2000:
40:
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2704:. The question of being infinite is logically separate from the question of having boundaries. The two-dimensional surface of the Earth, for example, is finite, yet has no edge. By travelling in a straight line with respect to the Earth's curvature, one will eventually return to the exact spot one started from. The universe, at least in principle, might have a similar
2075:. 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".
2766:
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
2102:
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
2468:
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
271:
Other 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. Finally, it has been maintained that a reflection on infinity, far from eliciting a "horror of the
4929:
H. Jerome
Keisler: Elementary Calculus: An Approach Using Infinitesimals. First edition 1976; 2nd edition 1986. This book is now out of print. The publisher has reverted the copyright to the author, who has made available the 2nd edition in .pdf format available for downloading at
1829:. Arithmetic operations similar to those given above for the extended real numbers can also be defined, though there is no distinction in the signs (which leads to the one exception that infinity cannot be added to itself). On the other hand, this kind of infinity enables
2516:
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.
2113:. Cantor's views prevailed and modern mathematics accepts actual infinity as part of a consistent and coherent theory. Certain extended number systems, such as the hyperreal numbers, incorporate the ordinary (finite) numbers and infinite numbers of different sizes.
2726:
However, the universe could be finite, even if its curvature is flat. An easy way to understand this is to consider two-dimensional examples, such as video games where items that leave one edge of the screen reappear on the other. The topology of such games is
297:
430 BC) did not advance any views concerning the infinite. Nevertheless, his paradoxes, especially "Achilles and the
Tortoise", were important contributions in that they made clear the inadequacy of popular conceptions. The paradoxes were described by
830:, speculated widely about infinite numbers and their use in mathematics. To Leibniz, both infinitesimals and infinite quantities were ideal entities, not of the same nature as appreciable quantities, but enjoying the same properties in accordance with the
335:
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.
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603:
266:
If a straight line falling across two straight lines makes internal angles on the same side less than two right angles, then the two straight lines, being produced to infinity, meet on that side that the is less than two right
472:
2602:
object is reiterated in its magnifications. Fractals can be magnified indefinitely without losing their structure and becoming "smooth"; they have infinite perimeters and can have infinite or finite areas. One such
2903:
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,...>.
4842:
2245:
476:
Suppose that
Achilles is running at 10 meters per second, the tortoise is walking at 0.1 meters per second, and the latter has a 100-meter head start. The duration of the chase fits Cauchy's pattern with
1217:
1308:
2314:
708:(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."
254:(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." It has also been maintained, that, in proving the
1510:
2571:, generally two or three. However, this is not implied by the abstract definition of a vector space, and vector spaces of infinite dimension can be considered. This is typically the case in
1084:
536:
147:
The mathematical concept of infinity refines and extends the old philosophical concept, in particular by introducing infinitely many different sizes of infinite sets. Among the axioms of
1426:
2686:: "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."
702:
155:, which guarantees the existence of infinite sets. The mathematical concept of infinity and the manipulation of infinite sets are widely used in mathematics, even in areas such as
1732:
894:
5112:, by Peter Suber. How Cantor's mathematics of the infinite solves a handful of ancient philosophical problems of the infinite. From the St. John's Review, XLIV, 2 (1998) 1–59.
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the greatest and least elements. In languages that do not provide explicit access to such values from the initial state of the program but do implement the floating-point
2147:
1865:
1002:
1917:
at the poles. The domain of a complex-valued function may be extended to include the point at infinity as well. One important example of such functions is the group of
2444:, 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.
1622:
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1973:. For example, if H is an infinite number in this sense, then H + H = 2H and H + 1 are distinct infinite numbers. This approach to
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83:
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sets, or counting carried on to any stopping point, including points after an infinite number have already been counted. Generalizing finite and (ordinary) infinite
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The first published proposal that the universe is infinite came from Thomas Digges in 1576. Eight years later, in 1584, the
Italian philosopher and astronomer
717:
112:) regarded as infinitely small quantities, but infinity continued to be associated with endless processes. As mathematicians struggled with the foundation of
5172:
4984:
2879:, located at an infinite distance from the observer. This allows artists to create paintings that realistically render space, distances, and forms. Artist
2508:
intersecting "at infinity". Mathematically, points at infinity have the advantage of allowing one to not consider some special cases. For example, in a
3042:
2622:
was skeptical of the notion of infinity and how his fellow mathematicians were using it in the 1870s and 1880s. This skepticism was developed in the
272:
infinite", underlay all of early Greek philosophy and that
Aristotle's "potential infinity" is an aberration from the general trend of this period.
4375:
5152:
4964:
132:, showing that they can be of various sizes. For example, if a line is viewed as the set of all of its points, their infinite number (i.e., the
4565:
3002:
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246:, which he regarded as impossible due to the various paradoxes it seemed to produce. It has been argued that, in line with this view, the
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quantities. In the second half of the 20th century, it was shown that this treatment could be put on a rigorous footing through various
5117:
4500:
2708:. If so, one might eventually return to one's starting point after travelling in a straight line through the universe for long enough.
640:
In the 17th century, European mathematicians started using infinite numbers and infinite expressions in a systematic fashion. In 1655,
96:
has been the subject of many discussions among philosophers. In the 17th century, with the introduction of the infinity symbol and the
5944:
208:
did not define infinity in precise formalism as does modern mathematics, and instead approached infinity as a philosophical concept.
4566:"Review of "Where Mathematics comes from: How the Embodied Mind Brings Mathematics Into Being" By George Lakoff and Rafael E. Nunez"
1965:. In the latter, infinitesimals are invertible, and their inverses are infinite numbers. The infinities in this sense are part of a
4108:
2806:, allow the programmer an explicit access to the positive and negative infinity values as language constants. These can be used as
2193:
1672:, the complex plane can be "wrapped" onto a sphere, with the top point of the sphere corresponding to infinity. This is called the
5564:
5280:
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2532:. With the universal use of set theory in mathematics, the point of view has dramatically changed: a line is now considered as
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Apparently, Achilles never overtakes the tortoise, since however many steps he completes, the tortoise remains ahead of him.
4305:"Ingenious: Paul J. Steinhardt – The Princeton physicist on what's wrong with inflation theory and his view of the Big Bang"
3603:
2040:. This modern mathematical conception of the quantitative infinite developed in the late 19th century from works by Cantor,
2255:
4542:
Evans, C.D.A; Joel David
Hamkins; Norman Lewis Perlmutter (2015). "A position in infinite chess with game value $ ω^4$ ".
1253:
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5058:
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In addition to defining a limit, infinity can be also used as a value in the extended real number system. Points labeled
3377:
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since both are infinite sets.) An infinite set can simply be defined as one having the same size as at least one of its
6180:
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spacecraft hints that the universe has a flat topology. This would be consistent with an infinite physical universe.
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5187:
4999:
4129:
In Search of the
Multiverse: Parallel Worlds, Hidden Dimensions, and the Ultimate Quest for the Frontiers of Reality
258:, Euclid "was the first to overcome the horror of the infinite". There is a similar controversy concerning Euclid's
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Step #4: Achilles advances to where the tortoise was at the end of Step #3 while the tortoise goes yet further.
