – c. 546 BC) a [pre-Socratic](https://bliptext.com/articles/pre-socratic-philosophy) Greek philosopher.\nHe used the word [apeiron](https://bliptext.com/articles/apeiron), which means \"unbounded\", \"indefinite\", and perhaps can be translated as \"infinite\".\n[Aristotle](https://bliptext.com/articles/aristotle) (350 BC) distinguished potential infinity from [actual infinity](https://bliptext.com/articles/actual-infinity), which he regarded as impossible due to the various [paradoxes](https://bliptext.com/articles/paradoxes) it seemed to produce.\nIt has been argued that, in line with this view, the [Hellenistic](https://bliptext.com/articles/hellenistic) Greeks had a \"horror of the infinite\" which would, for example, explain why [Euclid](https://bliptext.com/articles/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.\"\nIt has also been maintained, that, in proving the [infinitude of the prime numbers](https://bliptext.com/articles/infinitude-of-the-prime-numbers), Euclid \"was the first to overcome the horror of the infinite\".\nThere is a similar controversy concerning Euclid's [parallel postulate](https://bliptext.com/articles/parallel-postulate), sometimes translated:\n\"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.\"\nOther translators, however, prefer the translation \"the two straight lines, if produced indefinitely ...\", thus avoiding the implication that Euclid was comfortable with the notion of infinity.\nFinally, 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.\n\n## Zeno: Achilles and the tortoise\n[Zeno of Elea](https://bliptext.com/articles/zeno-of-elea) (c. 495 – c. 430 BC) did not advance any views concerning the infinite.\nNevertheless, his paradoxes, especially \"Achilles and the Tortoise\", were important contributions in that they made clear the inadequacy of popular conceptions.\nThe paradoxes were described by [Bertrand Russell](https://bliptext.com/articles/bertrand-russell) as \"immeasurably subtle and profound\".\n[Achilles](https://bliptext.com/articles/achilles) races a tortoise, giving the latter a head start.\nEtc.\nApparently, Achilles never overtakes the tortoise, since however many steps he completes, the tortoise remains ahead of him.\nZeno was not attempting to make a point about infinity.\nAs a member of the [Eleatics](https://bliptext.com/articles/eleatic) school which regarded motion as an illusion, he saw it as a mistake to suppose that Achilles could run at all.\nSubsequent thinkers, finding this solution unacceptable, struggled for over two millennia to find other weaknesses in the argument.\nFinally, in 1821, [Augustin-Louis Cauchy](https://bliptext.com/articles/augustin-louis-cauchy) provided both a satisfactory definition of a limit and a proof that, for\n0 < x < 1\n,\nSuppose 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.\nThe duration of the chase fits Cauchy's pattern with\na = 10 seconds\nand\nx = 0.01\n. Achilles does overtake the tortoise; it takes him\n\n## Early Indian\nThe [Jain mathematical](https://bliptext.com/articles/indian-mathematics) text Surya Prajnapti (c. 4th–3rd century BCE) classifies all numbers into three sets: [enumerable](https://bliptext.com/articles/enumerable), innumerable, and infinite.\nEach of these was further subdivided into three orders:\n\n## 17th century\nIn the 17th century, European mathematicians started using infinite numbers and infinite expressions in a systematic fashion.\nIn 1655, [John Wallis](https://bliptext.com/articles/john-wallis) first used the notation \\infty for such a number in his De sectionibus conicis, and exploited it in area calculations by dividing the region into [infinitesimal](https://bliptext.com/articles/infinitesimal) strips of width on the order of But in Arithmetica infinitorum (1656), 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.\"\nIn 1699, [Isaac Newton](https://bliptext.com/articles/isaac-newton) wrote about equations with an infinite number of terms in his work [De analysi per aequationes numero terminorum infinitas](https://bliptext.com/articles/de-analysi-per-aequationes-numero-terminorum-infinitas).\n\n# Mathematics\n[Hermann Weyl](https://bliptext.com/articles/hermann-weyl) opened a mathematico-philosophic address given in 1930 with:\n\"Mathematics is the science of the infinite.\"\n\n## Symbol\nThe infinity symbol \\infty (sometimes called the [lemniscate](https://bliptext.com/articles/lemniscate)) is a mathematical symbol representing the concept of infinity.\nThe symbol is encoded in [Unicode](https://bliptext.com/articles/unicode) at and in [LaTeX](https://bliptext.com/articles/latex) as.\nIt was introduced in 1655 by [John Wallis](https://bliptext.com/articles/john-wallis), and since its introduction, it has also been used outside mathematics in modern mysticism and literary [symbology](https://bliptext.com/articles/symbology).\n\n## Calculus\n[Gottfried Leibniz](https://bliptext.com/articles/gottfried-wilhelm-leibniz), one of the co-inventors of [infinitesimal calculus](https://bliptext.com/articles/infinitesimal-calculus), speculated widely about infinite numbers and their use in mathematics.\nTo 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](https://bliptext.com/articles/law-of-continuity).\n\n### Real analysis\nIn [real analysis](https://bliptext.com/articles/real-analysis), the symbol \\infty, called \"infinity\", is used to denote an unbounded [limit](https://bliptext.com/articles/limit-of-a-function).\nThe notation means that x increases without bound, and means that x decreases without bound.\nFor example, if f(t)\\ge 0 for every t, then\nInfinity can also be used to describe [infinite series](https://bliptext.com/articles/infinite-series), as follows:\nIn addition to defining a limit, infinity can be also used as a value in the extended real number system.\nPoints labeled +\\infty and -\\infty can be added to the [topological space](https://bliptext.com/articles/topological-space) of the real numbers, producing the two-point [compactification](https://bliptext.com/articles/compactification-mathematics) of the real numbers.\nAdding algebraic properties to this gives us the [extended real numbers](https://bliptext.com/articles/extended-real-number).\nWe can also treat +\\infty and -\\infty as the same, leading to the [one-point compactification](https://bliptext.com/articles/one-point-compactification) of the real numbers, which is the [real projective line](https://bliptext.com/articles/real-projective-line).\n[Projective geometry](https://bliptext.com/articles/projective-geometry) also refers to a [line at infinity](https://bliptext.com/articles/line-at-infinity) in plane geometry, a [plane at infinity](https://bliptext.com/articles/plane-at-infinity) in three-dimensional space, and a hyper[plane at infinity](https://bliptext.com/articles/plane-at-infinity) for general [dimensions](https://bliptext.com/articles/dimension-mathematics-and-physics), each consisting of [points at infinity](https://bliptext.com/articles/point-at-infinity).\n\n### Complex analysis\nIn [complex analysis](https://bliptext.com/articles/complex-analysis) the symbol \\infty, called \"infinity\", denotes an unsigned infinite [limit](https://bliptext.com/articles/limit-mathematics).\nThe expression means that the magnitude |x| of x grows beyond any assigned value.\nA [point labeled \\infty](https://bliptext.com/articles/point-at-infinity) can be added to the complex plane as a [topological space](https://bliptext.com/articles/topological-space) giving the [one-point compactification](https://bliptext.com/articles/one-point-compactification) of the complex plane.\nWhen this is done, the resulting space is a one-dimensional [complex manifold](https://bliptext.com/articles/complex-manifold), or [Riemann surface](https://bliptext.com/articles/riemann-surface), called the extended complex plane or the [Riemann sphere](https://bliptext.com/articles/riemann-sphere).\nArithmetic 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).\nOn the other hand, this kind of infinity enables [division by zero](https://bliptext.com/articles/division-by-zero), namely for any nonzero [complex number](https://bliptext.com/articles/complex-number) z.\nIn this context, it is often useful to consider [meromorphic functions](https://bliptext.com/articles/meromorphic-function) as maps into the Riemann sphere taking the value of \\infty at the poles.\nThe domain of a complex-valued function may be extended to include the point at infinity as well.\nOne important example of such functions is the group of [Möbius transformations](https://bliptext.com/articles/m-bius-transformation) (see [Möbius transformation § Overview](https://bliptext.com/articles/m-bius-transformation)).\n# [By 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. | upload.wikimedia.org/wikipedia/commons/3/32/Riemann///sphere1.svg]\n\n## Nonstandard analysis\nThe original formulation of [infinitesimal calculus](https://bliptext.com/articles/infinitesimal-calculus) by [Isaac Newton](https://bliptext.com/articles/isaac-newton) and Gottfried Leibniz used infinitesimal quantities.\nIn the second half of the 20th century, it was shown that this treatment could be put on a rigorous footing through various [logical systems](https://bliptext.com/articles/logical-system), including [smooth infinitesimal analysis](https://bliptext.com/articles/smooth-infinitesimal-analysis) and [nonstandard analysis](https://bliptext.com/articles/nonstandard-analysis).\nIn the latter, infinitesimals are invertible, and their inverses are infinite numbers.\nThe infinities in this sense are part of a [hyperreal field](https://bliptext.com/articles/hyperreal-number); there is no equivalence between them as with the Cantorian [transfinites](https://bliptext.com/articles/transfinite-number).\nFor example, if H is an infinite number in this sense, then H + H = 2H and H + 1 are distinct infinite numbers.\nThis approach to [non-standard calculus](https://bliptext.com/articles/non-standard-calculus) is fully developed in.\n# [450px | upload.wikimedia.org/wikipedia/commons/5/53/N%C3%BAmeros///hiperreales.png]\n\n## Set theory\nA different form of \"infinity\" is the [ordinal](https://bliptext.com/articles/ordinal-number) and [cardinal](https://bliptext.com/articles/cardinal-number) infinities of set theory—a system of [transfinite numbers](https://bliptext.com/articles/transfinite-number) first developed by [Georg Cantor](https://bliptext.com/articles/georg-cantor).\nIn this system, the first transfinite cardinal is [aleph-null](https://bliptext.com/articles/aleph-null) ( ℵ 0 ), the cardinality of the set of [natural numbers](https://bliptext.com/articles/natural-number).\nThis modern mathematical conception of the quantitative infinite developed in the late 19th century from works by Cantor, [Gottlob Frege](https://bliptext.com/articles/gottlob-frege), [Richard Dedekind](https://bliptext.com/articles/richard-dedekind) and others—using the idea of collections or sets.\nDedekind's approach was essentially to adopt the idea of [one-to-one correspondence](https://bliptext.com/articles/one-to-one-correspondence) as a standard for comparing the size of sets, and to reject the view of Galileo (derived from [Euclid](https://bliptext.com/articles/euclid)) that the whole cannot be the same size as the part.\n(However, see [Galileo's paradox](https://bliptext.com/articles/galileo-s-paradox) where Galileo concludes that positive integers cannot be compared to the subset of positive [square integers](https://bliptext.com/articles/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](https://bliptext.com/articles/proper-subset) parts; this notion of infinity is called [Dedekind infinite](https://bliptext.com/articles/dedekind-infinite).\nThe 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\".\nCantor defined two kinds of infinite numbers: [ordinal numbers](https://bliptext.com/articles/ordinal-number) and [cardinal numbers](https://bliptext.com/articles/cardinal-number).\nOrdinal numbers characterize [well-ordered](https://bliptext.com/articles/well-ordered) sets, or counting carried on to any stopping point, including points after an infinite number have already been counted.\nGeneralizing finite and (ordinary) infinite [sequences](https://bliptext.com/articles/sequence) which are maps from the positive [integers](https://bliptext.com/articles/integers) leads to [mappings](https://bliptext.com/articles/function-mathematics) from ordinal numbers to transfinite [sequences](https://bliptext.com/articles/sequence).\nCardinal 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.\nThe smallest ordinal infinity is that of the positive integers, and any set which has the cardinality of the integers is [countably infinite](https://bliptext.com/articles/countable-set).\nIf a set is too large to be put in one-to-one correspondence with the positive integers, it is called [uncountable](https://bliptext.com/articles/uncountable-set).\nCantor's views prevailed and modern mathematics accepts actual infinity as part of a consistent and coherent theory.\nCertain extended number systems, such as the hyperreal numbers, incorporate the ordinary (finite) numbers and infinite numbers of different sizes.\n\n### Cardinality of the continuum\nOne 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\nR\nthan natural numbers\nN\n. Namely, Cantor showed that.\nThe [continuum hypothesis](https://bliptext.com/articles/continuum-hypothesis) states that there is no [cardinal number](https://bliptext.com/articles/cardinal-number) between the cardinality of the reals and the cardinality of the natural numbers, that is, .This hypothesis cannot be proved or disproved within the widely accepted [Zermelo–Fraenkel set theory](https://bliptext.com/articles/zermelo-fraenkel-set-theory), even assuming the [Axiom of Choice](https://bliptext.com/articles/axiom-of-choice).\n[Cardinal arithmetic](https://bliptext.com/articles/cardinal-arithmetic) can be used to show not only that the number of points in a [real number line](https://bliptext.com/articles/real-number-line) is equal to the number of points in any [segment of that line](https://bliptext.com/articles/line-segment), but also that this is equal to the number of points on a plane and, indeed, in any [finite-dimensional](https://bliptext.com/articles/finite-dimensional) space.\nThe first of these results is apparent by considering, for instance, the [tangent](https://bliptext.com/articles/tangent-trigonometric-function) function, which provides a [one-to-one correspondence](https://bliptext.com/articles/one-to-one-correspondence) between the [interval](https://bliptext.com/articles/interval-mathematics) (\n−π⁄2, π⁄2\n) and\nR\n.The second result was proved by Cantor in 1878, but only became intuitively apparent in 1890, when [Giuseppe Peano](https://bliptext.com/articles/giuseppe-peano) introduced the [space-filling curves](https://bliptext.com/articles/space-filling-curve), curved lines that twist and turn enough to fill the whole of any square, or [cube](https://bliptext.com/articles/cube), or hyper[cube](https://bliptext.com/articles/cube), or finite-dimensional space.\nThese 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.\n# [The first three steps of a fractal construction whose limit is a space-filling curve, showing that there are as many points in a one-dimensional line as in a two-dimensional square | upload.wikimedia.org/wikipedia/commons/6/64/Peanocurve.svg]\n\n## Geometry\nUntil the end of the 19th century, infinity was rarely discussed in [geometry](https://bliptext.com/articles/geometry), except in the context of processes that could be continued without any limit.