Movatterモバイル変換

[0]ホーム
URL:

Jump to content
WikipediaThe Free Encyclopedia
Search

Infinity

From Wikipedia, the free encyclopedia
Mathematical concept
For the symbol, seeInfinity symbol. For other uses of "Infinity" and "Infinite", seeInfinity (disambiguation).Not to be confused withInfiniti.
TheSierpiński triangle contains infinitely many (scaled-down) copies of itself.

Infinity is something which is boundless, limitless, endless, or larger than anynatural number. It is denoted by∞, called theinfinity symbol.

From the time of theancient Greeks, thephilosophical nature of infinity has been the subject of many discussions. In the 17th century, with the introduction of the infinity symbol[1] and theinfinitesimal calculus, mathematicians began to work withinfinite series and what some mathematicians (includingl'Hôpital andBernoulli)[2] regarded as infinitely small quantities, but infinity continued to be associated with endless processes. As mathematicians struggled with the foundation ofcalculus, it remained unclear whether infinity could be considered as a number ormagnitude and, if so, how this could be done.[1] At the end of the 19th century,Georg Cantor enlarged the mathematical study of infinity by studyinginfinite sets andinfinite numbers, showing that they can be of various sizes.[1][3] For example, if a line is viewed as the set of all of its points, their infinite number (i.e., thecardinality of the line) is larger than the number ofintegers.[4] In this usage, infinity is a mathematical concept, and infinitemathematical objects can be studied, manipulated, and used just like any other mathematical object.

The mathematical concept of infinity refines and extends the old philosophical concept, in particular by introducing infinitely many different sizes of infinite sets. Among the axioms ofZermelo–Fraenkel set theory, on which most of modern mathematics can be developed, is theaxiom of infinity, which guarantees the existence of infinite sets.[1] The mathematical concept of infinity and the manipulation of infinite sets are widely used in mathematics, even in areas such ascombinatorics that may seem to have nothing to do with them. For example,Wiles's proof of Fermat's Last Theorem implicitly relies on the existence ofGrothendieck universes, very large infinite sets,[5] for solving a long-standing problem that is stated in terms ofelementary arithmetic.

Inphysics andcosmology, it is an open questionwhether the universe is spatially infinite or not.

History

[edit]
Further information:Infinity (philosophy)

Ancient cultures had various ideas about the nature of infinity. Theancient Indians and theGreeks did not define infinity in precise formalism as does modern mathematics, and instead approached infinity as a philosophical concept.

Early Greek

[edit]

The earliest recorded idea of infinity in Greece may be that ofAnaximander (c. 610 – c. 546 BC) apre-Socratic Greek philosopher. He used the wordapeiron, which means "unbounded", "indefinite", and perhaps can be translated as "infinite".[1][6]

Aristotle (350 BC) distinguishedpotential infinity fromactual infinity, which he regarded as impossible due to the variousparadoxes it seemed to produce.[7] It has been argued that, in line with this view, theHellenistic Greeks had a "horror of the infinite"[8][9] which would, for example, explain whyEuclid (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."[10] It has also been maintained, that, in proving theinfinitude of the prime numbers, Euclid "was the first to overcome the horror of the infinite".[11] There is a similar controversy concerning Euclid'sparallel postulate, sometimes translated:

If a straight line falling across two [other] straight lines makes internal angles on the same side [of itself whose sum is] less than two right angles, then the two [other] straight lines, being produced to infinity, meet on that side [of the original straight line] that the [sum of the internal angles] is less than two right angles.[12]

Other translators, however, prefer the translation "the two straight lines, if produced indefinitely ...",[13] thus avoiding the implication that Euclid was comfortable with the notion of infinity. Finally, it has been maintained that a reflection on infinity, far from eliciting a "horror of the infinite", underlay all of early Greek philosophy and that Aristotle's "potential infinity" is an aberration from the general trend of this period.[14]

Zeno: Achilles and the tortoise

[edit]
Main article:Zeno's paradoxes § Achilles and the tortoise

Zeno of Elea (c. 495 –c. 430 BC) did not advance any views concerning the infinite. Nevertheless, his paradoxes,[15] especially "Achilles and the Tortoise", were important contributions in that they made clear the inadequacy of popular conceptions. The paradoxes were described byBertrand Russell as "immeasurably subtle and profound".[16]

Achilles races a tortoise, giving the latter a head start.