318:
Step #3: Achilles advances to where the tortoise was at the end of Step #2 while the tortoise goes yet further.
315:
Step #2: Achilles advances to where the tortoise was at the end of Step #1 while the tortoise goes yet further.
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The second result was proved by Cantor in 1878, but only became intuitively apparent in 1890, when
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1958:
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and the geometry is flat. Many possible bounded, flat possibilities also exist for three-dimensional space.
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811:, and since its introduction, it has also been used outside mathematics in modern mysticism and literary
4367:
2107:. If a set is too large to be put in one-to-one correspondence with the positive integers, it is called
6165:
5848:
5070:
3225:
Goodman, Nicolas D. (1981). "Reflections on Bishop's philosophy of mathematics". In
Richman, F. (ed.).
1814:
1625:
5521:
5149:
4961:
4825:
3560:
2630:, an extreme form of mathematical philosophy in the general philosophical and mathematical schools of
2477:
that satisfies some property" (singular), where modern mathematicians would generally say "the set of
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5698:
5516:
3872:
These uses of infinity for integrals and series can be found in any standard calculus text, such as,
3373:
3043:"Leibniz on the Foundations of the Calculus: The Question of the Reality of Infinitesimal Magnitudes"
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2525:
2371:
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823:
25:
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1918:
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221:
117:
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is specifically known for employing the concept of infinity in his work in this and other ways.
2750:, resulting in an infinite variety of universes after each Big Bang event in an infinite cycle.
2362:, showing that there are as many points in a one-dimensional line as in a two-dimensional square
972:
6108:
6008:
5906:
5801:
5550:
5223:
5093:, by Peter Suber. From the St. John's Review, XLIV, 2 (1998) 1–59. The stand-alone appendix to
3214:. Translated by Hardie, R. P.; Gaye, R. K. The Internet Classics Archive. Book 3, Chapters 5–8.
3159:"What Does it Take to Prove Fermat's Last Theorem? Grothendieck and the Logic of Number Theory"
2939:
2810:, as they compare (respectively) greater than or less than all other values. They have uses as
2456:, except in the context of processes that could be continued without any limit. For example, a
2375:
2099:
1942:
1645:
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97:
4453:
4437:
4194:
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2849:, the infinity values may still be accessible and usable as the result of certain operations.
2055:
as a standard for comparing the size of sets, and to reject the view of
Galileo (derived from
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6060:
6056:
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5954:
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Bell, J.L.: Continuity and infinitesimals. Stanford Encyclopedia of philosophy. Revised 2009.
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4150:
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2696:: Are there an infinite number of stars? Does the universe have infinite volume? Does space "
1974:
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where Galileo concludes that positive integers cannot be compared to the subset of positive
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1962:
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167:, very large infinite sets, for solving a long-standing problem that is stated in terms of
2268:
between the cardinality of the reals and the cardinality of the natural numbers, that is,
1313:
1222:
1089:
775:) is a mathematical symbol representing the concept of infinity. The symbol is encoded in
598:{\displaystyle ={\frac {10}{1-0.01}}={\frac {10}{0.99}}=10.10101\ldots {\text{ seconds}}.}
8:
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Step #1: Achilles runs to the tortoise's starting point while the tortoise walks forward.
141:
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4855:
3988:
Complex Analysis: An Invitation : a Concise Introduction to Complex Function Theory
3735:, Lecture Notes in Computer Science, vol. 417, Berlin: Springer, pp. 146–197,
1435:
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2016:
1970:
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1008:
951:
900:
614:
259:
129:
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4528:
Evans, C.D.A; Joel David Hamkins (2013). "Transfinite game values in infinite chess".
3669:
617:
text Surya Prajnapti (c. 4th–3rd century BCE) classifies all numbers into three sets:
230:, which means "unbounded", "indefinite", and perhaps can be translated as "infinite".
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2347:, but also that this is equal to the number of points on a plane and, indeed, in any
2072:
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1787:
1653:
1641:
1567:
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621:, innumerable, and infinite. Each of these was further subdivided into three orders:
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89:
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3182:
183:
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5893:
5713:
5663:
5531:
5526:
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5311:
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Singh, Navjyoti (1988). "Jaina Theory of Actual Infinity and Transfinite Numbers".
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of the complex plane. When this is done, the resulting space is a one-dimensional
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4118:– Application Note – Axicons – 2. Intensity Distribution. Retrieved 7 April 2014.
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3352:. Translated by Fitzpatrick, Richard. Lulu.com. p. 6 (Book I, Postulate 5).
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2827:
2779:
standard (IEEE 754) specifies a positive and a negative infinity value (and also
2513:
2497:
2485:
2457:
2330:
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One of Cantor's most important results was that the cardinality of the continuum
2109:
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2012:
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242:
101:
63:
44:
21:
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5205:(compilation of articles about infinity in physics, mathematics, and philosophy)
5198:
The Mystery Of The Aleph: Mathematics, the Kabbalah, and the Search for Infinity
4804:
The Mystery of the Aleph: Mathematics, the Kabbalah, and the Search for Infinity
2464:, with the proviso that one can extend it as far as one wants; but extending it
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5774:
5673:
5454:
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5403:
5398:
4843:
Proceedings of the National Academy of Sciences of the United States of America
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2929:
2891:
2811:
2679:
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measurements (i.e., counting). Concepts of infinite things such as an infinite
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2608:
2576:
2505:
2429:
2079:
2037:
2008:
1994:
1954:
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1826:
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Innumerable: nearly innumerable, truly innumerable, and innumerably innumerable
205:
144:
can be studied, manipulated, and used just like any other mathematical object.
59:
4931:
4647:
4541:
4516:
2436:, curved lines that twist and turn enough to fill the whole of any square, or
1937:
Infinitesimals (ε) and infinities (ω) on the hyperreal number line (1/ε = ω/1)
6159:
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5511:
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1950:
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3116:. Imre Leader, Princeton University. Princeton: Princeton University Press.
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2103:
positive integers, and any set which has the cardinality of the integers is
250:
Greeks had a "horror of the infinite" which would, for example, explain why
5878:
5506:
5418:
5362:
4883:
4864:
4820:
4675:
3229:. Lecture Notes in Mathematics. Vol. 873. Springer. pp. 135–145.
2949:
2919:
2743:
2742:, posits that there are an infinite number and variety of universes. Also,
2635:
2560:
2461:
2344:
2087:
2020:
1946:
1933:
729:
712:
331:
Zeno was not attempting to make a point about infinity. As a member of the
286:
201:
125:
121:
4526:
PBS Infinite Series, with academic sources by J. Hamkins (infinite chess:
4152:
Alien Life Imagined: Communicating the Science and Culture of Astrobiology
3313:
The origins of science; an inquiry into the foundations of Western thought
3111:
2528:, points and lines were viewed as distinct entities, and a point could be
467:{\displaystyle a+ax+ax^{2}+ax^{3}+ax^{4}+ax^{5}+\cdots ={\frac {a}{1-x}}.}
116:, it remained unclear whether infinity could be considered as a number or
5888:
5828:
5367:
5125:
5097:, below. A concise introduction to Cantor's mathematics of infinite sets.