\nFor example, a [line](https://bliptext.com/articles/line-geometry) was what is now called a [line](https://bliptext.com/articles/line-geometry) segment, with the proviso that one can extend it as far as one wants; but extending it infinitely was out of the question.\nSimilarly, a line was usually not considered to be composed of infinitely many points but was a location where a point may be placed.\nEven if there are infinitely many possible positions, only a finite number of points could be placed on a line.\nA witness of this is the expression \"the [locus](https://bliptext.com/articles/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).\nOne of the rare exceptions of a mathematical concept involving [actual infinity](https://bliptext.com/articles/actual-infinity) was [projective geometry](https://bliptext.com/articles/projective-geometry), where [points at infinity](https://bliptext.com/articles/points-at-infinity) are added to the [Euclidean space](https://bliptext.com/articles/euclidean-space) for modeling the [perspective](https://bliptext.com/articles/perspective-graphical) effect that shows [parallel lines](https://bliptext.com/articles/parallel-lines) intersecting \"at infinity\".\nMathematically, points at infinity have the advantage of allowing one to not consider some special cases.\nFor example, in a [projective plane](https://bliptext.com/articles/projective-plane), two distinct [lines](https://bliptext.com/articles/line-geometry) intersect in exactly one point, whereas without points at infinity, there are no intersection points for parallel [lines](https://bliptext.com/articles/line-geometry).\nSo, parallel and non-parallel lines must be studied separately in classical geometry, while they need not be distinguished in projective geometry.\nBefore the use of [set theory](https://bliptext.com/articles/set-theory) for the [foundation of mathematics](https://bliptext.com/articles/foundation-of-mathematics), points and lines were viewed as distinct entities, and a point could be located on a line.\nWith 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).\nIn particular, in modern mathematics, lines are infinite sets.\n\n## Infinite dimension\nThe [vector spaces](https://bliptext.com/articles/vector-space) that occur in classical [geometry](https://bliptext.com/articles/geometry) have always a finite [dimension](https://bliptext.com/articles/dimension-vector-space), generally two or three.\nHowever, this is not implied by the abstract definition of a vector space, and vector spaces of infinite dimension can be considered.\nThis is typically the case in [functional analysis](https://bliptext.com/articles/functional-analysis) where [function spaces](https://bliptext.com/articles/function-space) are generally vector spaces of infinite dimension.\nIn topology, some constructions can generate [topological spaces](https://bliptext.com/articles/topological-space) of infinite dimension.\nIn particular, this is the case of [iterated loop spaces](https://bliptext.com/articles/iterated-loop-space).\n\n## Fractals\nThe structure of a [fractal](https://bliptext.com/articles/fractal) object is reiterated in its magnifications.\nFractals can be magnified indefinitely without losing their structure and becoming \"smooth\"; they have infinite perimeters and can have infinite or finite areas.\nOne such [fractal curve](https://bliptext.com/articles/fractal-curve) with an infinite perimeter and finite area is the [Koch snowflake](https://bliptext.com/articles/koch-snowflake).\n\n## Mathematics without infinity\n[Leopold Kronecker](https://bliptext.com/articles/leopold-kronecker) was skeptical of the notion of infinity and how his fellow mathematicians were using it in the 1870s and 1880s.\nThis skepticism was developed in the [philosophy of mathematics](https://bliptext.com/articles/philosophy-of-mathematics) called [finitism](https://bliptext.com/articles/finitism), an extreme form of mathematical philosophy in the general philosophical and mathematical schools of [constructivism](https://bliptext.com/articles/mathematical-constructivism) and [intuitionism](https://bliptext.com/articles/intuitionism).\n\n# Physics\nIn [physics](https://bliptext.com/articles/physics), approximations of [real numbers](https://bliptext.com/articles/real-number) are used for [continuous](https://bliptext.com/articles/continuum-theory) measurements and [natural numbers](https://bliptext.com/articles/natural-number) are used for [discrete](https://bliptext.com/articles/countable) measurements (i.e., counting).\nConcepts of infinite things such as an infinite [plane wave](https://bliptext.com/articles/plane-wave) exist, but there are no experimental means to generate them.\n\n## Cosmology\nThe first published proposal that the universe is infinite came from Thomas Digges in 1576.\nEight years later, in 1584, the Italian philosopher and astronomer [Giordano Bruno](https://bliptext.com/articles/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.\"\n[Cosmologists](https://bliptext.com/articles/cosmology) have long sought to discover whether infinity exists in our physical [universe](https://bliptext.com/articles/universe): Are there an infinite number of stars?\nDoes the universe have infinite volume?\nDoes space \"[go on forever](https://bliptext.com/articles/shape-of-the-universe)\"?\nThis is still an open question of [cosmology](https://bliptext.com/articles/physical-cosmology).\nThe question of being infinite is logically separate from the question of having boundaries.\nThe two-dimensional surface of the Earth, for example, is finite, yet has no edge.\nBy travelling in a straight line with respect to the Earth's curvature, one will eventually return to the exact spot one started from.\nThe universe, at least in principle, might have a similar [topology](https://bliptext.com/articles/topology).\nIf so, one might eventually return to one's starting point after travelling in a straight line through the universe for long enough.\nThe curvature of the universe can be measured through [multipole moments](https://bliptext.com/articles/multipole-moments) in the spectrum of the [cosmic background radiation](https://bliptext.com/articles/cosmic-microwave-background-radiation).\nTo date, analysis of the radiation patterns recorded by the [WMAP](https://bliptext.com/articles/wmap) spacecraft hints that the universe has a flat topology.\nThis would be consistent with an infinite physical universe.\nHowever, the universe could be finite, even if its curvature is flat.\nAn 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.\nThe topology of such games is [toroidal](https://bliptext.com/articles/torus) and the geometry is flat.\nMany possible bounded, flat possibilities also exist for three-dimensional space.\nThe concept of infinity also extends to the [multiverse](https://bliptext.com/articles/multiverse) hypothesis, which, when explained by astrophysicists such as [Michio Kaku](https://bliptext.com/articles/michio-kaku), posits that there are an infinite number and variety of universes.\nAlso, [cyclic models](https://bliptext.com/articles/cyclic-model) posit an infinite amount of [Big Bangs](https://bliptext.com/articles/big-bang), resulting in an infinite variety of universes after each Big Bang event in an infinite cycle.\n\n# Logic\nIn [logic](https://bliptext.com/articles/logic), an [infinite regress](https://bliptext.com/articles/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.\"\n\n# Computing\nThe [IEEE floating-point](https://bliptext.com/articles/ieee-floating-point) standard (IEEE 754) specifies a positive and a negative infinity value (and also [indefinite](https://bliptext.com/articles/nan) values).\nThese are defined as the result of [arithmetic overflow](https://bliptext.com/articles/arithmetic-overflow), [division by zero](https://bliptext.com/articles/division-by-zero), and other exceptional operations.\nSome [programming languages](https://bliptext.com/articles/programming-language), such as [Java](https://bliptext.com/articles/java-programming-language) and [J](https://bliptext.com/articles/j-programming-language), allow the programmer an explicit access to the positive and negative infinity values as language constants.\nThese can be used as [greatest and least elements](https://bliptext.com/articles/greatest-element), as they compare (respectively) greater than or less than all other values.\nThey have uses as [sentinel values](https://bliptext.com/articles/sentinel-value) in [algorithms](https://bliptext.com/articles/algorithm) involving [sorting](https://bliptext.com/articles/sorting), [searching](https://bliptext.com/articles/search-algorithm), or [windowing](https://bliptext.com/articles/window-function).\nIn languages that do not have greatest and least elements but do allow [overloading](https://bliptext.com/articles/operator-overloading) of [relational operators](https://bliptext.com/articles/relational-operator), it is possible for a programmer to create the greatest and least elements.\nIn 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](https://bliptext.com/articles/data-type), the infinity values may still be accessible and usable as the result of certain operations.\nIn programming, an [infinite loop](https://bliptext.com/articles/infinite-[loop](https://bliptext.com/articles/loop-computing)) is a [loop](https://bliptext.com/articles/loop-computing) whose exit condition is never satisfied, thus executing indefinitely.\n\n# Arts, games, and cognitive sciences\n[Perspective](https://bliptext.com/articles/perspective-graphical) artwork uses the concept of [vanishing points](https://bliptext.com/articles/vanishing-point), roughly corresponding to mathematical [points at infinity](https://bliptext.com/articles/point-at-infinity), located at an infinite distance from the observer.\nThis allows artists to create paintings that realistically render space, distances, and forms.\nArtist [M.C. Escher](https://bliptext.com/articles/m-c-escher) is specifically known for employing the concept of infinity in his work in this and other ways.\nVariations of [chess](https://bliptext.com/articles/chess) played on an unbounded board are called infinite [chess](https://bliptext.com/articles/chess).\n[Cognitive scientist](https://bliptext.com/articles/cognitive-science) [George Lakoff](https://bliptext.com/articles/george-lakoff) considers the concept of infinity in mathematics and the sciences as a metaphor.\nThis perspective is based on the basic metaphor of infinity (BMI), defined as the ever-increasing sequence <1, 2, 3, …>.\n\n## Sources","wordIndex":1,"createdAt":"2026-01-27T12:07:36.971Z","user":{"id":"mfvswiH9fwHI8f36yNwHR40968SZeIKD","name":"den3den42069","image":"https://cdn.discordapp.com/avatars/975452809707278427/5e0588882214f34dc5f3b36f7e9c9ff5.png"}},{"id":"2bafc8e7-00c8-4cfa-9d83-57b53ac2c197","wordChanged":"**Infinity**","newWord":":::summary\n:::\n\n# \n4 is something which is boundless, endless, or larger than any [natural number](https://bliptext.com/articles/natural-number).\nIt is often denoted by the [infinity symbol](https://bliptext.com/articles/infinity-symbol) **INFINITY**\nFrom the time of the [ancient Greeks](https://bliptext.com/articles/greek-mathematics), the [philosophical nature of infinity](https://bliptext.com/articles/infinity-philosophy) has been the subject of many discussions among philosophers.\nIn the 17th century, with the introduction of the infinity symbol and the [infinitesimal calculus](https://bliptext.com/articles/infinitesimal-calculus), mathematicians began to work with [infinite series](https://bliptext.com/articles/infinite-series) and what some mathematicians (including [l'Hôpital](https://bliptext.com/articles/guillaume-de-l-h-pital) and [Bernoulli](https://bliptext.com/articles/johann-bernoulli)) regarded as infinitely small quantities, but infinity continued to be associated with endless processes.\nAs mathematicians struggled with the foundation of [calculus](https://bliptext.com/articles/calculus), it remained unclear whether infinity could be considered as a number or [magnitude](https://bliptext.com/articles/magnitude-mathematics) and, if so, how this could be done.\nAt the end of the 19th century, [Georg Cantor](https://bliptext.com/articles/georg-cantor) enlarged the mathematical study of infinity by studying [infinite sets](https://bliptext.com/articles/infinite-set) and [infinite numbers](https://bliptext.com/articles/transfinite-number), showing that they can be of various sizes.\nFor example, if a line is viewed as the set of all of its points, their infinite number (i.e., the [cardinality](https://bliptext.com/articles/cardinality) of the line) is larger than the number of [integers](https://bliptext.com/articles/integer).\nIn this usage, infinity is a mathematical concept, and infinite [mathematical objects](https://bliptext.com/articles/mathematical-object) can be studied, manipulated, and used just like any other mathematical object.\nThe mathematical concept of infinity refines and extends the old philosophical concept, in particular by introducing infinitely many different sizes of infinite sets.\nAmong the axioms of [Zermelo–Fraenkel set theory](https://bliptext.com/articles/zermelo-fraenkel-set-theory), on which most of modern mathematics can be developed, is the [axiom of infinity](https://bliptext.com/articles/axiom-of-infinity), which guarantees the existence of infinite sets.\nThe mathematical concept of infinity and the manipulation of infinite sets are widely used in mathematics, even in areas such as [combinatorics](https://bliptext.com/articles/combinatorics) that may seem to have nothing to do with them.\nFor example, [Wiles's proof of Fermat's Last Theorem](https://bliptext.com/articles/wiles-s-proof-of-fermat-s-last-theorem) implicitly relies on the existence of [Grothendieck universes](https://bliptext.com/articles/grothendieck-universe), very large infinite sets, for solving a long-standing problem that is stated in terms of [elementary arithmetic](https://bliptext.com/articles/elementary-arithmetic).\nIn [physics](https://bliptext.com/articles/physics) and [cosmology](https://bliptext.com/articles/cosmology), [whether the universe is spatially infinite or not](https://bliptext.com/articles/universe), is an open question.\n# [The Sierpiński Triangle contains an infinite amount of itself because of its recursive pattern. | upload.wikimedia.org/wikipedia/commons/c/c5/Sierpi%C5%84ski///Triangle.png]\n# [Due to the constant light reflection between opposing mirrors, it seems that there is a boundless amount of space and repetition inside of them. | upload.wikimedia.org/wikipedia/commons/9/9e/Reflections///1090029.jpg]\n\n# History\nAncient cultures had various ideas about the nature of infinity.\nThe [ancient Indians](https://bliptext.com/articles/vedic-period) and the [Greeks](https://bliptext.com/articles/ancient-greece) did not define infinity in precise formalism as does modern mathematics, and instead approached infinity as a philosophical concept.\n\n## Early Greek\nThe earliest recorded idea of infinity in Greece may be that of [Anaximander](https://bliptext.com/articles/anaximander) (c. 610 – c. 546 BC) a [pre-Socratic](https://bliptext.com/articles/pre-socratic-philosophy) Greek philosopher.\nHe used the word [apeiron](https://bliptext.com/articles/apeiron), which means \"unbounded\", \"indefinite\", and perhaps can be translated as \"infinite\".\n[Aristotle](https://bliptext.com/articles/aristotle) (350 BC) distinguished potential infinity from [actual infinity](https://bliptext.com/articles/actual-infinity), which he regarded as impossible due to the various [paradoxes](https://bliptext.com/articles/paradoxes) it seemed to produce.\nIt has been argued that, in line with this view, the [Hellenistic](https://bliptext.com/articles/hellenistic) Greeks had a \"horror of the infinite\" which would, for example, explain why [Euclid](https://bliptext.com/articles/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.