  • Step #1: Achilles runs to the tortoise's starting point while the tortoise walks forward.
  • Step #2: Achilles advances to where the tortoise was at the end of Step #1 while the tortoise goes yet farther.
  • Step #3: Achilles advances to where the tortoise was at the end of Step #2 while the tortoise goes yet farther.
  • Step #4: Achilles advances to where the tortoise was at the end of Step #3 while the tortoise goes yet farther.

Etc.

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

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

Finally, in 1821,Augustin-Louis Cauchy provided both a satisfactory definition of a limit and a proof that, for0 <x < 1,[17]a+ax+ax2+ax3+ax4+ax5+⋯=a1−x.{\displaystyle a+ax+ax^{2}+ax^{3}+ax^{4}+ax^{5}+\cdots ={\frac {a}{1-x}}.}

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

10+0.1+0.001+0.00001+⋯{\displaystyle 10+0.1+0.001+0.00001+\cdots }=101−0.01=100.99=10.10101… seconds.{\displaystyle ={\frac {10}{1-0.01}}={\frac {10}{0.99}}=10.10101\ldots {\text{ seconds}}.}

Early Indian

[edit]

TheJain mathematical textSurya Prajnapti (c. 4th–3rd century BCE) classifies all numbers into three sets:enumerable, innumerable, and infinite. Each of these was further subdivided into three orders:[18]

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

17th century

[edit]

In the 17th century, European mathematicians started using infinite numbers and infinite expressions in a systematic fashion. In 1655,John Wallis first used the notation∞{\displaystyle \infty } for such a number in hisDe sectionibus conicis,[19] and exploited it in area calculations by dividing the region intoinfinitesimal strips of width on the order of1∞.{\displaystyle {\tfrac {1}{\infty }}.}[20] But inArithmetica infinitorum (1656),[21] 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."[22]

In 1699,Isaac Newton wrote about equations with an infinite number of terms in his workDe analysi per aequationes numero terminorum infinitas.[23]

Symbol

[edit]
Main article:Infinity symbol

The infinity symbol∞{\displaystyle \infty } (sometimes called thelemniscate) is a mathematical symbol representing the concept of infinity. The symbol is encoded inUnicode atU+221E ∞INFINITY (&infin;)[24] and inLaTeX as\infty.[25]

It was introduced in 1655 byJohn Wallis,[26][27] and since its introduction, it has also been used outside mathematics in modern mysticism[28] and literarysymbology.[29]

Calculus

[edit]

Gottfried Leibniz, one of the co-inventors ofinfinitesimal calculus, speculated widely about infinite numbers and their use in mathematics. To Leibniz, both infinitesimals and infinite quantities were ideal entities, not of the same nature as appreciable quantities, but enjoying the same properties in accordance with theLaw of continuity.[30][2]

Real analysis

[edit]

Inreal analysis, the symbol∞{\displaystyle \infty }, called "infinity", is used to denote an unboundedlimit.[31] It is not a real number itself. The notationx→∞{\displaystyle x\rightarrow \infty } means that x{\displaystyle x} increases without bound, andx→−∞{\displaystyle x\to -\infty } means that x{\displaystyle x} decreases without bound. For example, iff(t)≥0{\displaystyle f(t)\geq 0} for every t{\displaystyle t}, then[32]

Infinity can also be used to describeinfinite series, as follows:

In addition to defining a limit, infinity can be also used as a value in the extended real number system. Points labeled+∞{\displaystyle +\infty } and−∞{\displaystyle -\infty } can be added to thetopological space of the real numbers, producing the two-pointcompactification of the real numbers. Adding algebraic properties to this gives us theextended real numbers.[34] We can also treat+∞{\displaystyle +\infty } and−∞{\displaystyle -\infty } as the same, leading to theone-point compactification of the real numbers, which is thereal projective line.[35]Projective geometry also refers to aline at infinity in plane geometry, aplane at infinity in three-dimensional space, and ahyperplane at infinity for generaldimensions, each consisting ofpoints at infinity.[36]

Complex analysis

[edit]
Bystereographic projection, the complex plane can be "wrapped" onto a sphere, with the top point of the sphere corresponding to infinity. This is called theRiemann sphere.