5010:
3060:
2880:
2739:
2651:
2366:
The first of these results is apparent by considering, for instance, the
2325:
This hypothesis cannot be proved or disproved within the widely accepted
1990:
808:
641:
343:
provided both a satisfactory definition of a limit and a proof that, for
247:
217:
133:
5202:
5976:
5678:
5326:
4659:
Mathematical Thinking and Writing: A Transition to Abstract Mathematics
3234:
2833:
In languages that do not have greatest and least elements but do allow
2735:
2667:
2521:
2024:
2003:
One-to-one correspondence between an infinite set and its proper subset
1516:
to infinity, in the sense that the partial sums increase without bound.
772:
618:
2860:
whose exit condition is never satisfied, thus executing indefinitely.
1574:
of the real numbers. Adding algebraic properties to this gives us the
39:
5784:
5779:
5613:
5449:
2964:
2846:
2815:
2692:
have long sought to discover whether infinity exists in our physical
2689:
2663:
2441:
812:
233:
200:
Ancient cultures had various ideas about the nature of infinity. The
179:
4899:
Jain, L.C. (1973). "Set theory in the Jaina school of mathematics",
4840:
Cohen, Paul (1963), "The Independence of the Continuum Hypothesis",
4182:(illustrated ed.). Jones & Bartlett Learning. p. 553.
3622:
Levels of Infinity / Selected Writings on Mathematics and Philosophy
2452:
Until the end of the 19th century, infinity was rarely discussed in
2059:) that the whole cannot be the same size as the part. (However, see
668:, and exploited it in area calculations by dividing the region into
120:
and, if so, how this could be done. At the end of the 19th century,
5971:
5966:
5347:
4937:
4548:
4293:
Kaku, M. (2006). Parallel worlds. Knopf Doubleday Publishing Group.
3288:
2914:
2747:
2705:
2693:
2627:
2564:
2453:
2095:
2091:
715:
wrote about equations with an infinite number of terms in his work
305:
113:
32:
5053:
4534:
1664:
140:. In this usage, infinity is a mathematical concept, and infinite
5608:
5265:
5015:
Infinity and the Mind: The Science and Philosophy of the Infinite
3916:(3rd ed.), San Diego, CA: Academic Press, Inc., p. 29,
2819:
2647:
2599:
2358:
The first three steps of a fractal construction whose limit is a
776:
332:
226:
175:
137:
5542:
4830:
4155:(illustrated ed.). Cambridge University Press. p. 63.
216:
The earliest recorded idea of infinity in Greece may be that of
5863:
5668:
5603:
5208:
3696:
The mathematical work of John Wallis, D.D., F.R.S., (1616–1703)
3471:. Libraires du Roi & de la Bibliothèque du Roi. p. 124
2484:
One of the rare exceptions of a mathematical concept involving
2056:
251:
151:, on which most of modern mathematics can be developed, is the
4608:
A History of Mathematical Notations (Two Volumes Bound as One)
2719:. To date, analysis of the radiation patterns recorded by the
631:
Infinite: nearly infinite, truly infinite, infinitely infinite
3316:. Internet Archive. London, Allen and Unwin. pp. 1–241.
2887:
2759:
2738:
hypothesis, which, when explained by astrophysicists such as
2728:
2670:
exist, but there are no experimental means to generate them.
2354:
2240:{\displaystyle \mathbf {c} =2^{\aleph _{0}}>{\aleph _{0}}}
1969:; there is no equivalence between them as with the Cantorian
797:
732:
opened a mathematico-philosophic address given in 1930 with:
58:
is something which is boundless, endless, or larger than any
48:
2339:
can be used to show not only that the number of points in a
159:
that may seem to have nothing to do with them. For example,
4909:
The Crest of the Peacock: Non-European Roots of Mathematics
2720:
2437:
2586:
of infinite dimension. In particular, this is the case of
1212:{\displaystyle \int _{-\infty }^{\infty }f(t)\,dt=\infty }
5952:
5593:
5076:
4682:, New York: Oxford University Press, pp. 1197–1198,
2780:
2051:
Dedekind's approach was essentially to adopt the idea of
4053:"Georg Cantor and the Battle for Transfinite Set Theory"
124:
enlarged the mathematical study of infinity by studying
4527:
4374:(Java SE 7 ed.). California: Oracle America, Inc.
3963:
Projective Geometry / from foundations to applications
3960:
2863:
2711:
The curvature of the universe can be measured through
718:
De analysi per aequationes numero terminorum infinitas
680:
4649:
Elementary Calculus: An Approach Using Infinitesimals
3910:
Aliprantis, Charalambos D.; Burkinshaw, Owen (1998),
3299:– via The University of Chicago Press Journals.
2274:
2196:
2155:
2133:
1903:
1878:
1839:
1794:
1771:
1740:
1714:
1690:
1607:
1584:
1549:
1526:
1462:
1438:
1378:
1345:
1316:
1256:
1225:
1165:
1141:
1121:
1092:
1035:
1011:
975:
954:
924:
903:
876:
852:
757:
678:
650:
543:
500:
491:. Achilles does overtake the tortoise; it takes him
357:
71:
4749:(Alternate ed.), Prindle, Weber & Schmidt,
3909:
3417:
3382:. Vol. v. 1. The University Press. p. 212.
2023:. In this system, the first transfinite cardinal is
1897:
as maps into the Riemann sphere taking the value of
1303:{\displaystyle \int _{-\infty }^{\infty }f(t)\,dt=a}
866:, called "infinity", is used to denote an unbounded
2579:are generally vector spaces of infinite dimension.
2309:{\displaystyle \mathbf {c} =\aleph _{1}=\beth _{1}}
1512:means that the sum of the infinite series properly
24:. For other uses of "Infinity" and "Infinite", see
5083:A Crash Course in the Mathematics of Infinite Sets
4980:O'Connor, John J. and Edmund F. Robertson (2000).
4960:O'Connor, John J. and Edmund F. Robertson (1998).
4777:Everything and More: A Compact History of Infinity
4429:
4266:
3589:Landmark Writings in Western Mathematics 1640-1940
3109:
2607:with an infinite perimeter and finite area is the
2308:
2239:
2170:
2141:
2048:and others—using the idea of collections or sets.
1909:
1893:. In this context, it is often useful to consider
1884:
1859:
1800:
1777:
1756:
1726:
1704:, called "infinity", denotes an unsigned infinite
1696:
1616:
1593:
1558:
1535:
1504:
1447:
1420:
1354:
1331:
1302:
1240:
1211:
1150:
1127:
1107:
1078:
1017:
996:
960:
939:
909:
888:
858:
763:
696:
656:
597:
530:
466:
308:races a tortoise, giving the latter a head start.