\"\nIt has also been maintained, that, in proving the [infinitude of the prime numbers](https://bliptext.com/articles/infinitude-of-the-prime-numbers), Euclid \"was the first to overcome the horror of the infinite\".\nThere is a similar controversy concerning Euclid's [parallel postulate](https://bliptext.com/articles/parallel-postulate), sometimes translated:\n\"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.\"\nOther translators, however, prefer the translation \"the two straight lines, if produced indefinitely ...\", thus avoiding the implication that Euclid was comfortable with the notion of infinity.\nFinally, 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.\n\n## Zeno: Achilles and the tortoise\n[Zeno of Elea](https://bliptext.com/articles/zeno-of-elea) (c. 495 – c. 430 BC) did not advance any views concerning the infinite.\nNevertheless, his paradoxes, especially \"Achilles and the Tortoise\", were important contributions in that they made clear the inadequacy of popular conceptions.\nThe paradoxes were described by [Bertrand Russell](https://bliptext.com/articles/bertrand-russell) as \"immeasurably subtle and profound\".\n[Achilles](https://bliptext.com/articles/achilles) races a tortoise, giving the latter a head start.\nEtc.\nApparently, Achilles never overtakes the tortoise, since however many steps he completes, the tortoise remains ahead of him.\nZeno was not attempting to make a point about infinity.\nAs a member of the [Eleatics](https://bliptext.com/articles/eleatic) school which regarded motion as an illusion, he saw it as a mistake to suppose that Achilles could run at all.\nSubsequent thinkers, finding this solution unacceptable, struggled for over two millennia to find other weaknesses in the argument.\nFinally, in 1821, [Augustin-Louis Cauchy](https://bliptext.com/articles/augustin-louis-cauchy) provided both a satisfactory definition of a limit and a proof that, for\n0 < x < 1\n,\nSuppose 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.\nThe duration of the chase fits Cauchy's pattern with\na = 10 seconds\nand\nx = 0.01\n. Achilles does overtake the tortoise; it takes him\n\n## Early Indian\nThe [Jain mathematical](https://bliptext.com/articles/indian-mathematics) text Surya Prajnapti (c. 4th–3rd century BCE) classifies all numbers into three sets: [enumerable](https://bliptext.com/articles/enumerable), innumerable, and infinite.\nEach of these was further subdivided into three orders:\n\n## 17th century\nIn the 17th century, European mathematicians started using infinite numbers and infinite expressions in a systematic fashion.\nIn 1655, [John Wallis](https://bliptext.com/articles/john-wallis) first used the notation \\infty for such a number in his De sectionibus conicis, and exploited it in area calculations by dividing the region into [infinitesimal](https://bliptext.com/articles/infinitesimal) strips of width on the order of But in Arithmetica infinitorum (1656), 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.\"\nIn 1699, [Isaac Newton](https://bliptext.com/articles/isaac-newton) wrote about equations with an infinite number of terms in his work [De analysi per aequationes numero terminorum infinitas](https://bliptext.com/articles/de-analysi-per-aequationes-numero-terminorum-infinitas).\n\n# Mathematics\n[Hermann Weyl](https://bliptext.com/articles/hermann-weyl) opened a mathematico-philosophic address given in 1930 with:\n\"Mathematics is the science of the infinite.\"\n\n## Symbol\nThe infinity symbol \\infty (sometimes called the [lemniscate](https://bliptext.com/articles/lemniscate)) is a mathematical symbol representing the concept of infinity.\nThe symbol is encoded in [Unicode](https://bliptext.com/articles/unicode) at and in [LaTeX](https://bliptext.com/articles/latex) as.\nIt was introduced in 1655 by [John Wallis](https://bliptext.com/articles/john-wallis), and since its introduction, it has also been used outside mathematics in modern mysticism and literary [symbology](https://bliptext.com/articles/symbology).\n\n## Calculus\n[Gottfried Leibniz](https://bliptext.com/articles/gottfried-wilhelm-leibniz), one of the co-inventors of [infinitesimal calculus](https://bliptext.com/articles/infinitesimal-calculus), speculated widely about infinite numbers and their use in mathematics.\nTo 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](https://bliptext.com/articles/law-of-continuity).\n\n### Real analysis\nIn [real analysis](https://bliptext.com/articles/real-analysis), the symbol \\infty, called \"infinity\", is used to denote an unbounded [limit](https://bliptext.com/articles/limit-of-a-function).\nThe notation means that x increases without bound, and means that x decreases without bound.\nFor example, if f(t)\\ge 0 for every t, then\nInfinity can also be used to describe [infinite series](https://bliptext.com/articles/infinite-series), as follows:\nIn addition to defining a limit, infinity can be also used as a value in the extended real number system.\nPoints labeled +\\infty and -\\infty can be added to the [topological space](https://bliptext.com/articles/topological-space) of the real numbers, producing the two-point [compactification](https://bliptext.com/articles/compactification-mathematics) of the real numbers.\nAdding algebraic properties to this gives us the [extended real numbers](https://bliptext.com/articles/extended-real-number).\nWe can also treat +\\infty and -\\infty as the same, leading to the [one-point compactification](https://bliptext.com/articles/one-point-compactification) of the real numbers, which is the [real projective line](https://bliptext.com/articles/real-projective-line).\n[Projective geometry](https://bliptext.com/articles/projective-geometry) also refers to a [line at infinity](https://bliptext.com/articles/line-at-infinity) in plane geometry, a [plane at infinity](https://bliptext.com/articles/plane-at-infinity) in three-dimensional space, and a hyper[plane at infinity](https://bliptext.com/articles/plane-at-infinity) for general [dimensions](https://bliptext.com/articles/dimension-mathematics-and-physics), each consisting of [points at infinity](https://bliptext.com/articles/point-at-infinity).\n\n### Complex analysis\nIn [complex analysis](https://bliptext.com/articles/complex-analysis) the symbol \\infty, called \"infinity\", denotes an unsigned infinite [limit](https://bliptext.com/articles/limit-mathematics).\nThe expression means that the magnitude |x| of x grows beyond any assigned value.\nA [point labeled \\infty](https://bliptext.com/articles/point-at-infinity) can be added to the complex plane as a [topological space](https://bliptext.com/articles/topological-space) giving the [one-point compactification](https://bliptext.com/articles/one-point-compactification) of the complex plane.\nWhen this is done, the resulting space is a one-dimensional [complex manifold](https://bliptext.com/articles/complex-manifold), or [Riemann surface](https://bliptext.com/articles/riemann-surface), called the extended complex plane or the [Riemann sphere](https://bliptext.com/articles/riemann-sphere).\nArithmetic 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).\nOn the other hand, this kind of infinity enables [division by zero](https://bliptext.com/articles/division-by-zero), namely for any nonzero [complex number](https://bliptext.com/articles/complex-number) z.\nIn this context, it is often useful to consider [meromorphic functions](https://bliptext.com/articles/meromorphic-function) as maps into the Riemann sphere taking the value of \\infty at the poles.\nThe domain of a complex-valued function may be extended to include the point at infinity as well.\nOne important example of such functions is the group of [Möbius transformations](https://bliptext.com/articles/m-bius-transformation) (see [Möbius transformation § Overview](https://bliptext.com/articles/m-bius-transformation)).\n# [By 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. | upload.wikimedia.org/wikipedia/commons/3/32/Riemann///sphere1.svg]\n\n## Nonstandard analysis\nThe original formulation of [infinitesimal calculus](https://bliptext.com/articles/infinitesimal-calculus) by [Isaac Newton](https://bliptext.com/articles/isaac-newton) and Gottfried Leibniz used infinitesimal quantities.\nIn the second half of the 20th century, it was shown that this treatment could be put on a rigorous footing through various [logical systems](https://bliptext.com/articles/logical-system), including [smooth infinitesimal analysis](https://bliptext.com/articles/smooth-infinitesimal-analysis) and [nonstandard analysis](https://bliptext.com/articles/nonstandard-analysis).\nIn the latter, infinitesimals are invertible, and their inverses are infinite numbers.\nThe infinities in this sense are part of a [hyperreal field](https://bliptext.com/articles/hyperreal-number); there is no equivalence between them as with the Cantorian [transfinites](https://bliptext.com/articles/transfinite-number).\nFor example, if H is an infinite number in this sense, then H + H = 2H and H + 1 are distinct infinite numbers.\nThis approach to [non-standard calculus](https://bliptext.com/articles/non-standard-calculus) is fully developed in.\n# [450px | upload.wikimedia.org/wikipedia/commons/5/53/N%C3%BAmeros///hiperreales.png]\n\n## Set theory\nA different form of \"infinity\" is the [ordinal](https://bliptext.com/articles/ordinal-number) and [cardinal](https://bliptext.com/articles/cardinal-number) infinities of set theory—a system of [transfinite numbers](https://bliptext.com/articles/transfinite-number) first developed by [Georg Cantor](https://bliptext.com/articles/georg-cantor).\nIn this system, the first transfinite cardinal is [aleph-null](https://bliptext.com/articles/aleph-null) ( ℵ 0 ), the cardinality of the set of [natural numbers](https://bliptext.com/articles/natural-number).\nThis modern mathematical conception of the quantitative infinite developed in the late 19th century from works by Cantor, [Gottlob Frege](https://bliptext.com/articles/gottlob-frege), [Richard Dedekind](https://bliptext.com/articles/richard-dedekind) and others—using the idea of collections or sets.\nDedekind's approach was essentially to adopt the idea of [one-to-one correspondence](https://bliptext.com/articles/one-to-one-correspondence) as a standard for comparing the size of sets, and to reject the view of Galileo (derived from [Euclid](https://bliptext.com/articles/euclid)) that the whole cannot be the same size as the part.\n(However, see [Galileo's paradox](https://bliptext.com/articles/galileo-s-paradox) where Galileo concludes that positive integers cannot be compared to the subset of positive [square integers](https://bliptext.com/articles/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](https://bliptext.com/articles/proper-subset) parts; this notion of infinity is called [Dedekind infinite](https://bliptext.com/articles/dedekind-infinite).\nThe 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\".\nCantor defined two kinds of infinite numbers: [ordinal numbers](https://bliptext.com/articles/ordinal-number) and [cardinal numbers](https://bliptext.com/articles/cardinal-number).\nOrdinal numbers characterize [well-ordered](https://bliptext.com/articles/well-ordered) sets, or counting carried on to any stopping point, including points after an infinite number have already been counted.\nGeneralizing finite and (ordinary) infinite [sequences](https://bliptext.com/articles/sequence) which are maps from the positive [integers](https://bliptext.com/articles/integers) leads to [mappings](https://bliptext.com/articles/function-mathematics) from ordinal numbers to transfinite [sequences](https://bliptext.com/articles/sequence).\nCardinal 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.\nThe smallest ordinal infinity is that of the positive integers, and any set which has the cardinality of the integers is [countably infinite](https://bliptext.com/articles/countable-set).\nIf a set is too large to be put in one-to-one correspondence with the positive integers, it is called [uncountable](https://bliptext.com/articles/uncountable-set).\nCantor's views prevailed and modern mathematics accepts actual infinity as part of a consistent and coherent theory.\nCertain extended number systems, such as the hyperreal numbers, incorporate the ordinary (finite) numbers and infinite numbers of different sizes.\n\n### Cardinality of the continuum\nOne 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\nR\nthan natural numbers\nN\n. Namely, Cantor showed that.\nThe [continuum hypothesis](https://bliptext.com/articles/continuum-hypothesis) states that there is no [cardinal number](https://bliptext.com/articles/cardinal-number) between the cardinality of the reals and the cardinality of the natural numbers, that is, .This hypothesis cannot be proved or disproved within the widely accepted [Zermelo–Fraenkel set theory](https://bliptext.com/articles/zermelo-fraenkel-set-theory), even assuming the [Axiom of Choice](https://bliptext.com/articles/axiom-of-choice).\n[Cardinal arithmetic](https://bliptext.com/articles/cardinal-arithmetic) can be used to show not only that the number of points in a [real number line](https://bliptext.com/articles/real-number-line) is equal to the number of points in any [segment of that line](https://bliptext.com/articles/line-segment), but also that this is equal to the number of points on a plane and, indeed, in any [finite-dimensional](https://bliptext.com/articles/finite-dimensional) space.\nThe first of these results is apparent by considering, for instance, the [tangent](https://bliptext.com/articles/tangent-trigonometric-function) function, which provides a [one-to-one correspondence](https://bliptext.com/articles/one-to-one-correspondence) between the [interval](https://bliptext.com/articles/interval-mathematics) (\n−π⁄2, π⁄2\n) and\nR\n.The second result was proved by Cantor in 1878, but only became intuitively apparent in 1890, when [Giuseppe Peano](https://bliptext.com/articles/giuseppe-peano) introduced the [space-filling curves](https://bliptext.com/articles/space-filling-curve), curved lines that twist and turn enough to fill the whole of any square, or [cube](https://bliptext.com/articles/cube), or hyper[cube](https://bliptext.com/articles/cube), or finite-dimensional space.\nThese 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.\n# [The first three steps of a fractal construction whose limit is a space-filling curve, showing that there are as many points in a one-dimensional line as in a two-dimensional square | upload.wikimedia.org/wikipedia/commons/6/64/Peanocurve.svg]\n\n## Geometry\nUntil the end of the 19th century, infinity was rarely discussed in [geometry](https://bliptext.com/articles/geometry), except in the context of processes that could be continued without any limit.\nFor example, a [line](https://bliptext.com/articles/line-geometry) was what is now called a [line](https://bliptext.com/articles/line-geometry) segment, with the proviso that one can extend it as far as one wants; but extending it infinitely was out of the question.\nSimilarly, a line was usually not considered to be composed of infinitely many points but was a location where a point may be placed.\nEven if there are infinitely many possible positions, only a finite number of points could be placed on a line.\nA witness of this is the expression \"the [locus](https://bliptext.com/articles/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).