Incomplex analysis the symbol∞{\displaystyle \infty }, called "infinity", denotes an unsigned infinitelimit. The expressionx→∞{\displaystyle x\rightarrow \infty } means that the magnitude |x|{\displaystyle |x|} of x{\displaystyle x} grows beyond any assigned value. Apoint labeled∞{\displaystyle \infty } can be added to the complex plane as atopological space giving theone-point compactification of the complex plane. When this is done, the resulting space is a one-dimensionalcomplex manifold, orRiemann surface, called the extended complex plane or theRiemann sphere.[37] Arithmetic operations similar to those given above for the extended real numbers can also be defined, though there is no distinction in the signs (which leads to the one exception that infinity cannot be added to itself). On the other hand, this kind of infinity enablesdivision by zero, namelyz/0=∞{\displaystyle z/0=\infty } for any nonzerocomplex number z{\displaystyle z}. In this context, it is often useful to considermeromorphic functions as maps into the Riemann sphere taking the value of∞{\displaystyle \infty } at the poles. The domain of a complex-valued function may be extended to include the point at infinity as well. One important example of such functions is the group ofMöbius transformations (seeMöbius transformation § Overview).

Nonstandard analysis

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

The original formulation ofinfinitesimal calculus byIsaac Newton and Gottfried Leibniz used infinitesimal quantities. In the second half of the 20th century, it was shown that this treatment could be put on a rigorous footing through variouslogical systems, includingsmooth infinitesimal analysis andnonstandard analysis. In the latter, infinitesimals are invertible, and their inverses are infinite numbers. The infinities in this sense are part of ahyperreal field; there is no equivalence between them as with the Cantoriantransfinites. For example, if H is an infinite number in this sense, then H + H = 2H and H + 1 are distinct infinite numbers. This approach tonon-standard calculus is fully developed inKeisler (1986).

Set theory

[edit]
Main articles:Cardinality andOrdinal number
One-to-one correspondence between an infinite set and its proper subset

A different form of "infinity" is theordinal andcardinal infinities of set theory—a system oftransfinite numbers first developed byGeorg Cantor. In this system, the first transfinite cardinal isaleph-null (ℵ0), the cardinality of the set ofnatural numbers. This modern mathematical conception of the quantitative infinite developed in the late 19th century from works by Cantor,Gottlob Frege,Richard Dedekind and others—using the idea of collections or sets.[1]

Dedekind's approach was essentially to adopt the idea ofone-to-one correspondence as a standard for comparing the size of sets, and to reject the view of Galileo (derived fromEuclid) that the whole cannot be the same size as the part. (However, seeGalileo's paradox where Galileo concludes that positive integers cannot be compared to the subset of positivesquare integers since both are infinite sets.) An infinite set can simply be defined as one having the same size as at least one of itsproper parts; this notion of infinity is calledDedekind infinite. The diagram to the right gives an example: viewing lines as infinite sets of points, the left half of the lower blue line can be mapped in a one-to-one manner (green correspondences) to the higher blue line, and, in turn, to the whole lower blue line (red correspondences); therefore the whole lower blue line and its left half have the same cardinality, i.e. "size".[38]

Cantor defined two kinds of infinite numbers:ordinal numbers andcardinal numbers. Ordinal numbers characterizewell-ordered sets, or counting carried on to any stopping point, including points after an infinite number have already been counted. Generalizing finite and (ordinary) infinitesequences which are maps from the positiveintegers leads tomappings from ordinal numbers to transfinite sequences. Cardinal numbers define the size of sets, meaning how many members they contain, and can be standardized by choosing the first ordinal number of a certain size to represent the cardinal number of that size. The smallest ordinal infinity is that of the positive integers, and any set which has the cardinality of the integers iscountably infinite. If a set is too large to be put in one-to-one correspondence with the positive integers, it is calleduncountable. Cantor's views prevailed and modern mathematics accepts actual infinity as part of a consistent and coherent theory.[39] Certain extended number systems, such as thehyperreal numbers, incorporate the ordinary (finite) numbers and infinite numbers of different sizes.[40]