275:
77:
5168:John J. O'Connor and Edmund F. Robertson (2000).
5148:John J. O'Connor and Edmund F. Robertson (1998).
4680:Mathematical Thought from Ancient to Modern Times
1505:{\displaystyle \sum _{i=0}^{\infty }f(i)=\infty }
282:Zeno's paradoxes § Achilles and the tortoise
6157:
4250:Marcus Y. Yoo (2011). "Unexpected connections".
3961:Beutelspacher, Albrecht; Rosenbaum, Ute (1998),
3585:
2547:In particular, in modern mathematics, lines are
4471:
3670:"List of LaTeX mathematical symbols - OeisWiki"
3468:Cours d'Analyse de l'École Royale Polytechnique
2614:
2116:
47:, it seems that there is a boundless amount of
3770:
2078:Cantor defined two kinds of infinite numbers:
5938:
5558:
5224:
4249:
2582:In topology, some constructions can generate
1570:of the real numbers, producing the two-point
1079:{\displaystyle \int _{a}^{b}f(t)\,dt=\infty }
625:Enumerable: lowest, intermediate, and highest
4366:Gosling, James; et al. (27 July 2012).
4296:
4175:
3777:, University of Chicago Press, p. 243,
3403:. Stuttgart, Germany: Franz Steiner Verlag.
3372:
3110:Gowers, Timothy; Barrow-Green, June (2008).
2783:values). These are defined as the result of
2734:The concept of infinity also extends to the
2544:(however, the latter phrase is still used).
2149:is greater than that of the natural numbers
531:{\displaystyle 10+0.1+0.001+0.00001+\cdots }
4932:http://www.math.wisc.edu/~keisler/calc.html
3887:"Properly Divergent Sequences - Mathonline"
3805:Nabokov: The Mystery of Literary Structures
3489:
1825:, called the extended complex plane or the
1421:{\displaystyle \sum _{i=0}^{\infty }f(i)=a}
736:Mathematics is the science of the infinite.
5945:
5931:
5565:
5551:
5231:
5217:
4826:Ancient Jaina Mathematics: an Introduction
4225:
4199:
3984:
3965:, Cambridge University Press, p. 27,
3727:
1428:means that the sum of the infinite series
184:whether the universe is spatially infinite
136:of the line) is larger than the number of
4873:
4863:
4742:
4624:
4547:
4533:
4498:Infinite chess at the Chess Variant Pages
3948:
3873:
3808:, Cornell University Press, p. 159,
3620:Weyl, Hermann (2012), Peter Pesic (ed.),
3527:. Vol. 1. Cosimo, Inc. p. 214.
3036:
3034:
2071:parts; this notion of infinity is called
1287:
1196:
1063:
969:decreases without bound. For example, if
4936:
3496:. Oxford University Press. p. 117.
2890:played on an unbounded board are called
2875:, roughly corresponding to mathematical
2353:
2343:is equal to the number of points in any
1998:
1932:
1663:
104:and what some mathematicians (including
38:
5192:MacTutor History of Mathematics archive
5181:MacTutor History of Mathematics archive
5162:MacTutor History of Mathematics archive
5150:'Georg Ferdinand Ludwig Philipp Cantor'
5004:MacTutor History of Mathematics archive
4993:MacTutor History of Mathematics archive
4974:MacTutor History of Mathematics archive
4962:'Georg Ferdinand Ludwig Philipp Cantor'
4771:
4695:
4642:
4564:Elglaly, Yasmine Nader; Quek, Francis.
4563:
4474:"The Mathematical Side of M. C. Escher"
4365:
3449:
3392:
3379:The Thirteen Books of Euclid's Elements
3271:The Thirteen Books of Euclid's Elements
3224:
3195:
3185:– via Cambridge University Press.
3156:
3041:Jesseph, Douglas Michael (1998-05-01).
3040:
2178:; that is, there are more real numbers
1978:
1928:
1809:can be added to the complex plane as a
302:as "immeasurably subtle and profound".
6158:
5009:
4906:
4762:
4656:
4604:
4436:. Courier Dover Publications. p.
4396:
4343:"Infinity and NaN (The GNU C Library)"
4176:Koupelis, Theo; Kuhn, Karl F. (2007).
4050:
3860:
3573:
3547:
3520:
3461:
3342:
3309:
3266:
3144:
3113:The Princeton companion to mathematics
3031:
2007:A different form of "infinity" is the
1365:Infinity can also be used to describe
697:{\displaystyle {\tfrac {1}{\infty }}.}
163:implicitly relies on the existence of
161:Wiles's proof of Fermat's Last Theorem
5926:
5546:
5212:
5031:
4839:
4801:
4724:
4674:
4517:"Infinite Chess, PBS Infinite Series"
4427:
4264:
4148:
4096:
4084:
4072:
4035:
3985:Rao, Murali; Stetkær, Henrik (1991).
3801:
3774:Dreams, Illusion, and Other Realities
3692:
3207:
3000:
2841:, it is possible for a programmer to
2700:"? This is still an open question of
2554:
2015:infinities of set theory—a system of
43:Due to the constant light reflection
5281:Hilbert's paradox of the Grand Hotel
4901:Indian Journal of History of Science
4890:
4779:, Norton, W.W. & Company, Inc.,
4432:Mathematics for the nonmathematician
3619:
2996:
2994:
2992:
2990:
2988:
2986:
2791:, and other exceptional operations.
2424:Hilbert's paradox of the Grand Hotel
1727:{\displaystyle x\rightarrow \infty }
889:{\displaystyle x\rightarrow \infty }
224:Greek philosopher. He used the word
100:, mathematicians began to work with
5118:"Infinity is bigger than you think"
5059:Internet Encyclopedia of Philosophy
4303:McKee, Maggie (25 September 2014).
4239:from the original on 10 April 2015.
4207:"Will the Universe expand forever?"
3847:Stanford Encyclopedia of Philosophy
3731:(1990), "Mathematics of infinity",
3524:A History of Mathematical Notations
3510:from the original on April 3, 2017.
3493:Infinity: a Very Short Introduction
3157:McLarty, Colin (15 January 2014) .
2864:Arts, games, and cognitive sciences
2684:On the Infinite Universe and Worlds
1786:grows beyond any assigned value. A
1659:
13:
4330:Cambridge Dictionary of Philosophy
3771:O'Flaherty, Wendy Doniger (1986),
3310:Hutten, Ernest Hirschlaff (1962).
2682:proposed an unbounded universe in
2481:that have the property" (plural).
2284:
2227:
2211:
2158:
1904:
1854:
1795:
1721:
1691:
1644:in three-dimensional space, and a
1628:of the real numbers, which is the
1611:
1588:
1553:
1530:
1499:
1479:
1395:
1270:
1265:
1206:
1179:
1174:
1115:does not bound a finite area from
1073:
934:
883:
853:
758:
685:
651:
72:
14:
6192:
5839:Indefinite and fictitious numbers
5572:
5476:Differential geometry of surfaces
5115:
5046:
4893:Exact Sciences from Jaina Sources
3991:. World Scientific. p. 113.