\nOne of the rare exceptions of a mathematical concept involving [actual infinity](https://bliptext.com/articles/actual-infinity) was [projective geometry](https://bliptext.com/articles/projective-geometry), where [points at infinity](https://bliptext.com/articles/points-at-infinity) are added to the [Euclidean space](https://bliptext.com/articles/euclidean-space) for modeling the [perspective](https://bliptext.com/articles/perspective-graphical) effect that shows [parallel lines](https://bliptext.com/articles/parallel-lines) intersecting \"at infinity\".\nMathematically, points at infinity have the advantage of allowing one to not consider some special cases.\nFor example, in a [projective plane](https://bliptext.com/articles/projective-plane), two distinct [lines](https://bliptext.com/articles/line-geometry) intersect in exactly one point, whereas without points at infinity, there are no intersection points for parallel [lines](https://bliptext.com/articles/line-geometry).\nSo, parallel and non-parallel lines must be studied separately in classical geometry, while they need not be distinguished in projective geometry.\nBefore the use of [set theory](https://bliptext.com/articles/set-theory) for the [foundation of mathematics](https://bliptext.com/articles/foundation-of-mathematics), points and lines were viewed as distinct entities, and a point could be located on a line.\nWith 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).\nIn particular, in modern mathematics, lines are infinite sets.\n\n## Infinite dimension\nThe [vector spaces](https://bliptext.com/articles/vector-space) that occur in classical [geometry](https://bliptext.com/articles/geometry) have always a finite [dimension](https://bliptext.com/articles/dimension-vector-space), generally two or three.\nHowever, this is not implied by the abstract definition of a vector space, and vector spaces of infinite dimension can be considered.\nThis is typically the case in [functional analysis](https://bliptext.com/articles/functional-analysis) where [function spaces](https://bliptext.com/articles/function-space) are generally vector spaces of infinite dimension.\nIn topology, some constructions can generate [topological spaces](https://bliptext.com/articles/topological-space) of infinite dimension.\nIn particular, this is the case of [iterated loop spaces](https://bliptext.com/articles/iterated-loop-space).\n\n## Fractals\nThe structure of a [fractal](https://bliptext.com/articles/fractal) object is reiterated in its magnifications.\nFractals can be magnified indefinitely without losing their structure and becoming \"smooth\"; they have infinite perimeters and can have infinite or finite areas.\nOne such [fractal curve](https://bliptext.com/articles/fractal-curve) with an infinite perimeter and finite area is the [Koch snowflake](https://bliptext.com/articles/koch-snowflake).\n\n## Mathematics without infinity\n[Leopold Kronecker](https://bliptext.com/articles/leopold-kronecker) was skeptical of the notion of infinity and how his fellow mathematicians were using it in the 1870s and 1880s.\nThis skepticism was developed in the [philosophy of mathematics](https://bliptext.com/articles/philosophy-of-mathematics) called [finitism](https://bliptext.com/articles/finitism), an extreme form of mathematical philosophy in the general philosophical and mathematical schools of [constructivism](https://bliptext.com/articles/mathematical-constructivism) and [intuitionism](https://bliptext.com/articles/intuitionism).\n\n# Physics\nIn [physics](https://bliptext.com/articles/physics), approximations of [real numbers](https://bliptext.com/articles/real-number) are used for [continuous](https://bliptext.com/articles/continuum-theory) measurements and [natural numbers](https://bliptext.com/articles/natural-number) are used for [discrete](https://bliptext.com/articles/countable) measurements (i.e., counting).\nConcepts of infinite things such as an infinite [plane wave](https://bliptext.com/articles/plane-wave) exist, but there are no experimental means to generate them.\n\n## Cosmology\nThe first published proposal that the universe is infinite came from Thomas Digges in 1576.\nEight years later, in 1584, the Italian philosopher and astronomer [Giordano Bruno](https://bliptext.com/articles/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.\"\n[Cosmologists](https://bliptext.com/articles/cosmology) have long sought to discover whether infinity exists in our physical [universe](https://bliptext.com/articles/universe): Are there an infinite number of stars?\nDoes the universe have infinite volume?\nDoes space \"[go on forever](https://bliptext.com/articles/shape-of-the-universe)\"?\nThis is still an open question of [cosmology](https://bliptext.com/articles/physical-cosmology).\nThe question of being infinite is logically separate from the question of having boundaries.\nThe two-dimensional surface of the Earth, for example, is finite, yet has no edge.\nBy travelling in a straight line with respect to the Earth's curvature, one will eventually return to the exact spot one started from.\nThe universe, at least in principle, might have a similar [topology](https://bliptext.com/articles/topology).\nIf so, one might eventually return to one's starting point after travelling in a straight line through the universe for long enough.\nThe curvature of the universe can be measured through [multipole moments](https://bliptext.com/articles/multipole-moments) in the spectrum of the [cosmic background radiation](https://bliptext.com/articles/cosmic-microwave-background-radiation).\nTo date, analysis of the radiation patterns recorded by the [WMAP](https://bliptext.com/articles/wmap) spacecraft hints that the universe has a flat topology.\nThis would be consistent with an infinite physical universe.\nHowever, the universe could be finite, even if its curvature is flat.\nAn 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.\nThe topology of such games is [toroidal](https://bliptext.com/articles/torus) and the geometry is flat.\nMany possible bounded, flat possibilities also exist for three-dimensional space.\nThe concept of infinity also extends to the [multiverse](https://bliptext.com/articles/multiverse) hypothesis, which, when explained by astrophysicists such as [Michio Kaku](https://bliptext.com/articles/michio-kaku), posits that there are an infinite number and variety of universes.\nAlso, [cyclic models](https://bliptext.com/articles/cyclic-model) posit an infinite amount of [Big Bangs](https://bliptext.com/articles/big-bang), resulting in an infinite variety of universes after each Big Bang event in an infinite cycle.\n\n# Logic\nIn [logic](https://bliptext.com/articles/logic), an [infinite regress](https://bliptext.com/articles/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.\"\n\n# Computing\nThe [IEEE floating-point](https://bliptext.com/articles/ieee-floating-point) standard (IEEE 754) specifies a positive and a negative infinity value (and also [indefinite](https://bliptext.com/articles/nan) values).\nThese are defined as the result of [arithmetic overflow](https://bliptext.com/articles/arithmetic-overflow), [division by zero](https://bliptext.com/articles/division-by-zero), and other exceptional operations.\nSome [programming languages](https://bliptext.com/articles/programming-language), such as [Java](https://bliptext.com/articles/java-programming-language) and [J](https://bliptext.com/articles/j-programming-language), allow the programmer an explicit access to the positive and negative infinity values as language constants.\nThese can be used as [greatest and least elements](https://bliptext.com/articles/greatest-element), as they compare (respectively) greater than or less than all other values.\nThey have uses as [sentinel values](https://bliptext.com/articles/sentinel-value) in [algorithms](https://bliptext.com/articles/algorithm) involving [sorting](https://bliptext.com/articles/sorting), [searching](https://bliptext.com/articles/search-algorithm), or [windowing](https://bliptext.com/articles/window-function).\nIn languages that do not have greatest and least elements but do allow [overloading](https://bliptext.com/articles/operator-overloading) of [relational operators](https://bliptext.com/articles/relational-operator), it is possible for a programmer to create the greatest and least elements.\nIn 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](https://bliptext.com/articles/data-type), the infinity values may still be accessible and usable as the result of certain operations.\nIn programming, an [infinite loop](https://bliptext.com/articles/infinite-[loop](https://bliptext.com/articles/loop-computing)) is a [loop](https://bliptext.com/articles/loop-computing) whose exit condition is never satisfied, thus executing indefinitely.\n\n# Arts, games, and cognitive sciences\n[Perspective](https://bliptext.com/articles/perspective-graphical) artwork uses the concept of [vanishing points](https://bliptext.com/articles/vanishing-point), roughly corresponding to mathematical [points at infinity](https://bliptext.com/articles/point-at-infinity), located at an infinite distance from the observer.\nThis allows artists to create paintings that realistically render space, distances, and forms.\nArtist [M.C. Escher](https://bliptext.com/articles/m-c-escher) is specifically known for employing the concept of infinity in his work in this and other ways.\nVariations of [chess](https://bliptext.com/articles/chess) played on an unbounded board are called infinite [chess](https://bliptext.com/articles/chess).\n[Cognitive scientist](https://bliptext.com/articles/cognitive-science) [George Lakoff](https://bliptext.com/articles/george-lakoff) considers the concept of infinity in mathematics and the sciences as a metaphor.\nThis perspective is based on the basic metaphor of infinity (BMI), defined as the ever-increasing sequence <1, 2, 3, …>.\n\n## Sources","wordIndex":1,"createdAt":"2026-01-27T12:06:59.990Z","user":{"id":"mfvswiH9fwHI8f36yNwHR40968SZeIKD","name":"den3den42069","image":"https://cdn.discordapp.com/avatars/975452809707278427/5e0588882214f34dc5f3b36f7e9c9ff5.png"}},{"id":"ad7d107c-7108-49b9-b0ac-b1a158a8aaf0","wordChanged":"\\infty.","newWord":":::summary\n:::\n\n# \n**Infinity** is something which is boundless, endless, or larger than any [natural number](https://bliptext.com/articles/natural-number).\nIt is often denoted by the [infinity symbol](https://bliptext.com/articles/infinity-symbol) **INFINITY**\nFrom the time of the [ancient Greeks](https://bliptext.com/articles/greek-mathematics), the [philosophical nature of infinity](https://bliptext.com/articles/infinity-philosophy) has been the subject of many discussions among philosophers.\nIn the 17th century, with the introduction of the infinity symbol and the [infinitesimal calculus](https://bliptext.com/articles/infinitesimal-calculus), mathematicians began to work with [infinite series](https://bliptext.com/articles/infinite-series) and what some mathematicians (including [l'Hôpital](https://bliptext.com/articles/guillaume-de-l-h-pital) and [Bernoulli](https://bliptext.com/articles/johann-bernoulli)) regarded as infinitely small quantities, but infinity continued to be associated with endless processes.\nAs mathematicians struggled with the foundation of [calculus](https://bliptext.com/articles/calculus), it remained unclear whether infinity could be considered as a number or [magnitude](https://bliptext.com/articles/magnitude-mathematics) and, if so, how this could be done.\nAt the end of the 19th century, [Georg Cantor](https://bliptext.com/articles/georg-cantor) enlarged the mathematical study of infinity by studying [infinite sets](https://bliptext.com/articles/infinite-set) and [infinite numbers](https://bliptext.com/articles/transfinite-number), showing that they can be of various sizes.\nFor example, if a line is viewed as the set of all of its points, their infinite number (i.e., the [cardinality](https://bliptext.com/articles/cardinality) of the line) is larger than the number of [integers](https://bliptext.com/articles/integer).\nIn this usage, infinity is a mathematical concept, and infinite [mathematical objects](https://bliptext.com/articles/mathematical-object) can be studied, manipulated, and used just like any other mathematical object.\nThe mathematical concept of infinity refines and extends the old philosophical concept, in particular by introducing infinitely many different sizes of infinite sets.\nAmong the axioms of [Zermelo–Fraenkel set theory](https://bliptext.com/articles/zermelo-fraenkel-set-theory), on which most of modern mathematics can be developed, is the [axiom of infinity](https://bliptext.com/articles/axiom-of-infinity), which guarantees the existence of infinite sets.\nThe mathematical concept of infinity and the manipulation of infinite sets are widely used in mathematics, even in areas such as [combinatorics](https://bliptext.com/articles/combinatorics) that may seem to have nothing to do with them.\nFor example, [Wiles's proof of Fermat's Last Theorem](https://bliptext.com/articles/wiles-s-proof-of-fermat-s-last-theorem) implicitly relies on the existence of [Grothendieck universes](https://bliptext.com/articles/grothendieck-universe), very large infinite sets, for solving a long-standing problem that is stated in terms of [elementary arithmetic](https://bliptext.com/articles/elementary-arithmetic).\nIn [physics](https://bliptext.com/articles/physics) and [cosmology](https://bliptext.com/articles/cosmology), [whether the universe is spatially infinite or not](https://bliptext.com/articles/universe), is an open question.\n# [The Sierpiński Triangle contains an infinite amount of itself because of its recursive pattern. | upload.wikimedia.org/wikipedia/commons/c/c5/Sierpi%C5%84ski///Triangle.png]\n# [Due to the constant light reflection between opposing mirrors, it seems that there is a boundless amount of space and repetition inside of them. | upload.wikimedia.org/wikipedia/commons/9/9e/Reflections///1090029.jpg]\n\n# History\nAncient cultures had various ideas about the nature of infinity.\nThe [ancient Indians](https://bliptext.com/articles/vedic-period) and the [Greeks](https://bliptext.com/articles/ancient-greece) did not define infinity in precise formalism as does modern mathematics, and instead approached infinity as a philosophical concept.\n\n## Early Greek\nThe earliest recorded idea of infinity in Greece may be that of [Anaximander](https://bliptext.com/articles/anaximander) (c. 610 – c. 546 BC) a [pre-Socratic](https://bliptext.com/articles/pre-socratic-philosophy) Greek philosopher.\nHe used the word [apeiron](https://bliptext.com/articles/apeiron), which means \"unbounded\", \"indefinite\", and perhaps can be translated as \"infinite\".\n[Aristotle](https://bliptext.com/articles/aristotle) (350 BC) distinguished potential infinity from [actual infinity](https://bliptext.com/articles/actual-infinity), which he regarded as impossible due to the various [paradoxes](https://bliptext.com/articles/paradoxes) it seemed to produce.\nIt has been argued that, in line with this view, the [Hellenistic](https://bliptext.com/articles/hellenistic) Greeks had a \"horror of the infinite\" which would, for example, explain why [Euclid](https://bliptext.com/articles/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.\"\nIt has also been maintained, that, in proving the [infinitude of the prime numbers](https://bliptext.com/articles/infinitude-of-the-prime-numbers), Euclid \"was the first to overcome the horror of the infinite\".\nThere is a similar controversy concerning Euclid's [parallel postulate](https://bliptext.com/articles/parallel-postulate), sometimes translated:\n\"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.\"\nOther translators, however, prefer the translation \"the two straight lines, if produced indefinitely ...