Cardinality of the continuum

[edit]
Main article:Cardinality of the continuum

One of Cantor's most important results was that the cardinality of the continuumc{\displaystyle \mathbf {c} } is greater than that of the natural numbersℵ0{\displaystyle {\aleph _{0}}}; that is, there are more real numbersR than natural numbersN. Namely, Cantor showed thatc=2ℵ0>ℵ0{\displaystyle \mathbf {c} =2^{\aleph _{0}}>{\aleph _{0}}}.[41]

Further information:Cantor's diagonal argument andCantor's first set theory article

Thecontinuum hypothesis states that there is nocardinal number between the cardinality of the reals and the cardinality of the natural numbers, that is,c=ℵ1=ℶ1{\displaystyle \mathbf {c} =\aleph _{1}=\beth _{1}}.

Further information:Beth number § Beth one

This hypothesis cannot be proved or disproved within the widely acceptedZermelo–Fraenkel set theory, even assuming theAxiom of Choice.[42]

Cardinal arithmetic can be used to show not only that the number of points in areal number line is equal to the number of points in anysegment of that line, but also that this is equal to the number of points on a plane and, indeed, in anyfinite-dimensional space.[43]

The first three steps of a fractal construction whose limit is aspace-filling curve, showing that there are as many points in a one-dimensional line as in a two-dimensional square

The first of these results is apparent by considering, for instance, thetangent function, which provides aone-to-one correspondence between theinterval (−⁠π/2⁠,⁠π/2⁠) andR.

See also:Hilbert's paradox of the Grand Hotel

The second result was proved by Cantor in 1878, but only became intuitively apparent in 1890, whenGiuseppe Peano introduced thespace-filling curves, curved lines that twist and turn enough to fill the whole of any square, orcube, orhypercube, or finite-dimensional space. These curves can be used to define a one-to-one correspondence between the points on one side of a square and the points in the square.[44]

Geometry

[edit]

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

One of the rare exceptions of a mathematical concept involvingactual infinity wasprojective geometry, wherepoints at infinity are added to theEuclidean space for modeling theperspective effect that showsparallel lines intersecting "at infinity". Mathematically, points at infinity have the advantage of allowing one to not consider some special cases. For example, in aprojective plane, two distinctlines intersect in exactly one point, whereas without points at infinity, there are no intersection points for parallel lines. So, parallel and non-parallel lines must be studied separately in classical geometry, while they need not be distinguished in projective geometry.

Before the use ofset theory for thefoundation of mathematics, points and lines were viewed as distinct entities, and a point could belocated on a line. With the universal use of set theory in mathematics, the point of view has dramatically changed: a line is now considered asthe set of its points, and one says that a pointbelongs to a line instead ofis located on a line (however, the latter phrase is still used).

In particular, in modern mathematics, lines areinfinite sets.

Infinite dimension

[edit]

Thevector spaces that occur in classicalgeometry have always a finitedimension, generally two or three. However, this is not implied by the abstract definition of a vector space, and vector spaces of infinite dimension can be considered. This is typically the case infunctional analysis wherefunction spaces are generally vector spaces of infinite dimension.

In topology, some constructions can generatetopological spaces of infinite dimension. In particular, this is the case ofiterated loop spaces.

Fractals

[edit]

The structure of afractal object is reiterated in its magnifications. Fractals can be magnified indefinitely without losing their structure and becoming "smooth"; they have infinite perimeters and can have infinite or finite areas. One suchfractal curve with an infinite perimeter and finite area is theKoch snowflake.[45]

Finitism

[edit]

Leopold Kronecker was skeptical of the notion of infinity and how his fellow mathematicians were using it in the 1870s and 1880s. This skepticism was developed in thephilosophy of mathematics calledfinitism, an extreme form of mathematical philosophy in the general philosophical and mathematical schools ofconstructivism andintuitionism.[46]

Logic

[edit]

Inlogic, aninfinite 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."[47]

Infirst-order logic, both thecompactness theorem andLöwenheim–Skolem theorems are used to constructnon-standard models with certain infinite properties.