2983:
2256:Cantor's first set theory article
2094:which are maps from the positive
2036:), the cardinality of the set of
5271:Controversy over Cantor's theory
5238:
4378:from the original on 9 June 2012
4311:. No. 17. NautilusThink Inc
4213:from the original on 1 June 2012
3832:
3693:Scott, Joseph Frederick (1981),
3646:"Unicode Character "∞" (U+221E)"
3606:from the original on 2016-06-03.
3376:; Heiberg, Johan Ludvig (1908).
2276:
2198:
2135:
1923:Möbius transformation § Overview
1310:means that the total area under
837:
672:strips of width on the order of
94:philosophical nature of infinity
6077:Least-squares spectral analysis
6004:Fundamental theorem of calculus
5332:Synthetic differential geometry
4746:Calculus with Analytic Geometry
4597:
4557:
4510:
4491:
4472:Schattschneider, Doris (2010).
4465:
4421:
4390:
4372:The Java Language Specification
4359:
4335:
4323:
4287:
4258:
4243:
4235:. FermiLab/SLAC. 7 April 2015.
4169:
4142:
4121:
4102:
4090:
4078:
4066:
4060:9th ACMS Conference Proceedings
4044:
4029:
4005:
3978:
3954:
3942:
3938:from the original on 2015-05-15
3903:
3879:
3866:
3854:
3838:"Continuity and Infinitesimals"
3826:
3822:from the original on 2016-05-09
3795:
3791:from the original on 2016-06-29
3764:
3721:
3717:from the original on 2016-05-09
3686:
3662:
3637:
3613:
3586:Grattan-Guinness, Ivor (2005).
3579:
3567:
3553:
3541:
3514:
3483:
3455:
3443:
3386:
3366:
3336:
3303:
3260:
2086:. Ordinal numbers characterize
635:
608:
276:Zeno: Achilles and the tortoise
256:infinitude of the prime numbers
5034:Journal of the Asiatic Society
5017:. Princeton University Press.
4946:. Princeton University Press.
4767:, Blaisdell Publishing Company
4625:Gemignani, Michael C. (1990),
4302:
3643:
3400:: Infinity in Greek Philosophy
3251:
3218:
3201:
3189:
3163:The Bulletin of Symbolic Logic
3150:
3138:
3103:
1750:
1742:
1734:means that the magnitude
1718:
1493:
1487:
1409:
1403:
1326:
1320:
1284:
1278:
1235:
1229:
1193:
1187:
1102:
1096:
1060:
1054:
985:
979:
928:
880:
724:
211:
51:and repetition inside of them.
1:
5754:Conway chained arrow notation
4701:The Principles of Mathematics
4332:, Second Edition, p. 429
3701:American Mathematical Society
3346:Euclid's Elements of Geometry
3267:Sarton, George (March 1928).
2976:
2190:. Namely, Cantor showed that
2171:{\displaystyle {\aleph _{0}}}
1984:
1959:smooth infinitesimal analysis
940:{\displaystyle x\to -\infty }
918:increases without bound, and
826:, one of the co-inventors of
807:It was introduced in 1655 by
294:
290:
62:. It is often denoted by the
5378:Cardinality of the continuum
2871:artwork uses the concept of
2770:
2746:posit an infinite amount of
2673:
2615:Mathematics without infinity
2536:, and one says that a point
2142:{\displaystyle \mathbf {c} }
2123:Cardinality of the continuum
2117:Cardinality of the continuum
1941:The original formulation of
1624:as the same, leading to the
7:
4743:Swokowski, Earl W. (1983),
4657:Maddox, Randall B. (2002),
4397:Stokes, Roger (July 2012).
3913:Principles of Real Analysis
3273:. Thomas L. Heath, Heiberg"
2907:
2808:greatest and least elements
2717:cosmic background radiation
2593:
2447:
2370:function, which provides a
2327:Zermelo–Fraenkel set theory
2321:Beth number § Beth one
1949:and Gottfried Leibniz used
1860:{\displaystyle z/0=\infty }
1339:is finite, and is equal to
818:
149:Zermelo–Fraenkel set theory
10:
6197:
5849:Largest known prime number
5341:Formalizations of infinity
5203:Dictionary of the Infinite
4907:Joseph, George G. (2000).
4806:. New York: Pocket Books.
4794:
3576:, Sec. 435, Vol. II, p. 58
3550:, Sec. 421, Vol. II, p. 44
2641:
2421:
2318:
2252:Cantor's diagonal argument
2249:
2120:
1988:
1815:one-point compactification
1626:one-point compactification
1219:means that the area under
997:{\displaystyle f(t)\geq 0}
744:
279:
193:
189:
18:
6181:Philosophy of mathematics
6142:
6042:
5961:
5902:
5834:Extended real number line
5814:
5767:
5749:Knuth's up-arrow notation
5736:
5727:
5580:
5517:Gottfried Wilhelm Leibniz
5499:
5468:
5340:
5304:
5253:
4763:Taylor, Angus E. (1955),
4605:Cajori, Florian (1993) ,
4507:An infinite chess scheme.
4454:Section 10-7, p. 229
4252:Engineering & Science
4209:. NASA. 24 January 2014.
3561:"Arithmetica Infinitorum"
3099:– via Project MUSE.
3010:Texas A&M Mathematics
3003:"The History of Infinity"
2624:philosophy of mathematics
2526:foundation of mathematics
2460:was what is now called a
2372:one-to-one correspondence
2053:one-to-one correspondence
740:
664:for such a number in his
26:Infinity (disambiguation)
5759:Steinhaus–Moser notation
4179:In Quest of the Universe
3741:10.1007/3-540-52335-9_54
3733:COLOG-88 (Tallinn, 1988)
3592:. Elsevier. p. 62.
3521:Cajori, Florian (2007).
3396:In the Beginning Was the
3374:Heath, Sir Thomas Little
3227:Constructive Mathematics
2777:IEEE floating-point
2753:
2563:that occur in classical
2264:states that there is no
1670:stereographic projection
1617:{\displaystyle -\infty }
1594:{\displaystyle +\infty }
1559:{\displaystyle -\infty }
1536:{\displaystyle +\infty }
644:first used the notation
262:, sometimes translated:
45:between opposing mirrors
31:Not to be confused with
5522:August Ferdinand Möbius
5305:Branches of mathematics
5296:Paradoxes of set theory
4802:Aczel, Amir D. (2001).
4629:(2nd ed.), Dover,
4265:Weeks, Jeffrey (2001).
4051:Dauben, Joseph (1993).
3048:Perspectives on Science
2945:Infinite monkey theorem
2715:in the spectrum of the
1910:{\displaystyle \infty }
1801:{\displaystyle \infty }
1697:{\displaystyle \infty }
859:{\displaystyle \infty }
764:{\displaystyle \infty }
706:Arithmetica infinitorum
657:{\displaystyle \infty }
236:(350 BC) distinguished
220:(c. 610 – c. 546 BC) a
78:{\displaystyle \infty }
6009:Calculus of variations
5982:Differential equations
5802:Fast-growing hierarchy
4943:To Infinity and Beyond
4865:10.1073/pnas.50.6.1143
4428:Kline, Morris (1985).