\", thus avoiding the implication that Euclid was comfortable with the notion of infinity.\nFinally, 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.\n\n## Zeno: Achilles and the tortoise\n[Zeno of Elea](https://bliptext.com/articles/zeno-of-elea) (c. 495 – c. 430 BC) did not advance any views concerning the infinite.\nNevertheless, his paradoxes, especially \"Achilles and the Tortoise\", were important contributions in that they made clear the inadequacy of popular conceptions.\nThe paradoxes were described by [Bertrand Russell](https://bliptext.com/articles/bertrand-russell) as \"immeasurably subtle and profound\".\n[Achilles](https://bliptext.com/articles/achilles) races a tortoise, giving the latter a head start.\nEtc.\nApparently, Achilles never overtakes the tortoise, since however many steps he completes, the tortoise remains ahead of him.\nZeno was not attempting to make a point about infinity.\nAs a member of the [Eleatics](https://bliptext.com/articles/eleatic) school which regarded motion as an illusion, he saw it as a mistake to suppose that Achilles could run at all.\nSubsequent thinkers, finding this solution unacceptable, struggled for over two millennia to find other weaknesses in the argument.\nFinally, in 1821, [Augustin-Louis Cauchy](https://bliptext.com/articles/augustin-louis-cauchy) provided both a satisfactory definition of a limit and a proof that, for\n0 < x < 1\n,\nSuppose 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.\nThe duration of the chase fits Cauchy's pattern with\na = 10 seconds\nand\nx = 0.01\n. Achilles does overtake the tortoise; it takes him\n\n## Early Indian\nThe [Jain mathematical](https://bliptext.com/articles/indian-mathematics) text Surya Prajnapti (c. 4th–3rd century BCE) classifies all numbers into three sets: [enumerable](https://bliptext.com/articles/enumerable), innumerable, and infinite.\nEach of these was further subdivided into three orders:\n\n## 17th century\nIn the 17th century, European mathematicians started using infinite numbers and infinite expressions in a systematic fashion.\nIn 1655, [John Wallis](https://bliptext.com/articles/john-wallis) first used the notation \\infty for such a number in his De sectionibus conicis, and exploited it in area calculations by dividing the region into [infinitesimal](https://bliptext.com/articles/infinitesimal) strips of width on the order of But in Arithmetica infinitorum (1656), 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.\"\nIn 1699, [Isaac Newton](https://bliptext.com/articles/isaac-newton) wrote about equations with an infinite number of terms in his work [De analysi per aequationes numero terminorum infinitas](https://bliptext.com/articles/de-analysi-per-aequationes-numero-terminorum-infinitas).\n\n# Mathematics\n[Hermann Weyl](https://bliptext.com/articles/hermann-weyl) opened a mathematico-philosophic address given in 1930 with:\n\"Mathematics is the science of the infinite.\"\n\n## Symbol\nThe infinity symbol \\infty (sometimes called the [lemniscate](https://bliptext.com/articles/lemniscate)) is a mathematical symbol representing the concept of infinity.\nThe symbol is encoded in [Unicode](https://bliptext.com/articles/unicode) at and in [LaTeX](https://bliptext.com/articles/latex) as.\nIt was introduced in 1655 by [John Wallis](https://bliptext.com/articles/john-wallis), and since its introduction, it has also been used outside mathematics in modern mysticism and literary [symbology](https://bliptext.com/articles/symbology).\n\n## Calculus\n[Gottfried Leibniz](https://bliptext.com/articles/gottfried-wilhelm-leibniz), one of the co-inventors of [infinitesimal calculus](https://bliptext.com/articles/infinitesimal-calculus), speculated widely about infinite numbers and their use in mathematics.\nTo 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](https://bliptext.com/articles/law-of-continuity).\n\n### Real analysis\nIn [real analysis](https://bliptext.com/articles/real-analysis), the symbol \\infty, called \"infinity\", is used to denote an unbounded [limit](https://bliptext.com/articles/limit-of-a-function).\nThe notation means that x increases without bound, and means that x decreases without bound.\nFor example, if f(t)\\ge 0 for every t, then\nInfinity can also be used to describe [infinite series](https://bliptext.com/articles/infinite-series), as follows:\nIn addition to defining a limit, infinity can be also used as a value in the extended real number system.\nPoints labeled +\\infty and -\\infty can be added to the [topological space](https://bliptext.com/articles/topological-space) of the real numbers, producing the two-point [compactification](https://bliptext.com/articles/compactification-mathematics) of the real numbers.\nAdding algebraic properties to this gives us the [extended real numbers](https://bliptext.com/articles/extended-real-number).\nWe can also treat +\\infty and -\\infty as the same, leading to the [one-point compactification](https://bliptext.com/articles/one-point-compactification) of the real numbers, which is the [real projective line](https://bliptext.com/articles/real-projective-line).\n[Projective geometry](https://bliptext.com/articles/projective-geometry) also refers to a [line at infinity](https://bliptext.com/articles/line-at-infinity) in plane geometry, a [plane at infinity](https://bliptext.com/articles/plane-at-infinity) in three-dimensional space, and a hyper[plane at infinity](https://bliptext.com/articles/plane-at-infinity) for general [dimensions](https://bliptext.com/articles/dimension-mathematics-and-physics), each consisting of [points at infinity](https://bliptext.com/articles/point-at-infinity).\n\n### Complex analysis\nIn [complex analysis](https://bliptext.com/articles/complex-analysis) the symbol \\infty, called \"infinity\", denotes an unsigned infinite [limit](https://bliptext.com/articles/limit-mathematics).\nThe expression means that the magnitude |x| of x grows beyond any assigned value.\nA [point labeled \\infty](https://bliptext.com/articles/point-at-infinity) can be added to the complex plane as a [topological space](https://bliptext.com/articles/topological-space) giving the [one-point compactification](https://bliptext.com/articles/one-point-compactification) of the complex plane.\nWhen this is done, the resulting space is a one-dimensional [complex manifold](https://bliptext.com/articles/complex-manifold), or [Riemann surface](https://bliptext.com/articles/riemann-surface), called the extended complex plane or the [Riemann sphere](https://bliptext.com/articles/riemann-sphere).\nArithmetic 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).\nOn the other hand, this kind of infinity enables [division by zero](https://bliptext.com/articles/division-by-zero), namely for any nonzero [complex number](https://bliptext.com/articles/complex-number) z.\nIn this context, it is often useful to consider [meromorphic functions](https://bliptext.com/articles/meromorphic-function) as maps into the Riemann sphere taking the value of \\infty at the poles.\nThe domain of a complex-valued function may be extended to include the point at infinity as well.\nOne important example of such functions is the group of [Möbius transformations](https://bliptext.com/articles/m-bius-transformation) (see [Möbius transformation § Overview](https://bliptext.com/articles/m-bius-transformation)).\n# [By 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. | upload.wikimedia.org/wikipedia/commons/3/32/Riemann///sphere1.svg]\n\n## Nonstandard analysis\nThe original formulation of [infinitesimal calculus](https://bliptext.com/articles/infinitesimal-calculus) by [Isaac Newton](https://bliptext.com/articles/isaac-newton) and Gottfried Leibniz used infinitesimal quantities.\nIn the second half of the 20th century, it was shown that this treatment could be put on a rigorous footing through various [logical systems](https://bliptext.com/articles/logical-system), including [smooth infinitesimal analysis](https://bliptext.com/articles/smooth-infinitesimal-analysis) and [nonstandard analysis](https://bliptext.com/articles/nonstandard-analysis).\nIn the latter, infinitesimals are invertible, and their inverses are infinite numbers.\nThe infinities in this sense are part of a [hyperreal field](https://bliptext.com/articles/hyperreal-number); there is no equivalence between them as with the Cantorian [transfinites](https://bliptext.com/articles/transfinite-number).\nFor example, if H is an infinite number in this sense, then H + H = 2H and H + 1 are distinct infinite numbers.\nThis approach to [non-standard calculus](https://bliptext.com/articles/non-standard-calculus) is fully developed in.\n# [450px | upload.wikimedia.org/wikipedia/commons/5/53/N%C3%BAmeros///hiperreales.png]\n\n## Set theory\nA different form of \"infinity\" is the [ordinal](https://bliptext.com/articles/ordinal-number) and [cardinal](https://bliptext.com/articles/cardinal-number) infinities of set theory—a system of [transfinite numbers](https://bliptext.com/articles/transfinite-number) first developed by [Georg Cantor](https://bliptext.com/articles/georg-cantor).\nIn this system, the first transfinite cardinal is [aleph-null](https://bliptext.com/articles/aleph-null) ( ℵ 0 ), the cardinality of the set of [natural numbers](https://bliptext.com/articles/natural-number).\nThis modern mathematical conception of the quantitative infinite developed in the late 19th century from works by Cantor, [Gottlob Frege](https://bliptext.com/articles/gottlob-frege), [Richard Dedekind](https://bliptext.com/articles/richard-dedekind) and others—using the idea of collections or sets.\nDedekind's approach was essentially to adopt the idea of [one-to-one correspondence](https://bliptext.com/articles/one-to-one-correspondence) as a standard for comparing the size of sets, and to reject the view of Galileo (derived from [Euclid](https://bliptext.com/articles/euclid)) that the whole cannot be the same size as the part.\n(However, see [Galileo's paradox](https://bliptext.com/articles/galileo-s-paradox) where Galileo concludes that positive integers cannot be compared to the subset of positive [square integers](https://bliptext.com/articles/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](https://bliptext.com/articles/proper-subset) parts; this notion of infinity is called [Dedekind infinite](https://bliptext.com/articles/dedekind-infinite).\nThe 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\".\nCantor defined two kinds of infinite numbers: [ordinal numbers](https://bliptext.com/articles/ordinal-number) and [cardinal numbers](https://bliptext.com/articles/cardinal-number).\nOrdinal numbers characterize [well-ordered](https://bliptext.com/articles/well-ordered) sets, or counting carried on to any stopping point, including points after an infinite number have already been counted.\nGeneralizing finite and (ordinary) infinite [sequences](https://bliptext.com/articles/sequence) which are maps from the positive [integers](https://bliptext.com/articles/integers) leads to [mappings](https://bliptext.com/articles/function-mathematics) from ordinal numbers to transfinite [sequences](https://bliptext.com/articles/sequence).\nCardinal 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.\nThe smallest ordinal infinity is that of the positive integers, and any set which has the cardinality of the integers is [countably infinite](https://bliptext.com/articles/countable-set).\nIf a set is too large to be put in one-to-one correspondence with the positive integers, it is called [uncountable](https://bliptext.com/articles/uncountable-set).\nCantor's views prevailed and modern mathematics accepts actual infinity as part of a consistent and coherent theory.\nCertain extended number systems, such as the hyperreal numbers, incorporate the ordinary (finite) numbers and infinite numbers of different sizes.\n\n### Cardinality of the continuum\nOne 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\nR\nthan natural numbers\nN\n. Namely, Cantor showed that.\nThe [continuum hypothesis](https://bliptext.com/articles/continuum-hypothesis) states that there is no [cardinal number](https://bliptext.com/articles/cardinal-number) between the cardinality of the reals and the cardinality of the natural numbers, that is, .This hypothesis cannot be proved or disproved within the widely accepted [Zermelo–Fraenkel set theory](https://bliptext.com/articles/zermelo-fraenkel-set-theory), even assuming the [Axiom of Choice](https://bliptext.com/articles/axiom-of-choice).\n[Cardinal arithmetic](https://bliptext.com/articles/cardinal-arithmetic) can be used to show not only that the number of points in a [real number line](https://bliptext.com/articles/real-number-line) is equal to the number of points in any [segment of that line](https://bliptext.com/articles/line-segment), but also that this is equal to the number of points on a plane and, indeed, in any [finite-dimensional](https://bliptext.com/articles/finite-dimensional) space.\nThe first of these results is apparent by considering, for instance, the [tangent](https://bliptext.com/articles/tangent-trigonometric-function) function, which provides a [one-to-one correspondence](https://bliptext.com/articles/one-to-one-correspondence) between the [interval](https://bliptext.com/articles/interval-mathematics) (\n−π⁄2, π⁄2\n) and\nR\n.The second result was proved by Cantor in 1878, but only became intuitively apparent in 1890, when [Giuseppe Peano](https://bliptext.com/articles/giuseppe-peano) introduced the [space-filling curves](https://bliptext.com/articles/space-filling-curve), curved lines that twist and turn enough to fill the whole of any square, or [cube](https://bliptext.com/articles/cube), or hyper[cube](https://bliptext.com/articles/cube), or finite-dimensional space.\nThese 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.\n# [The first three steps of a fractal construction whose limit is a space-filling curve, showing that there are as many points in a one-dimensional line as in a two-dimensional square | upload.wikimedia.org/wikipedia/commons/6/64/Peanocurve.svg]\n\n## Geometry\nUntil the end of the 19th century, infinity was rarely discussed in [geometry](https://bliptext.com/articles/geometry), except in the context of processes that could be continued without any limit.\nFor example, a [line](https://bliptext.com/articles/line-geometry) was what is now called a [line](https://bliptext.com/articles/line-geometry) segment, with the proviso that one can extend it as far as one wants; but extending it infinitely was out of the question.\nSimilarly, a line was usually not considered to be composed of infinitely many points but was a location where a point may be placed.\nEven if there are infinitely many possible positions, only a finite number of points could be placed on a line.\nA witness of this is the expression \"the [locus](https://bliptext.com/articles/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).\nOne of the rare exceptions of a mathematical concept involving [actual infinity](https://bliptext.com/articles/actual-infinity) was [projective geometry](https://bliptext.com/articles/projective-geometry), where [points at infinity](https://bliptext.com/articles/points-at-infinity) are added to the [Euclidean space](https://bliptext.com/articles/euclidean-space) for modeling the [perspective](https://bliptext.com/articles/perspective-graphical) effect that shows [parallel lines](https://bliptext.com/articles/parallel-lines) intersecting \"at infinity\".\nMathematically, points at infinity have the advantage of allowing one to not consider some special cases.\nFor example, in a [projective plane](https://bliptext.com/articles/projective-plane), two distinct [lines](https://bliptext.com/articles/line-geometry) intersect in exactly one point, whereas without points at infinity, there are no intersection points for parallel [lines](https://bliptext.com/articles/line-geometry).