Applications

[edit]

Physics

[edit]

Inphysics, approximations ofreal numbers are used forcontinuous measurements andnatural numbers are used fordiscrete measurements (i.e., counting). Concepts of infinite things such as an infiniteplane wave exist, but there are no experimental means to generate them.[48]

Cosmology

[edit]

The first published proposal that the universe is infinite came from Thomas Digges in 1576.[49] Eight years later, in 1584, the Italian philosopher and astronomerGiordano Bruno proposed an unbounded universe inOn 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."[50]

Cosmologists have long sought to discover whether infinity exists in our physicaluniverse: Are there an infinite number of stars? Does the universe have infinite volume? Does space "go on forever"? This is still an open question ofcosmology. The question of being infinite is logically separate from the question of having boundaries. The two-dimensional surface of the Earth, for example, is finite, yet has no edge. By travelling in a straight line with respect to the Earth's curvature, one will eventually return to the exact spot one started from. The universe, at least in principle, might have a similartopology. If so, one might eventually return to one's starting point after travelling in a straight line through the universe for long enough.[51]

The curvature of the universe can be measured throughmultipole moments in the spectrum of thecosmic background radiation. To date, analysis of the radiation patterns recorded by theWMAP spacecraft hints that the universe has a flat topology. This would be consistent with an infinite physical universe.[52][53][54]

However, the universe could be finite, even if its curvature is flat. An easy way to understand this is to consider two-dimensional examples, such as video games where items that leave one edge of the screen reappear on the other. The topology of such games istoroidal and the geometry is flat. Many possible bounded, flat possibilities also exist for three-dimensional space.[55]

The concept of infinity also extends to themultiverse hypothesis, which, when explained by astrophysicists such asMichio Kaku, posits that there are an infinite number and variety of universes.[56] Also,cyclic models posit an infinite amount ofBig Bangs, resulting in an infinite variety of universes after each Big Bang event in an infinite cycle.[57]

Computing

[edit]

TheIEEE floating-point standard (IEEE 754) specifies a positive and a negative infinity value (and alsoindefinite values). These are defined as the result ofarithmetic overflow,division by zero, and other exceptional operations.[58]

Someprogramming languages, such asJava[59] andJ,[60] allow the programmer an explicit access to the positive and negative infinity values as language constants. These can be used asgreatest and least elements, as they compare (respectively) greater than or less than all other values. They have uses assentinel values inalgorithms involvingsorting,searching, orwindowing.[citation needed]

In languages that do not have greatest and least elements but do allowoverloading ofrelational operators, it is possible for a programmer tocreate the greatest and least elements. In languages that do not provide explicit access to such values from the initial state of the program but do implement the floating-pointdata type, the infinity values may still be accessible and usable as the result of certain operations.[citation needed]

In programming, aninfinite loop is aloop whose exit condition is never satisfied, thus executing indefinitely.

Arts, games, and cognitive sciences

[edit]

Perspective artwork uses the concept ofvanishing points, roughly corresponding to mathematicalpoints at infinity, located at an infinite distance from the observer. This allows artists to create paintings that realistically render space, distances, and forms.[61] ArtistM.C. Escher is specifically known for employing the concept of infinity in his work in this and other ways.[62]

Variations ofchess played on an unbounded board are calledinfinite chess.[63][64]

Cognitive scientistGeorge Lakoff considers the concept of infinity in mathematics and the sciences as a metaphor. This perspective is based on the basic metaphor of infinity (BMI), defined as the ever-increasing sequence <1, 2, 3, …>.[65]

See also

[edit]