4233:"Our universe is Flat"
3891:mathonline.wikidot.com
3463:Cauchy, Augustin-Louis
3393:Drozdek, Adam (2008).
3175:10.2178/bsl/1286284558
3001:Allen, Donald (2003).
2940:Names of large numbers
2363:
2310:
2241:
2172:
2143:
2004:
1977:is fully developed in
1943:infinitesimal calculus
1938:
1919:Möbius transformations
1911:
1886:
1861:
1802:
1779:
1758:
1728:
1698:
1677:
1646:hyperplane at infinity
1618:
1595:
1560:
1537:
1506:
1483:
1449:
1422:
1399:
1356:
1333:
1304:
1242:
1213:
1152:
1129:
1109:
1080:
1019:
998:
962:
941:
911:
890:
860:
828:infinitesimal calculus
771:(sometimes called the
765:
738:
698:
666:De sectionibus conicis
658:
599:
532:
468:
269:
165:Grothendieck universes
98:infinitesimal calculus
79:
52:
6102:Representation theory
6061:quaternionic analysis
6057:Hypercomplex analysis
5955:mathematical analysis
5859:Long and short scales
5797:Grzegorczyk hierarchy
5486:Möbius transformation
5383:Dedekind-infinite set
5291:Paradoxes of infinity
5286:Infinity (philosophy)
4998:Pearce, Ian. (2002).
4773:Wallace, David Foster
4127:John Gribbin (2009),
3802:Toker, Leona (1989),
3624:, Dover, p. 17,
2960:Paradoxes of infinity
2796:programming languages
2567:have always a finite
2534:the set of its points
2357:
2319:Further information:
2311:
2250:Further information:
2242:
2184:than natural numbers
2173:
2144:
2002:
1975:non-standard calculus
1936:
1912:
1895:meromorphic functions
1887:
1862:
1803:
1780:
1759:
1729:
1699:
1667:
1652:, each consisting of
1640:in plane geometry, a
1619:
1596:
1576:extended real numbers
1561:
1538:
1507:
1463:
1450:
1423:
1379:
1357:
1334:
1305:
1243:
1214:
1153:
1130:
1110:
1081:
1020:
999:
963:
942:
912:
891:
861:
766:
734:
699:
659:
600:
533:
469:
341:Augustin-Louis Cauchy
264:
196:Infinity (philosophy)
194:Further information:
186:is an open question.
169:elementary arithmetic
88:From the time of the
80:
42:
6176:Mathematical objects
6034:Table of derivatives
5322:Nonstandard analysis
5102:Infinite Reflections
5095:Infinite Reflections
4727:Space-Filling Curves
4725:Sagan, Hans (1994),
4703:, New York: Norton,
4149:Brake, Mark (2013).
4099:, pp. 1197–1198
4036:Moore, A.W. (1991).
3490:Ian Stewart (2017).
3061:10.1162/posc_a_00543
2839:relational operators
2650:, approximations of
2588:iterated loop spaces
2542:is located on a line
2434:space-filling curves
2345:segment of that line
2329:, even assuming the
2272:
2262:continuum hypothesis
2194:
2153:
2131:
1963:nonstandard analysis
1929:Nonstandard analysis
1901:
1876:
1837:
1792:
1769:
1738:
1712:
1688:
1630:real projective line
1605:
1582:
1578:. We can also treat
1566:can be added to the
1547:
1524:
1460:
1436:
1376:
1343:
1332:{\displaystyle f(t)}
1314:
1254:
1241:{\displaystyle f(t)}
1223:
1163:
1139:
1119:
1108:{\displaystyle f(t)}
1090:
1033:
1009:
973:
952:
922:
901:
874:
850:
755:
751:The infinity symbol
676:
648:
541:
498:
355:
142:mathematical objects
69:
20:For the symbol, see
16:Mathematical concept
6114:Continuous function
6067:Functional analysis
5874:Orders of magnitude
5744:Scientific notation
5491:Riemannian manifold
5460:Transfinite numbers
5317:Internal set theory
5186:Ian Pearce (2002).
5170:'Jaina mathematics'
4982:'Jaina mathematics'
4891:Jain, L.C. (1982).
4856:1963PNAS...50.1143C
4831:Infinity Foundation
4627:Elementary Topology
3429:Stanford University
2898:Cognitive scientist
2852:In programming, an
2785:arithmetic overflow
2767:supposed to play."
2598:The structure of a
2573:functional analysis
2490:projective geometry
2360:space-filling curve
2337:Cardinal arithmetic
2019:first developed by
2017:transfinite numbers
1757:{\displaystyle |x|}
1634:Projective geometry
1432:to some real value
1274:
1183:
1050:
6146:Mathematics portal
6029:Lists of integrals
5792:Ackermann function
5444:Sphere at infinity
5395:(Complex infinity)
5175:2008-12-20 at the
5155:2006-09-16 at the
5107:2009-11-05 at the
5088:2010-02-27 at the
4987:2008-12-20 at the
4967:2006-09-16 at the
4661:, Academic Press,
4644:Keisler, H. Jerome
4522:2017-04-07 at the
4503:2017-04-02 at the
4481:Notices of the AMS
4459:2016-05-16 at the
4269:The Shape of Space
4114:2013-01-24 at the
4017:math.dartmouth.edu
3876:, pp. 468–510
3431:. October 15, 2010
3425:"Zeno's Paradoxes"
3235:10.1007/BFb0090732
3091:on 11 January 2012
2935:Indeterminate form
2877:points at infinity
2584:topological spaces
2555:Infinite dimension
2520:Before the use of
2504:effect that shows
2494:points at infinity
2364:
2349:finite-dimensional
2306:
2237:
2168:
2139:
2105:countably infinite
2005:
1939:
1907:
1882:
1857:
1798:
1775:
1754:
1724:
1694:
1678:
1654:points at infinity
1614:
1591:
1556:
1533:
1502:
1448:{\displaystyle a.}
1445:
1418:
1355:{\displaystyle a.}
1352:
1329:
1300:
1257:
1238:
1209:
1166:
1151:{\displaystyle b.}
1148:
1125:
1105:
1076:
1036:
1015:
994:
958:
937:
907:
886:
856:
761:
694:
689:
654:
595:
528:
464:
339:Finally, in 1821,
260:parallel postulate
238:potential infinity
75:
53:
6166:Concepts in logic
6153:
6152:
6119:Special functions
6082:Harmonic analysis
5920:
5919:
5810:
5809:
5540:
5539:
5434:Point at infinity
5414:Hyperreal numbers
5388:Directed infinity
5353:Absolute infinite
5276:Galileo's paradox
5261:Ananta (infinite)
5024:978-0-691-00172-2
4953:978-0-691-02511-7
4922:978-0-14-027778-4
4813:978-0-7434-2299-4
4786:978-0-393-32629-1
4765:Advanced Calculus
4756:978-0-87150-341-1
4736:978-1-4612-0871-6
4710:978-0-393-31404-5
4697:Russell, Bertrand
4689:978-0-19-506135-2
4668:978-0-12-464976-7
4636:978-0-486-66522-1
4618:978-0-486-67766-8
4447:978-0-486-24823-3
4280:978-0-8247-0709-5
4195:Extract of p. 553
4189:978-0-7637-4387-1
4162:978-0-521-49129-7
4137:978-0-470-61352-8
3972:978-0-521-48364-3
3923:978-0-12-050257-8
3815:978-0-8014-2211-9
3784:978-0-226-61855-5
3750:978-3-540-52335-2
3710:978-0-8284-0314-6
3631:978-0-486-48903-2
3599:978-0-08-045744-4
3503:978-0-19-875523-4
3410:978-3-515-09258-6
3359:978-0-6151-7984-1
3323:978-0-04-946007-2
3244:978-3-540-10850-4
3123:978-1-4008-3039-8
3055:(1&2): 6–40.