\nSo, parallel and non-parallel lines must be studied separately in classical geometry, while they need not be distinguished in projective geometry.\nBefore the use of [set theory](https://bliptext.com/articles/set-theory) for the [foundation of mathematics](https://bliptext.com/articles/foundation-of-mathematics), points and lines were viewed as distinct entities, and a point could be located on a line.\nWith 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).\nIn particular, in modern mathematics, lines are infinite sets.\n\n## Infinite dimension\nThe [vector spaces](https://bliptext.com/articles/vector-space) that occur in classical [geometry](https://bliptext.com/articles/geometry) have always a finite [dimension](https://bliptext.com/articles/dimension-vector-space), generally two or three.\nHowever, this is not implied by the abstract definition of a vector space, and vector spaces of infinite dimension can be considered.\nThis is typically the case in [functional analysis](https://bliptext.com/articles/functional-analysis) where [function spaces](https://bliptext.com/articles/function-space) are generally vector spaces of infinite dimension.\nIn topology, some constructions can generate [topological spaces](https://bliptext.com/articles/topological-space) of infinite dimension.\nIn particular, this is the case of [iterated loop spaces](https://bliptext.com/articles/iterated-loop-space).\n\n## Fractals\nThe structure of a [fractal](https://bliptext.com/articles/fractal) object is reiterated in its magnifications.\nFractals can be magnified indefinitely without losing their structure and becoming \"smooth\"; they have infinite perimeters and can have infinite or finite areas.\nOne such [fractal curve](https://bliptext.com/articles/fractal-curve) with an infinite perimeter and finite area is the [Koch snowflake](https://bliptext.com/articles/koch-snowflake).\n\n## Mathematics without infinity\n[Leopold Kronecker](https://bliptext.com/articles/leopold-kronecker) was skeptical of the notion of infinity and how his fellow mathematicians were using it in the 1870s and 1880s.\nThis skepticism was developed in the [philosophy of mathematics](https://bliptext.com/articles/philosophy-of-mathematics) called [finitism](https://bliptext.com/articles/finitism), an extreme form of mathematical philosophy in the general philosophical and mathematical schools of [constructivism](https://bliptext.com/articles/mathematical-constructivism) and [intuitionism](https://bliptext.com/articles/intuitionism).\n\n# Physics\nIn [physics](https://bliptext.com/articles/physics), approximations of [real numbers](https://bliptext.com/articles/real-number) are used for [continuous](https://bliptext.com/articles/continuum-theory) measurements and [natural numbers](https://bliptext.com/articles/natural-number) are used for [discrete](https://bliptext.com/articles/countable) measurements (i.e., counting).\nConcepts of infinite things such as an infinite [plane wave](https://bliptext.com/articles/plane-wave) exist, but there are no experimental means to generate them.\n\n## Cosmology\nThe first published proposal that the universe is infinite came from Thomas Digges in 1576.\nEight years later, in 1584, the Italian philosopher and astronomer [Giordano Bruno](https://bliptext.com/articles/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.\"\n[Cosmologists](https://bliptext.com/articles/cosmology) have long sought to discover whether infinity exists in our physical [universe](https://bliptext.com/articles/universe): Are there an infinite number of stars?\nDoes the universe have infinite volume?\nDoes space \"[go on forever](https://bliptext.com/articles/shape-of-the-universe)\"?\nThis is still an open question of [cosmology](https://bliptext.com/articles/physical-cosmology).\nThe question of being infinite is logically separate from the question of having boundaries.\nThe two-dimensional surface of the Earth, for example, is finite, yet has no edge.\nBy travelling in a straight line with respect to the Earth's curvature, one will eventually return to the exact spot one started from.\nThe universe, at least in principle, might have a similar [topology](https://bliptext.com/articles/topology).\nIf so, one might eventually return to one's starting point after travelling in a straight line through the universe for long enough.\nThe curvature of the universe can be measured through [multipole moments](https://bliptext.com/articles/multipole-moments) in the spectrum of the [cosmic background radiation](https://bliptext.com/articles/cosmic-microwave-background-radiation).\nTo date, analysis of the radiation patterns recorded by the [WMAP](https://bliptext.com/articles/wmap) spacecraft hints that the universe has a flat topology.\nThis would be consistent with an infinite physical universe.\nHowever, the universe could be finite, even if its curvature is flat.\nAn 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.\nThe topology of such games is [toroidal](https://bliptext.com/articles/torus) and the geometry is flat.\nMany possible bounded, flat possibilities also exist for three-dimensional space.\nThe concept of infinity also extends to the [multiverse](https://bliptext.com/articles/multiverse) hypothesis, which, when explained by astrophysicists such as [Michio Kaku](https://bliptext.com/articles/michio-kaku), posits that there are an infinite number and variety of universes.\nAlso, [cyclic models](https://bliptext.com/articles/cyclic-model) posit an infinite amount of [Big Bangs](https://bliptext.com/articles/big-bang), resulting in an infinite variety of universes after each Big Bang event in an infinite cycle.\n\n# Logic\nIn [logic](https://bliptext.com/articles/logic), an [infinite regress](https://bliptext.com/articles/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.\"\n\n# Computing\nThe [IEEE floating-point](https://bliptext.com/articles/ieee-floating-point) standard (IEEE 754) specifies a positive and a negative infinity value (and also [indefinite](https://bliptext.com/articles/nan) values).\nThese are defined as the result of [arithmetic overflow](https://bliptext.com/articles/arithmetic-overflow), [division by zero](https://bliptext.com/articles/division-by-zero), and other exceptional operations.\nSome [programming languages](https://bliptext.com/articles/programming-language), such as [Java](https://bliptext.com/articles/java-programming-language) and [J](https://bliptext.com/articles/j-programming-language), allow the programmer an explicit access to the positive and negative infinity values as language constants.\nThese can be used as [greatest and least elements](https://bliptext.com/articles/greatest-element), as they compare (respectively) greater than or less than all other values.\nThey have uses as [sentinel values](https://bliptext.com/articles/sentinel-value) in [algorithms](https://bliptext.com/articles/algorithm) involving [sorting](https://bliptext.com/articles/sorting), [searching](https://bliptext.com/articles/search-algorithm), or [windowing](https://bliptext.com/articles/window-function).\nIn languages that do not have greatest and least elements but do allow [overloading](https://bliptext.com/articles/operator-overloading) of [relational operators](https://bliptext.com/articles/relational-operator), it is possible for a programmer to create the greatest and least elements.\nIn 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](https://bliptext.com/articles/data-type), the infinity values may still be accessible and usable as the result of certain operations.\nIn programming, an [infinite loop](https://bliptext.com/articles/infinite-[loop](https://bliptext.com/articles/loop-computing)) is a [loop](https://bliptext.com/articles/loop-computing) whose exit condition is never satisfied, thus executing indefinitely.\n\n# Arts, games, and cognitive sciences\n[Perspective](https://bliptext.com/articles/perspective-graphical) artwork uses the concept of [vanishing points](https://bliptext.com/articles/vanishing-point), roughly corresponding to mathematical [points at infinity](https://bliptext.com/articles/point-at-infinity), located at an infinite distance from the observer.\nThis allows artists to create paintings that realistically render space, distances, and forms.\nArtist [M.C. Escher](https://bliptext.com/articles/m-c-escher) is specifically known for employing the concept of infinity in his work in this and other ways.\nVariations of [chess](https://bliptext.com/articles/chess) played on an unbounded board are called infinite [chess](https://bliptext.com/articles/chess).\n[Cognitive scientist](https://bliptext.com/articles/cognitive-science) [George Lakoff](https://bliptext.com/articles/george-lakoff) considers the concept of infinity in mathematics and the sciences as a metaphor.\nThis perspective is based on the basic metaphor of infinity (BMI), defined as the ever-increasing sequence <1, 2, 3, …>.\n\n## Sources","wordIndex":21,"createdAt":"2025-11-18T21:53:30.924Z","user":{"id":"B8O0pNxNjW11Vo2NiZAfS3EHtkC2ajZZ","name":"Elliot Bloom","image":"https://lh3.googleusercontent.com/a/ACg8ocLbPBvuaxRk1WMozlDlt4zVsg3tHfdHJE3bD8GaPMvqSKOCfGUt=s96-c"}}],"article":{"id":"f89a6eed-44b6-4cb3-a312-896ec97aa6f0","title":"Infinity","slug":"infinity","content":":::summary\n:::\n\n# \n**4** is something which is boundless, endless, or larger than any [natural number](https://bliptext.com/articles/natural-number).\nIt is often denoted by the [infinity symbol](https://bliptext.com/articles/infinity-symbol) **INFINITY**\nFrom the time of the [ancient Greeks](https://bliptext.com/articles/greek-mathematics), the [philosophical nature of infinity](https://bliptext.com/articles/infinity-philosophy) has been the subject of many discussions among philosophers.\nIn the 17th century, with the introduction of the infinity symbol and the [infinitesimal calculus](https://bliptext.com/articles/infinitesimal-calculus), mathematicians began to work with [infinite series](https://bliptext.com/articles/infinite-series) and what some mathematicians (including [l'Hôpital](https://bliptext.com/articles/guillaume-de-l-h-pital) and [Bernoulli](https://bliptext.com/articles/johann-bernoulli)) regarded as infinitely small quantities, but infinity continued to be associated with endless processes.\nAs mathematicians struggled with the foundation of [calculus](https://bliptext.com/articles/calculus), it remained unclear whether infinity could be considered as a number or [magnitude](https://bliptext.com/articles/magnitude-mathematics) and, if so, how this could be done.\nAt the end of the 19th century, [Georg Cantor](https://bliptext.com/articles/georg-cantor) enlarged the mathematical study of infinity by studying [infinite sets](https://bliptext.com/articles/infinite-set) and [infinite numbers](https://bliptext.com/articles/transfinite-number), showing that they can be of various sizes.\nFor example, if a line is viewed as the set of all of its points, their infinite number (i.e., the [cardinality](https://bliptext.com/articles/cardinality) of the line) is larger than the number of [integers](https://bliptext.com/articles/integer).\nIn this usage, infinity is a mathematical concept, and infinite [mathematical objects](https://bliptext.com/articles/mathematical-object) can be studied, manipulated, and used just like any other mathematical object.\nThe mathematical concept of infinity refines and extends the old philosophical concept, in particular by introducing infinitely many different sizes of infinite sets.\nAmong the axioms of [Zermelo–Fraenkel set theory](https://bliptext.com/articles/zermelo-fraenkel-set-theory), on which most of modern mathematics can be developed, is the [axiom of infinity](https://bliptext.com/articles/axiom-of-infinity), which guarantees the existence of infinite sets.\nThe mathematical concept of infinity and the manipulation of infinite sets are widely used in mathematics, even in areas such as [combinatorics](https://bliptext.com/articles/combinatorics) that may seem to have nothing to do with them.\nFor example, [Wiles's proof of Fermat's Last Theorem](https://bliptext.com/articles/wiles-s-proof-of-fermat-s-last-theorem) implicitly relies on the existence of [Grothendieck universes](https://bliptext.com/articles/grothendieck-universe), very large infinite sets, for solving a long-standing problem that is stated in terms of [elementary arithmetic](https://bliptext.com/articles/elementary-arithmetic).\nIn [physics](https://bliptext.com/articles/physics) and [cosmology](https://bliptext.com/articles/cosmology), [whether the universe is spatially infinite or not](https://bliptext.com/articles/universe), is an open question.\n# [The Sierpiński Triangle contains an infinite amount of itself because of its recursive pattern. | upload.wikimedia.org/wikipedia/commons/c/c5/Sierpi%C5%84ski///Triangle.png]\n# [Due to the constant light reflection between opposing mirrors, it seems that there is a boundless amount of space and repetition inside of them. | upload.wikimedia.org/wikipedia/commons/9/9e/Reflections///1090029.jpg]\n\n# History\nAncient cultures had various ideas about the nature of infinity.\nThe [ancient Indians](https://bliptext.com/articles/vedic-period) and the [Greeks](https://bliptext.com/articles/ancient-greece) did not define infinity in precise formalism as does modern mathematics, and instead approached infinity as a philosophical concept.\n\n## Early Greek\nThe earliest recorded idea of infinity in Greece may be that of [Anaximander](https://bliptext.com/articles/anaximander) (c. 610 – c. 546 BC) a [pre-Socratic](https://bliptext.com/articles/pre-socratic-philosophy) Greek philosopher.\nHe used the word [apeiron](https://bliptext.com/articles/apeiron), which means \"unbounded\", \"indefinite\", and perhaps can be translated as \"infinite\".\n[Aristotle](https://bliptext.com/articles/aristotle) (350 BC) distinguished potential infinity from [actual infinity](https://bliptext.com/articles/actual-infinity), which he regarded as impossible due to the various [paradoxes](https://bliptext.com/articles/paradoxes) it seemed to produce.\nIt has been argued that, in line with this view, the [Hellenistic](https://bliptext.com/articles/hellenistic) Greeks had a \"horror of the infinite\" which would, for example, explain why [Euclid](https://bliptext.com/articles/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.\"\nIt has also been maintained, that, in proving the [infinitude of the prime numbers](https://bliptext.com/articles/infinitude-of-the-prime-numbers), Euclid \"was the first to overcome the horror of the infinite\".\nThere is a similar controversy concerning Euclid's [parallel postulate](https://bliptext.com/articles/parallel-postulate), sometimes translated:\n\"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.\"\nOther translators, however, prefer the translation \"the two straight lines, if produced indefinitely ...\", thus avoiding the implication that Euclid was comfortable with the notion of infinity.\nFinally, 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.\n\n## Zeno: Achilles and the tortoise\n[Zeno of Elea](https://bliptext.com/articles/zeno-of-elea) (c. 495 – c. 430 BC) did not advance any views concerning the infinite.\nNevertheless, his paradoxes, especially \"Achilles and the Tortoise\", were important contributions in that they made clear the inadequacy of popular conceptions.\nThe paradoxes were described by [Bertrand Russell](https://bliptext.com/articles/bertrand-russell) as \"immeasurably subtle and profound\".\n[Achilles](https://bliptext.com/articles/achilles) races a tortoise, giving the latter a head start.