References

[edit]
  1. ^abcdefAllen, Donald (2003)."The History of Infinity"(PDF).Texas A&M Mathematics. Archived fromthe original(PDF) on August 1, 2020. RetrievedNov 15, 2019.
  2. ^abJesseph, Douglas Michael (1998-05-01)."Leibniz on the Foundations of the Calculus: The Question of the Reality of Infinitesimal Magnitudes".Perspectives on Science.6 (1&2):6–40.doi:10.1162/posc_a_00543.ISSN 1063-6145.OCLC 42413222.S2CID 118227996. Archived fromthe original on 11 January 2012. Retrieved1 November 2019 – via Project MUSE.
  3. ^Gowers, Timothy; Barrow-Green, June (2008).The Princeton companion to mathematics. Imre Leader, Princeton University. Princeton: Princeton University Press.ISBN 978-1-4008-3039-8.OCLC 659590835.
  4. ^Maddox 2002, pp. 113–117
  5. ^McLarty, Colin (15 January 2014) [September 2010]."What Does it Take to Prove Fermat's Last Theorem? Grothendieck and the Logic of Number Theory".The Bulletin of Symbolic Logic.16 (3):359–377.doi:10.2178/bsl/1286284558.S2CID 13475845 – via Cambridge University Press.
  6. ^Wallace 2004, p. 44
  7. ^Aristotle.Physics. Translated by Hardie, R. P.; Gaye, R. K. The Internet Classics Archive. Book 3, Chapters 5–8.
  8. ^Goodman, Nicolas D. (1981). "Reflections on Bishop's philosophy of mathematics". In Richman, F. (ed.).Constructive Mathematics. Lecture Notes in Mathematics. Vol. 873. Springer. pp. 135–145.doi:10.1007/BFb0090732.ISBN 978-3-540-10850-4.
  9. ^Maor, p. 3
  10. ^Sarton, George (March 1928)."The Thirteen Books of Euclid's Elements. Thomas L. Heath, Heiberg".Isis.10 (1):60–62.doi:10.1086/346308.ISSN 0021-1753 – via The University of Chicago Press Journals.
  11. ^Hutten, Ernest Hirschlaff (1962).The origins of science; an inquiry into the foundations of Western thought. Internet Archive. London, Allen and Unwin. pp. 1–241.ISBN 978-0-04-946007-2. Retrieved2020-01-09.{{cite book}}:ISBN / Date incompatibility (help)
  12. ^Euclid (2008) [c. 300 BC].Euclid's Elements of Geometry(PDF). Translated by Fitzpatrick, Richard. Lulu.com. p. 6 (Book I, Postulate 5).ISBN 978-0-6151-7984-1.
  13. ^Heath, Sir Thomas Little; Heiberg, Johan Ludvig (1908).The Thirteen Books of Euclid's Elements. Vol. v. 1. The University Press. p. 212.
  14. ^Drozdek, Adam (2008).In the Beginning Was the Apeiron: Infinity in Greek Philosophy. Stuttgart, Germany: Franz Steiner Verlag.ISBN 978-3-515-09258-6.
  15. ^"Zeno's Paradoxes".Stanford University. October 15, 2010. RetrievedApril 3, 2017.
  16. ^Russell 1996, p. 347
  17. ^Cauchy, Augustin-Louis (1821).Cours d'Analyse de l'École Royale Polytechnique. Libraires du Roi & de la Bibliothèque du Roi. p. 124. RetrievedOctober 12, 2019.
  18. ^Ian Stewart (2017).Infinity: a Very Short Introduction. Oxford University Press. p. 117.ISBN 978-0-19-875523-4.Archived from the original on April 3, 2017.
  19. ^Cajori, Florian (2007).A History of Mathematical Notations. Vol. 1. Cosimo, Inc. p. 214.ISBN 9781602066854.
  20. ^Cajori 1993, Sec. 421, Vol. II, p. 44
  21. ^"Arithmetica Infinitorum".
  22. ^Cajori 1993, Sec. 435, Vol. II, p. 58
  23. ^Grattan-Guinness, Ivor (2005).Landmark Writings in Western Mathematics 1640-1940. Elsevier. p. 62.ISBN 978-0-08-045744-4.Archived from the original on 2016-06-03.Extract of p. 62
  24. ^AG, Compart."Unicode Character "∞" (U+221E)".Compart.com. Retrieved2019-11-15.
  25. ^"List of LaTeX mathematical symbols - OeisWiki".oeis.org. Retrieved2019-11-15.
  26. ^Scott, Joseph Frederick (1981),The mathematical work of John Wallis, D.D., F.R.S., (1616–1703) (2 ed.),American Mathematical Society, p. 24,ISBN 978-0-8284-0314-6,archived from the original on 2016-05-09
  27. ^Martin-Löf, Per (1990), "Mathematics of infinity",COLOG-88 (Tallinn, 1988), Lecture Notes in Computer Science, vol. 417, Berlin: Springer, pp. 146–197,doi:10.1007/3-540-52335-9_54,ISBN 978-3-540-52335-2,MR 1064143
  28. ^O'Flaherty, Wendy Doniger (1986),Dreams, Illusion, and Other Realities, University of Chicago Press, p. 243,ISBN 978-0-226-61855-5,archived from the original on 2016-06-29
  29. ^Toker, Leona (1989),Nabokov: The Mystery of Literary Structures, Cornell University Press, p. 159,ISBN 978-0-8014-2211-9,archived from the original on 2016-05-09
  30. ^Bell, John Lane."Continuity and Infinitesimals". InZalta, Edward N. (ed.).Stanford Encyclopedia of Philosophy.
  31. ^Taylor 1955, p. 63
  32. ^These uses of infinity for integrals and series can be found in any standard calculus text, such as,Swokowski 1983, pp. 468–510
  33. ^"Properly Divergent Sequences - Mathonline".mathonline.wikidot.com. Retrieved2019-11-15.
  34. ^Aliprantis, Charalambos D.; Burkinshaw, Owen (1998),Principles of Real Analysis (3rd ed.), San Diego, CA: Academic Press, Inc., p. 29,ISBN 978-0-12-050257-8,MR 1669668,archived from the original on 2015-05-15
  35. ^Gemignani 1990, p. 177
  36. ^Beutelspacher, Albrecht; Rosenbaum, Ute (1998),Projective Geometry / from foundations to applications, Cambridge University Press, p. 27,ISBN 978-0-521-48364-3
  37. ^Rao, Murali; Stetkær, Henrik (1991).Complex Analysis: An Invitation : a Concise Introduction to Complex Function Theory. World Scientific. p. 113.ISBN 9789810203757.
  38. ^Eric Schechter (2005).Classical and Nonclassical Logics: An Introduction to the Mathematics of Propositions (illustrated ed.). Princeton University Press. p. 118.ISBN 978-0-691-12279-3.Extract of page 118
  39. ^A.W. Moore (2012).The Infinite (2nd, revised ed.). Routledge. p. xiv.ISBN 978-1-134-91213-1.Extract of page xiv
  40. ^Rudolf V Rucker (2019).Infinity and the Mind: The Science and Philosophy of the Infinite (illustrated ed.). Princeton University Press. p. 85.ISBN 978-0-691-19125-6.Extract of page 85
  41. ^Dauben, Joseph (1993)."Georg Cantor and the Battle for Transfinite Set Theory"(PDF).9th ACMS Conference Proceedings: 4.
  42. ^Cohen 1963, p. 1143
  43. ^Felix Hausdorff (2021).Set Theory. American Mathematical Soc. p. 44.ISBN 978-1-4704-6494-3.Extract of page 44
  44. ^Sagan 1994, pp. 10–12
  45. ^Michael Frame; Benoit Mandelbrot (2002).Fractals, Graphics, and Mathematics Education (illustrated ed.). Cambridge University Press. p. 36.ISBN 978-0-88385-169-2.Extract of page 36
  46. ^Kline 1972, pp. 1197–1198
  47. ^Cambridge Dictionary of Philosophy, Second Edition, p. 429
  48. ^Doric LensesArchived 2013-01-24 at theWayback Machine – Application Note – Axicons – 2. Intensity Distribution. Retrieved 7 April 2014.
  49. ^John Gribbin (2009),In Search of the Multiverse: Parallel Worlds, Hidden Dimensions, and the Ultimate Quest for the Frontiers of Reality,ISBN 978-0-470-61352-8. p. 88