\nEtc.\nApparently, Achilles never overtakes the tortoise, since however many steps he completes, the tortoise remains ahead of him.\nZeno was not attempting to make a point about infinity.\nAs a member of the [Eleatics](https://bliptext.com/articles/eleatic) school which regarded motion as an illusion, he saw it as a mistake to suppose that Achilles could run at all.\nSubsequent thinkers, finding this solution unacceptable, struggled for over two millennia to find other weaknesses in the argument.\nFinally, in 1821, [Augustin-Louis Cauchy](https://bliptext.com/articles/augustin-louis-cauchy) provided both a satisfactory definition of a limit and a proof that, for\n0 < x < 1\n,\nSuppose 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.\nThe duration of the chase fits Cauchy's pattern with\na = 10 seconds\nand\nx = 0.01\n. Achilles does overtake the tortoise; it takes him\n\n## Early Indian\nThe [Jain mathematical](https://bliptext.com/articles/indian-mathematics) text Surya Prajnapti (c. 4th–3rd century BCE) classifies all numbers into three sets: [enumerable](https://bliptext.com/articles/enumerable), innumerable, and infinite.\nEach of these was further subdivided into three orders:\n\n## 17th century\nIn the 17th century, European mathematicians started using infinite numbers and infinite expressions in a systematic fashion.\nIn 1655, [John Wallis](https://bliptext.com/articles/john-wallis) first used the notation \\infty for such a number in his De sectionibus conicis, and exploited it in area calculations by dividing the region into [infinitesimal](https://bliptext.com/articles/infinitesimal) strips of width on the order of But in Arithmetica infinitorum (1656), 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.\"\nIn 1699, [Isaac Newton](https://bliptext.com/articles/isaac-newton) wrote about equations with an infinite number of terms in his work [De analysi per aequationes numero terminorum infinitas](https://bliptext.com/articles/de-analysi-per-aequationes-numero-terminorum-infinitas).\n\n# Mathematics\n[Hermann Weyl](https://bliptext.com/articles/hermann-weyl) opened a mathematico-philosophic address given in 1930 with:\n\"Mathematics is the science of the infinite.\"\n\n## Symbol\nThe infinity symbol \\infty (sometimes called the [lemniscate](https://bliptext.com/articles/lemniscate)) is a mathematical symbol representing the concept of infinity.\nThe symbol is encoded in [Unicode](https://bliptext.com/articles/unicode) at and in [LaTeX](https://bliptext.com/articles/latex) as.\nIt was introduced in 1655 by [John Wallis](https://bliptext.com/articles/john-wallis), and since its introduction, it has also been used outside mathematics in modern mysticism and literary [symbology](https://bliptext.com/articles/symbology).\n\n## Calculus\n[Gottfried Leibniz](https://bliptext.com/articles/gottfried-wilhelm-leibniz), one of the co-inventors of [infinitesimal calculus](https://bliptext.com/articles/infinitesimal-calculus), speculated widely about infinite numbers and their use in mathematics.\nTo 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](https://bliptext.com/articles/law-of-continuity).\n\n### Real analysis\nIn [real analysis](https://bliptext.com/articles/real-analysis), the symbol \\infty, called \"infinity\", is used to denote an unbounded [limit](https://bliptext.com/articles/limit-of-a-function).\nThe notation means that x increases without bound, and means that x decreases without bound.\nFor example, if f(t)\\ge 0 for every t, then\nInfinity can also be used to describe [infinite series](https://bliptext.com/articles/infinite-series), as follows:\nIn addition to defining a limit, infinity can be also used as a value in the extended real number system.\nPoints labeled +\\infty and -\\infty can be added to the [topological space](https://bliptext.com/articles/topological-space) of the real numbers, producing the two-point [compactification](https://bliptext.com/articles/compactification-mathematics) of the real numbers.\nAdding algebraic properties to this gives us the [extended real numbers](https://bliptext.com/articles/extended-real-number).\nWe can also treat +\\infty and -\\infty as the same, leading to the [one-point compactification](https://bliptext.com/articles/one-point-compactification) of the real numbers, which is the [real projective line](https://bliptext.com/articles/real-projective-line).\n[Projective geometry](https://bliptext.com/articles/projective-geometry) also refers to a [line at infinity](https://bliptext.com/articles/line-at-infinity) in plane geometry, a [plane at infinity](https://bliptext.com/articles/plane-at-infinity) in three-dimensional space, and a hyper[plane at infinity](https://bliptext.com/articles/plane-at-infinity) for general [dimensions](https://bliptext.com/articles/dimension-mathematics-and-physics), each consisting of [points at infinity](https://bliptext.com/articles/point-at-infinity).\n\n### Complex analysis\nIn [complex analysis](https://bliptext.com/articles/complex-analysis) the symbol \\infty, called \"infinity\", denotes an unsigned infinite [limit](https://bliptext.com/articles/limit-mathematics).\nThe expression means that the magnitude |x| of x grows beyond any assigned value.\nA [point labeled \\infty](https://bliptext.com/articles/point-at-infinity) can be added to the complex plane as a [topological space](https://bliptext.com/articles/topological-space) giving the [one-point compactification](https://bliptext.com/articles/one-point-compactification) of the complex plane.\nWhen this is done, the resulting space is a one-dimensional [complex manifold](https://bliptext.com/articles/complex-manifold), or [Riemann surface](https://bliptext.com/articles/riemann-surface), called the extended complex plane or the [Riemann sphere](https://bliptext.com/articles/riemann-sphere).\nArithmetic 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).\nOn the other hand, this kind of infinity enables [division by zero](https://bliptext.com/articles/division-by-zero), namely for any nonzero [complex number](https://bliptext.com/articles/complex-number) z.\nIn this context, it is often useful to consider [meromorphic functions](https://bliptext.com/articles/meromorphic-function) as maps into the Riemann sphere taking the value of \\infty at the poles.\nThe domain of a complex-valued function may be extended to include the point at infinity as well.\nOne important example of such functions is the group of [Möbius transformations](https://bliptext.com/articles/m-bius-transformation) (see [Möbius transformation § Overview](https://bliptext.com/articles/m-bius-transformation)).\n# [By 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. | upload.wikimedia.org/wikipedia/commons/3/32/Riemann///sphere1.svg]\n\n## Nonstandard analysis\nThe original formulation of [infinitesimal calculus](https://bliptext.com/articles/infinitesimal-calculus) by [Isaac Newton](https://bliptext.com/articles/isaac-newton) and Gottfried Leibniz used infinitesimal quantities.\nIn the second half of the 20th century, it was shown that this treatment could be put on a rigorous footing through various [logical systems](https://bliptext.com/articles/logical-system), including [smooth infinitesimal analysis](https://bliptext.com/articles/smooth-infinitesimal-analysis) and [nonstandard analysis](https://bliptext.com/articles/nonstandard-analysis).\nIn the latter, infinitesimals are invertible, and their inverses are infinite numbers.\nThe infinities in this sense are part of a [hyperreal field](https://bliptext.com/articles/hyperreal-number); there is no equivalence between them as with the Cantorian [transfinites](https://bliptext.com/articles/transfinite-number).\nFor example, if H is an infinite number in this sense, then H + H = 2H and H + 1 are distinct infinite numbers.\nThis approach to [non-standard calculus](https://bliptext.com/articles/non-standard-calculus) is fully developed in.\n# [450px | upload.wikimedia.org/wikipedia/commons/5/53/N%C3%BAmeros///hiperreales.png]\n\n## Set theory\nA different form of \"infinity\" is the [ordinal](https://bliptext.com/articles/ordinal-number) and [cardinal](https://bliptext.com/articles/cardinal-number) infinities of set theory—a system of [transfinite numbers](https://bliptext.com/articles/transfinite-number) first developed by [Georg Cantor](https://bliptext.com/articles/georg-cantor).\nIn this system, the first transfinite cardinal is [aleph-null](https://bliptext.com/articles/aleph-null) ( ℵ 0 ), the cardinality of the set of [natural numbers](https://bliptext.com/articles/natural-number).\nThis modern mathematical conception of the quantitative infinite developed in the late 19th century from works by Cantor, [Gottlob Frege](https://bliptext.com/articles/gottlob-frege), [Richard Dedekind](https://bliptext.com/articles/richard-dedekind) and others—using the idea of collections or sets.\nDedekind's approach was essentially to adopt the idea of [one-to-one correspondence](https://bliptext.com/articles/one-to-one-correspondence) as a standard for comparing the size of sets, and to reject the view of Galileo (derived from [Euclid](https://bliptext.com/articles/euclid)) that the whole cannot be the same size as the part.\n(However, see [Galileo's paradox](https://bliptext.com/articles/galileo-s-paradox) where Galileo concludes that positive integers cannot be compared to the subset of positive [square integers](https://bliptext.com/articles/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](https://bliptext.com/articles/proper-subset) parts; this notion of infinity is called [Dedekind infinite](https://bliptext.com/articles/dedekind-infinite).\nThe 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\".\nCantor defined two kinds of infinite numbers: [ordinal numbers](https://bliptext.com/articles/ordinal-number) and [cardinal numbers](https://bliptext.com/articles/cardinal-number).\nOrdinal numbers characterize [well-ordered](https://bliptext.com/articles/well-ordered) sets, or counting carried on to any stopping point, including points after an infinite number have already been counted.\nGeneralizing finite and (ordinary) infinite [sequences](https://bliptext.com/articles/sequence) which are maps from the positive [integers](https://bliptext.com/articles/integers) leads to [mappings](https://bliptext.com/articles/function-mathematics) from ordinal numbers to transfinite [sequences](https://bliptext.com/articles/sequence).\nCardinal 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.\nThe smallest ordinal infinity is that of the positive integers, and any set which has the cardinality of the integers is [countably infinite](https://bliptext.com/articles/countable-set).\nIf a set is too large to be put in one-to-one correspondence with the positive integers, it is called [uncountable](https://bliptext.com/articles/uncountable-set).\nCantor's views prevailed and modern mathematics accepts actual infinity as part of a consistent and coherent theory.\nCertain extended number systems, such as the hyperreal numbers, incorporate the ordinary (finite) numbers and infinite numbers of different sizes.\n\n### Cardinality of the continuum\nOne 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\nR\nthan natural numbers\nN\n. Namely, Cantor showed that.\nThe [continuum hypothesis](https://bliptext.com/articles/continuum-hypothesis) states that there is no [cardinal number](https://bliptext.com/articles/cardinal-number) between the cardinality of the reals and the cardinality of the natural numbers, that is, .This hypothesis cannot be proved or disproved within the widely accepted [Zermelo–Fraenkel set theory](https://bliptext.com/articles/zermelo-fraenkel-set-theory), even assuming the [Axiom of Choice](https://bliptext.com/articles/axiom-of-choice).\n[Cardinal arithmetic](https://bliptext.com/articles/cardinal-arithmetic) can be used to show not only that the number of points in a [real number line](https://bliptext.com/articles/real-number-line) is equal to the number of points in any [segment of that line](https://bliptext.com/articles/line-segment), but also that this is equal to the number of points on a plane and, indeed, in any [finite-dimensional](https://bliptext.com/articles/finite-dimensional) space.\nThe first of these results is apparent by considering, for instance, the [tangent](https://bliptext.com/articles/tangent-trigonometric-function) function, which provides a [one-to-one correspondence](https://bliptext.com/articles/one-to-one-correspondence) between the [interval](https://bliptext.com/articles/interval-mathematics) (\n−π⁄2, π⁄2\n) and\nR\n.The second result was proved by Cantor in 1878, but only became intuitively apparent in 1890, when [Giuseppe Peano](https://bliptext.com/articles/giuseppe-peano) introduced the [space-filling curves](https://bliptext.com/articles/space-filling-curve), curved lines that twist and turn enough to fill the whole of any square, or [cube](https://bliptext.com/articles/cube), or hyper[cube](https://bliptext.com/articles/cube), or finite-dimensional space.\nThese 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.\n# [The first three steps of a fractal construction whose limit is a space-filling curve, showing that there are as many points in a one-dimensional line as in a two-dimensional square | upload.wikimedia.org/wikipedia/commons/6/64/Peanocurve.svg]\n\n## Geometry\nUntil the end of the 19th century, infinity was rarely discussed in [geometry](https://bliptext.com/articles/geometry), except in the context of processes that could be continued without any limit.\nFor example, a [line](https://bliptext.com/articles/line-geometry) was what is now called a [line](https://bliptext.com/articles/line-geometry) segment, with the proviso that one can extend it as far as one wants; but extending it infinitely was out of the question.\nSimilarly, a line was usually not considered to be composed of infinitely many points but was a location where a point may be placed.\nEven if there are infinitely many possible positions, only a finite number of points could be placed on a line.\nA witness of this is the expression \"the [locus](https://bliptext.com/articles/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).\nOne of the rare exceptions of a mathematical concept involving [actual infinity](https://bliptext.com/articles/actual-infinity) was [projective geometry](https://bliptext.com/articles/projective-geometry), where [points at infinity](https://bliptext.com/articles/points-at-infinity) are added to the [Euclidean space](https://bliptext.com/articles/euclidean-space) for modeling the [perspective](https://bliptext.com/articles/perspective-graphical) effect that shows [parallel lines](https://bliptext.com/articles/parallel-lines) intersecting \"at infinity\".\nMathematically, points at infinity have the advantage of allowing one to not consider some special cases.\nFor example, in a [projective plane](https://bliptext.com/articles/projective-plane), two distinct [lines](https://bliptext.com/articles/line-geometry) intersect in exactly one point, whereas without points at infinity, there are no intersection points for parallel [lines](https://bliptext.com/articles/line-geometry).\nSo, parallel and non-parallel lines must be studied separately in classical geometry, while they need not be distinguished in projective geometry.\nBefore the use of [set theory](https://bliptext.com/articles/set-theory) for the [foundation of mathematics](https://bliptext.com/articles/foundation-of-mathematics), points and lines were viewed as distinct entities, and a point could be located on a line.\nWith 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).\nIn particular, in modern mathematics, lines are infinite sets.