  50. ^Brake, Mark (2013).Alien Life Imagined: Communicating the Science and Culture of Astrobiology (illustrated ed.). Cambridge University Press. p. 63.ISBN 978-0-521-49129-7.
  51. ^Koupelis, Theo; Kuhn, Karl F. (2007).In Quest of the Universe (illustrated ed.). Jones & Bartlett Learning. p. 553.ISBN 978-0-7637-4387-1.Extract of p. 553
  52. ^"Will the Universe expand forever?". NASA. 24 January 2014.Archived from the original on 1 June 2012. Retrieved16 March 2015.
  53. ^"Our universe is Flat". FermiLab/SLAC. 7 April 2015.Archived from the original on 10 April 2015.
  54. ^Marcus Y. Yoo (2011). "Unexpected connections".Engineering & Science. LXXIV1: 30.
  55. ^Weeks, Jeffrey (2001).The Shape of Space. CRC Press.ISBN 978-0-8247-0709-5.
  56. ^Kaku, M. (2006). Parallel worlds. Knopf Doubleday Publishing Group.
  57. ^McKee, Maggie (25 September 2014)."Ingenious: Paul J. Steinhardt – The Princeton physicist on what's wrong with inflation theory and his view of the Big Bang".Nautilus. No. 17. NautilusThink Inc. Retrieved31 March 2017.
  58. ^"Infinity and NaN (The GNU C Library)".www.gnu.org. Retrieved2021-03-15.
  59. ^Gosling, James; et al. (27 July 2012)."4.2.3.".The Java Language Specification (Java SE 7 ed.). California: Oracle America, Inc.Archived from the original on 9 June 2012. Retrieved6 September 2012.
  60. ^Stokes, Roger (July 2012)."19.2.1".Learning J. Archived fromthe original on 25 March 2012. Retrieved6 September 2012.
  61. ^Morris Kline (1985).Mathematics for the Nonmathematician (illustrated, unabridged, reprinted ed.). Courier Corporation. p. 229.ISBN 978-0-486-24823-3.Extract of page 229, Section 10-7
  62. ^Schattschneider, Doris (2010)."The Mathematical Side of M. C. Escher"(PDF).Notices of the AMS.57 (6):706–718.
  63. ^Infinite chess at the Chess Variant PagesArchived 2017-04-02 at theWayback Machine An infinite chess scheme.
  64. ^"Infinite Chess, PBS Infinite Series"Archived 2017-04-07 at theWayback Machine PBS Infinite Series, with academic sources by J. Hamkins (infinite chess:Evans, C.D.A; Joel David Hamkins (2013). "Transfinite game values in infinite chess".arXiv:1302.4377 [math.LO]. andEvans, C.D.A; Joel David Hamkins; Norman Lewis Perlmutter (2015). "A position in infinite chess with game value $ω^4$".arXiv:1510.08155 [math.LO].).
  65. ^Elglaly, Yasmine Nader; Quek, Francis."Review of "Where Mathematics comes from: How the Embodied Mind Brings Mathematics Into Being" By George Lakoff and Rafael E. Nunez"(PDF).CHI 2009. Archived fromthe original(PDF) on 2020-02-26. Retrieved2021-03-25.

Bibliography

[edit]

Sources

[edit]

External links

[edit]
Infinity at Wikipedia'ssister projects
Look upinfinity in Wiktionary, the free dictionary.
Wikibooks has a book on the topic of:Infinity is not a number
History
Branches of mathematics
Formalizations of infinity
Geometries
Mathematicians
Examples
in
numerical
order
Expression
methods
Notations
Operators
Related
articles
(alphabetical
order)
Major topics inmathematical analysis
International
National
Retrieved from "https://en.wikipedia.org/w/index.php?title=Infinity&oldid=1322875146"
Categories:
Hidden categories:
[8]ページ先頭
©2009-2025 Movatter.jp