\n\n## Infinite dimension\nThe [vector spaces](https://bliptext.com/articles/vector-space) that occur in classical [geometry](https://bliptext.com/articles/geometry) have always a finite [dimension](https://bliptext.com/articles/dimension-vector-space), generally two or three.\nHowever, this is not implied by the abstract definition of a vector space, and vector spaces of infinite dimension can be considered.\nThis is typically the case in [functional analysis](https://bliptext.com/articles/functional-analysis) where [function spaces](https://bliptext.com/articles/function-space) are generally vector spaces of infinite dimension.\nIn topology, some constructions can generate [topological spaces](https://bliptext.com/articles/topological-space) of infinite dimension.\nIn particular, this is the case of [iterated loop spaces](https://bliptext.com/articles/iterated-loop-space).\n\n## Fractals\nThe structure of a [fractal](https://bliptext.com/articles/fractal) object is reiterated in its magnifications.\nFractals can be magnified indefinitely without losing their structure and becoming \"smooth\"; they have infinite perimeters and can have infinite or finite areas.\nOne such [fractal curve](https://bliptext.com/articles/fractal-curve) with an infinite perimeter and finite area is the [Koch snowflake](https://bliptext.com/articles/koch-snowflake).\n\n## Mathematics without infinity\n[Leopold Kronecker](https://bliptext.com/articles/leopold-kronecker) was skeptical of the notion of infinity and how his fellow mathematicians were using it in the 1870s and 1880s.\nThis skepticism was developed in the [philosophy of mathematics](https://bliptext.com/articles/philosophy-of-mathematics) called [finitism](https://bliptext.com/articles/finitism), an extreme form of mathematical philosophy in the general philosophical and mathematical schools of [constructivism](https://bliptext.com/articles/mathematical-constructivism) and [intuitionism](https://bliptext.com/articles/intuitionism).\n\n# Physics\nIn [physics](https://bliptext.com/articles/physics), approximations of [real numbers](https://bliptext.com/articles/real-number) are used for [continuous](https://bliptext.com/articles/continuum-theory) measurements and [natural numbers](https://bliptext.com/articles/natural-number) are used for [discrete](https://bliptext.com/articles/countable) measurements (i.e., counting).\nConcepts of infinite things such as an infinite [plane wave](https://bliptext.com/articles/plane-wave) exist, but there are no experimental means to generate them.\n\n## Cosmology\nThe first published proposal that the universe is infinite came from Thomas Digges in 1576.\nEight years later, in 1584, the Italian philosopher and astronomer [Giordano Bruno](https://bliptext.com/articles/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.\"\n[Cosmologists](https://bliptext.com/articles/cosmology) have long sought to discover whether infinity exists in our physical [universe](https://bliptext.com/articles/universe): Are there an infinite number of stars?\nDoes the universe have infinite volume?\nDoes space \"[go on forever](https://bliptext.com/articles/shape-of-the-universe)\"?\nThis is still an open question of [cosmology](https://bliptext.com/articles/physical-cosmology).\nThe question of being infinite is logically separate from the question of having boundaries.\nThe two-dimensional surface of the Earth, for example, is finite, yet has no edge.\nBy travelling in a straight line with respect to the Earth's curvature, one will eventually return to the exact spot one started from.\nThe universe, at least in principle, might have a similar [topology](https://bliptext.com/articles/topology).\nIf so, one might eventually return to one's starting point after travelling in a straight line through the universe for long enough.\nThe curvature of the universe can be measured through [multipole moments](https://bliptext.com/articles/multipole-moments) in the spectrum of the [cosmic background radiation](https://bliptext.com/articles/cosmic-microwave-background-radiation).\nTo date, analysis of the radiation patterns recorded by the [WMAP](https://bliptext.com/articles/wmap) spacecraft hints that the universe has a flat topology.\nThis would be consistent with an infinite physical universe.\nHowever, the universe could be finite, even if its curvature is flat.\nAn 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.\nThe topology of such games is [toroidal](https://bliptext.com/articles/torus) and the geometry is flat.\nMany possible bounded, flat possibilities also exist for three-dimensional space.\nThe concept of infinity also extends to the [multiverse](https://bliptext.com/articles/multiverse) hypothesis, which, when explained by astrophysicists such as [Michio Kaku](https://bliptext.com/articles/michio-kaku), posits that there are an infinite number and variety of universes.\nAlso, [cyclic models](https://bliptext.com/articles/cyclic-model) posit an infinite amount of [Big Bangs](https://bliptext.com/articles/big-bang), resulting in an infinite variety of universes after each Big Bang event in an infinite cycle.\n\n# Logic\nIn [logic](https://bliptext.com/articles/logic), an [infinite regress](https://bliptext.com/articles/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.\"\n\n# Computing\nThe [IEEE floating-point](https://bliptext.com/articles/ieee-floating-point) standard (IEEE 754) specifies a positive and a negative infinity value (and also [indefinite](https://bliptext.com/articles/nan) values).\nThese are defined as the result of [arithmetic overflow](https://bliptext.com/articles/arithmetic-overflow), [division by zero](https://bliptext.com/articles/division-by-zero), and other exceptional operations.\nSome [programming languages](https://bliptext.com/articles/programming-language), such as [Java](https://bliptext.com/articles/java-programming-language) and [J](https://bliptext.com/articles/j-programming-language), allow the programmer an explicit access to the positive and negative infinity values as language constants.\nThese can be used as [greatest and least elements](https://bliptext.com/articles/greatest-element), as they compare (respectively) greater than or less than all other values.\nThey have uses as [sentinel values](https://bliptext.com/articles/sentinel-value) in [algorithms](https://bliptext.com/articles/algorithm) involving [sorting](https://bliptext.com/articles/sorting), [searching](https://bliptext.com/articles/search-algorithm), or [windowing](https://bliptext.com/articles/window-function).\nIn languages that do not have greatest and least elements but do allow [overloading](https://bliptext.com/articles/operator-overloading) of [relational operators](https://bliptext.com/articles/relational-operator), it is possible for a programmer to create the greatest and least elements.\nIn 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](https://bliptext.com/articles/data-type), the infinity values may still be accessible and usable as the result of certain operations.\nIn programming, an [infinite loop](https://bliptext.com/articles/infinite-[loop](https://bliptext.com/articles/loop-computing)) is a [loop](https://bliptext.com/articles/loop-computing) whose exit condition is never satisfied, thus executing indefinitely.\n\n# Arts, games, and cognitive sciences\n[Perspective](https://bliptext.com/articles/perspective-graphical) artwork uses the concept of [vanishing points](https://bliptext.com/articles/vanishing-point), roughly corresponding to mathematical [points at infinity](https://bliptext.com/articles/point-at-infinity), located at an infinite distance from the observer.\nThis allows artists to create paintings that realistically render space, distances, and forms.\nArtist [M.C. Escher](https://bliptext.com/articles/m-c-escher) is specifically known for employing the concept of infinity in his work in this and other ways.\nVariations of [chess](https://bliptext.com/articles/chess) played on an unbounded board are called infinite [chess](https://bliptext.com/articles/chess).\n[Cognitive scientist](https://bliptext.com/articles/cognitive-science) [George Lakoff](https://bliptext.com/articles/george-lakoff) considers the concept of infinity in mathematics and the sciences as a metaphor.\nThis perspective is based on the basic metaphor of infinity (BMI), defined as the ever-increasing sequence <1, 2, 3, …>.\n\n## Sources","current_revision":"d42f635a-9111-4841-ac77-1847b1edbed6","created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-26T11:14:37.380Z","updated_at":"2026-01-27T12:07:36.978Z","search_vector":"'infin':1","search_content":" \n \n\n# \n**Infinity** is something which is boundless, endless, or larger than any [natural number](https://bliptext.com/articles/natural-number).\nIt is often denoted by the [infinity symbol](https://bliptext.com/articles/infinity-symbol) \\infty.\nFrom the time of the [ancient Greeks](https://bliptext.com/articles/greek-mathematics), the [philosophical nature of infinity](https://bliptext.com/articles/infinity-philosophy) has been the subject of many discussions among philosophers.\nIn the 17th century, with the introduction of the infinity symbol and the [infinitesimal calculus](https://bliptext.com/articles/infinitesimal-calculus), mathematicians began to work with [infinite series](https://bliptext.com/articles/infinite-series) and what some mathematicians (including [l'Hôpital](https://bliptext.com/articles/guillaume-de-l-h-pital) and [Bernoulli](https://bliptext.com/articles/johann-bernoulli)) regarded as infinitely small quantities, but infinity continued to be associated with endless processes.\nAs mathematicians struggled with the foundation of [calculus](https://bliptext.com/articles/calculus), it remained unclear whether infinity could be considered as a number or [magnitude](https://bliptext.com/articles/magnitude-mathematics) and, if so, how this could be done.\nAt the end of the 19th century, [Georg Cantor](https://bliptext.com/articles/georg-cantor) enlarged the mathematical study of infinity by studying [infinite sets](https://bliptext.com/articles/infinite-set) and [infinite numbers](https://bliptext.com/articles/transfinite-number), showing that they can be of various sizes.\nFor example, if a line is viewed as the set of all of its points, their infinite number (i.e., the [cardinality](https://bliptext.com/articles/cardinality) of the line) is larger than the number of [integers](https://bliptext.com/articles/integer).\nIn this usage, infinity is a mathematical concept, and infinite [mathematical objects](https://bliptext.com/articles/mathematical-object) can be studied, manipulated, and used just like any other mathematical object.\nThe mathematical concept of infinity refines and extends the old philosophical concept, in particular by introducing infinitely many different sizes of infinite sets.\nAmong the axioms of [Zermelo–Fraenkel set theory](https://bliptext.com/articles/zermelo-fraenkel-set-theory), on which most of modern mathematics can be developed, is the [axiom of infinity](https://bliptext.com/articles/axiom-of-infinity), which guarantees the existence of infinite sets.\nThe mathematical concept of infinity and the manipulation of infinite sets are widely used in mathematics, even in areas such as [combinatorics](https://bliptext.com/articles/combinatorics) that may seem to have nothing to do with them.\nFor example, [Wiles's proof of Fermat's Last Theorem](https://bliptext.com/articles/wiles-s-proof-of-fermat-s-last-theorem) implicitly relies on the existence of [Grothendieck universes](https://bliptext.com/articles/grothendieck-universe), very large infinite sets, for solving a long-standing problem that is stated in terms of [elementary arithmetic](https://bliptext.com/articles/elementary-arithmetic).\nIn [physics](https://bliptext.com/articles/physics) and [cosmology](https://bliptext.com/articles/cosmology), [whether the universe is spatially infinite or not](https://bliptext.com/articles/universe), is an open question.\n# [The Sierpiński Triangle contains an infinite amount of itself because of its recursive pattern. | upload.wikimedia.org/wikipedia/commons/c/c5/Sierpi%C5%84ski///Triangle.png]\n# [Due to the constant light reflection between opposing mirrors, it seems that there is a boundless amount of space and repetition inside of them. | upload.wikimedia.org/wikipedia/commons/9/9e/Reflections///1090029.jpg]\n\n# History\nAncient cultures had various ideas about the nature of infinity.\nThe [ancient Indians](https://bliptext.com/articles/vedic-period) and the [Greeks](https://bliptext.com/articles/ancient-greece) did not define infinity in precise formalism as does modern mathematics, and instead approached infinity as a philosophical concept.\n\n## Early Greek\nThe earliest recorded idea of infinity in Greece may be that of [Anaximander](https://bliptext.com/articles/anaximander) (c. 610 – c. 546 BC) a [pre-Socratic](https://bliptext.com/articles/pre-socratic-philosophy) Greek philosopher.\nHe used the word [apeiron](https://bliptext.com/articles/apeiron), which means \"unbounded\", \"indefinite\", and perhaps can be translated as \"infinite\".\n[Aristotle](https://bliptext.com/articles/aristotle) (350 BC) distinguished potential infinity from [actual infinity](https://bliptext.com/articles/actual-infinity), which he regarded as impossible due to the various [paradoxes](https://bliptext.com/articles/paradoxes) it seemed to produce.\nIt has been argued that, in line with this view, the [Hellenistic](https://bliptext.com/articles/hellenistic) Greeks had a \"horror of the infinite\" which would, for example, explain why [Euclid](https://bliptext.com/articles/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.\"\nIt has also been maintained, that, in proving the [infinitude of the prime numbers](https://bliptext.com/articles/infinitude-of-the-prime-numbers), Euclid \"was the first to overcome the horror of the infinite\".\nThere is a similar controversy concerning Euclid's [parallel postulate](https://bliptext.com/articles/parallel-postulate), sometimes translated:\n\"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.\"\nOther translators, however, prefer the translation \"the two straight lines, if produced indefinitely ...\", thus avoiding the implication that Euclid was comfortable with the notion of infinity.\nFinally, 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.\n\n## Zeno: Achilles and the tortoise\n[Zeno of Elea](https://bliptext.com/articles/zeno-of-elea) (c. 495 – c. 430 BC) did not advance any views concerning the infinite.\nNevertheless, his paradoxes, especially \"Achilles and the Tortoise\", were important contributions in that they made clear the inadequacy of popular conceptions.\nThe paradoxes were described by [Bertrand Russell](https://bliptext.com/articles/bertrand-russell) as \"immeasurably subtle and profound\".\n[Achilles](https://bliptext.com/articles/achilles) races a tortoise, giving the latter a head start.\nEtc.\nApparently, Achilles never overtakes the tortoise, since however many steps he completes, the tortoise remains ahead of him.\nZeno was not attempting to make a point about infinity.\nAs a member of the [Eleatics](https://bliptext.com/articles/eleatic) school which regarded motion as an illusion, he saw it as a mistake to suppose that Achilles could run at all.\nSubsequent thinkers, finding this solution unacceptable, struggled for over two millennia to find other weaknesses in the argument.\nFinally, in 1821, [Augustin-Louis Cauchy](https://bliptext.com/articles/augustin-louis-cauchy) provided both a satisfactory definition of a limit and a proof that, for\n0 .\n\n## Sources","revisionCount":3},"hasMore":false,"nextCursor":null}