3019:on August 1, 2020
2713:multipole moments
2658:measurements and
2620:Leopold Kronecker
2538:belongs to a line
2530:located on a line
2500:for modeling the
2496:are added to the
2073:Dedekind infinite
2061:Galileo's paradox
1885:{\displaystyle z}
1811:topological space
1778:{\displaystyle x}
1708:. The expression
1642:plane at infinity
1636:also refers to a
1568:topological space
1128:{\displaystyle a}
1018:{\displaystyle t}
961:{\displaystyle x}
910:{\displaystyle x}
832:Law of continuity
824:Gottfried Leibniz
688:
615:Jain mathematical
590:
576:
563:
459:
153:axiom of infinity
6188:
6072:Fourier analysis
6052:Complex analysis
5953:Major topics in
5947:
5940:
5933:
5924:
5923:
5734:
5733:
5664:Eddington number
5609:Hundred thousand
5567:
5560:
5553:
5544:
5543:
5532:Abraham Robinson
5527:Bernhard Riemann
5446:(Kleinian group)
5439:Regular cardinal
5393:Division by zero
5373:Cardinal numbers
5312:Complex analysis
5247:
5233:
5226:
5219:
5210:
5209:
5140:
5138:
5137:
5128:. Archived from
5063:
5041:
5028:
4957:
4926:
4911:(2nd ed.).
4896:
4886:
4877:
4867:
4850:(6): 1143–1148,
4817:
4789:
4768:
4759:
4739:
4721:
4692:
4671:
4653:
4639:
4621:
4591:
4590:
4588:
4587:
4581:
4575:. Archived from
4570:
4561:
4555:
4553:
4551:
4539:
4537:
4514:
4508:
4495:
4489:
4488:
4478:
4469:
4463:
4451:
4435:
4425:
4419:
4418:
4416:
4414:
4409:on 25 March 2012
4405:. Archived from
4394:
4388:
4387:
4385:
4383:
4363:
4357:
4356:
4354:
4353:
4339:
4333:
4327:
4321:
4320:
4318:
4316:
4300:
4294:
4291:
4285:
4284:
4272:
4262:
4256:
4255:
4247:
4241:
4240:
4229:
4223:
4222:
4220:
4218:
4203:
4197:
4193:
4173:
4167:
4166:
4146:
4140:
4125:
4119:
4106:
4100:
4094:
4088:
4087:, pp. 10–12
4082:
4076:
4070:
4064:
4063:
4057:
4048:
4042:
4041:
4033:
4027:
4026:
4024:
4023:
4009:
4003:
4002:
3982:
3976:
3975:
3958:
3952:
3946:
3940:
3939:
3907:
3901:
3900:
3898:
3897:
3883:
3877:
3870:
3864:
3858:
3852:
3851:
3842:Zalta, Edward N.
3830:
3824:
3823:
3799:
3793:
3792:
3768:
3762:
3761:
3725:
3719:
3718:
3690:
3684:
3683:
3681:
3680:
3666:
3660:
3659:
3657:
3656:
3641:
3635:
3634:
3617:
3611:
3609:Extract of p. 62
3607:
3583:
3577:
3571:
3565:
3564:
3557:
3551:
3545:
3539:
3538:
3518:
3512:
3511:
3487:
3481:
3480:
3478:
3476:
3459:
3453:
3447:
3441:
3440:
3438:
3436:
3421:
3415:
3414:
3390:
3384:
3383:
3370:
3364:
3363:
3351:
3343:Euclid (2008) .
3340:
3334:
3333:
3331:
3330:
3307:
3301:
3300:
3264:
3258:
3255:
3249:
3248:
3222:
3216:
3215:
3205:
3199:
3193:
3187:
3186:
3154:
3148:
3142:
3136:
3135:
3107:
3101:
3100:
3098:
3096:
3087:. Archived from
3038:
3029:
3028:
3026:
3024:
3018:
3012:. Archived from
3007:
2998:
2873:vanishing points
2789:division by zero
2764:infinite regress
2510:projective plane
2469:expression "the
2418:
2412:
2411:
2409:
2408:
2405:
2402:
2395:
2393:
2392:
2389:
2386:
2341:real number line
2315:
2313:
2312:
2307:
2305:
2304:
2292:
2291:
2279:
2246:
2244:
2243:
2238:
2236:
2235:
2234:
2221:
2220:
2219:
2218:
2201:
2189:
2183:
2177:
2175:
2174:
2169:
2167:
2166:
2165:
2148:
2146:
2145:
2140:
2138:
2084:cardinal numbers
2046:Richard Dedekind
2035:
1916:
1914:
1913:
1908:
1891:
1889:
1888:
1883:
1867:for any nonzero
1866:
1864:
1863:
1858:
1847:
1831:division by zero
1819:complex manifold
1807:
1805:
1804:
1799:
1784:
1782:
1781:
1776:
1763:
1761:
1760:
1755:
1753:
1745:
1733:
1731:
1730:
1725:
1703:
1701:
1700:
1695:
1682:complex analysis
1660:Complex analysis
1638:line at infinity
1623:
1621:
1620:
1615:
1600:
1598:
1597:
1592:
1572:compactification
1565:
1563:
1562:
1557:
1542:
1540:
1539:
1534:
1511:
1509:
1508:
1503:
1482:
1477:
1454:
1452:
1451:
1446:
1427:
1425:
1424:
1419:
1398:
1393:
1361:
1359:
1358:
1353:
1338:
1336:
1335:
1330:
1309:
1307:
1306:
1301:
1273:
1268:
1247:
1245:
1244:
1239:
1218:
1216:
1215:
1210:
1182:
1177:
1157:
1155:
1154:
1149:
1134:
1132:
1131:
1126:
1114:
1112:
1111:
1106:
1085:
1083:
1082:
1077:
1049:
1044:
1024:
1022:
1021:
1016:
1003:
1001:
1000:
995:
967:
965:
964:
959:
947:means that
946:
944:
943:
938:
916:
914:
913:
908:
896:means that
895:
893:
892:
887:
865:
863:
862:
857:
803:
795:
791:
788:
785:
783:
770:
768:
767:
762:
703:
701:
700:
695:
690:
681:
663:
661:
660:
655:
604:
602:
601:
596:
591:
588:
577:
569:
564:
562:
548:
537:
535:
534:
529:
490:
483:
473:
471:
470:
465:
460:
458:
444:
433:
432:
417:
416:
401:
400:
385:
384:
350:
300:Bertrand Russell
296:
292:
130:infinite numbers
84:
82:
81:
76:
36:
29:
6196:
6195:
6191:
6190:
6189:
6187:
6186:
6185:
6156:
6155:
6154:
6149:
6138:
6087:P-adic analysis
6038:
6024:Matrix calculus
6019:Tensor calculus
6014:Vector calculus
5977:Differentiation
5957:
5951:
5921:
5916:
5898:
5854:List of numbers
5822:
5820:
5818:
5816:
5806:
5763:
5729:
5723:
5694:Graham's number
5684:Skewes's number
5586:
5584:
5582:
5576:
5571:
5541:
5536:
5495:
5464:
5455:Surreal numbers
5429:Ordinal numbers
5358:Actual infinity
5336:
5300:
5249:
5243:
5237:
5177:Wayback Machine
5157:Wayback Machine
5135:
5133:
5109:Wayback Machine
5090:Wayback Machine
5052:
5049:
5044:
5025:
4989:Wayback Machine
4969:Wayback Machine
4954:
4923:
4814:
4797:
4792:
4787:
4757:
4737:
4711:
4690:
4669:
4637:
4619:
4600:
4595:
4594:
4585:
4583:
4579:
4568:
4562:
4558:
4524:Wayback Machine
4515:
4511:
4505:Wayback Machine
4496:
4492:
4476:
4470:
4466:
4461:Wayback Machine
4448:
4426:
4422:
4412:
4410:
4395:
4391:
4381:
4379:
4364:
4360:
4351:
4349:
4341:
4340:
4336:
4328:
4324:
4314:
4312:
4301:
4297:
4292:
4288:
4281:
4263:
4259:
4248:
4244:
4231:
4230:
4226:
4216:
4214:
4205:
4204:
4200:
4190:
4174:
4170:
4163:
4147:
4143:
4126:
4122:
4116:Wayback Machine
4107:
4103:
4095:
4091:
4083:
4079:
4071:
4067:
4055:
4049:
4045:
4034:
4030:
4021:
4019:
4011:
4010:
4006:
3999:
3983:
3979:
3973:
3959:
3955:
3947:
3943:
3924:
3908:
3904:
3895:
3893:
3885:
3884:
3880:
3871:
3867:
3859:
3855:
3834:Bell, John Lane
3831:
3827:
3816:
3800:
3796:
3785:
3769:
3765:
3751:
3729:Martin-Löf, Per
3726:
3722:
3711:
3691:
3687:
3678:
3676:
3668:
3667:
3663:
3654:
3652:
3642:
3638:
3632:
3618:
3614:
3600:
3584:
3580:
3572:
3568:
3559:
3558:
3554:
3546:
3542:
3535:
3519:
3515:
3504:
3488:
3484:
3474:
3472:
3460:
3456:
3448:
3444:
3434:
3432:
3423:
3422:
3418:
3411:
3391:
3387:
3371:
3367:
3360:
3349:
3341:
3337:
3328:
3326:
3324:
3308:
3304:
3265:
3261:
3256:
3252:
3245:
3223:
3219:
3206:
3202:
3194:
3190:
3155:
3151:
3143:
3139:
3124:
3108:
3104:
3094:
3092:
3039:
3032:
3022:
3020:
3016:
3005:
2999:
2984:
2979:
2974:
2910:
2866:
2812:sentinel values
2773:
2756:
2676:
2660:natural numbers
2644:
2617:
2596:
2577:function spaces
2557:
2512:, two distinct
2498:Euclidean space
2486:actual infinity
2450:
2432:introduced the
2426:
2414:
2406:
2403:
2400:
2399:
2397:
2390:
2387:
2384:
2383:
2381:
2379:
2331:Axiom of Choice
2323:
2300:
2296:
2287:
2283:
2275:
2273:
2270:
2269:
2266:cardinal number
2258:
2230:
2226:
2225:
2214:
2210:
2209:
2205:
2197:
2195:
2192:
2191:
2185:
2179:
2161:
2157:
2156:
2154:
2151:
2150:
2134:
2132:
2129:
2128:
2125:
2119:
2080:ordinal numbers
2065:square integers
2038:natural numbers
2034:
2031:
2028:
1997:
1989:Main articles:
1987:
1967:hyperreal field
1955:logical systems
1931:
1902:
1899:
1898:
1877:
1874:
1873:
1843:
1838:
1835:
1834:
1823:Riemann surface
1793:
1790:
1789:
1770:
1767:
1766:
1749:
1741:
1739:
1736:
1735:
1713:
1710:
1709:
1689:
1686:
1685:
1662:
1606:
1603:
1602:
1583:
1580:
1579:
1548:
1545:
1544:
1525:
1522:
1521:
1478:
1467:
1461:
1458:
1457:
1437:
1434:
1433:
1394:
1383:
1377:
1374:
1373:
1367:infinite series
1344:
1341:
1340:
1315:
1312:
1311:
1269:
1261:
1255:
1252:
1251:
1224:
1221:
1220:
1178:
1170:
1164:
1161:
1160:
1140:
1137:
1136:
1120:
1117:
1116:
1091:
1088:
1087:
1045:
1040:
1034:
1031:
1030:
1010:
1007:
1006:
1004:for every
974:
971:
970:
953:
950:
949:
923:
920:
919:
902:
899:
898:
875:
872:
871:
870:. The notation
851:
848:
847:
840:
821:
801:
793:
789:
786:
781:
780:
756:
753:
752:
749:
747:Infinity symbol
743:
727:
679:
677:
674:
673:
649:
646:
645:
638:
611:
587:
568:
552:
547:
542:
539:
538:
499:
496:
495:
485:
478:
448:
443:
428:
424:
412:
408:
396:
392:
380:
376:
356:
353:
352:
344:
284:
278:
243:actual infinity
214:
202:ancient Indians
198:
192:
102:infinite series
70:
67:
66:
64:infinity symbol
37:
30:
22:Infinity symbol
19:
17:
12:
11:
5:
6194:
6184:
6183:
6178:
6173:
6168:
6151:
6150:
6143:
6140:
6139:
6137:
6136:
6131:
6126:
6121:
6116:
6111:
6105:
6104:
6099:
6097:Measure theory
6094:
6091:P-adic numbers
6084:
6079:
6074:
6069:
6064:
6054:
6049:
6043:
6040:
6039:
6037:
6036:
6031:
6026:
6021:
6016:
6011:
6006:
6001:
6000:
5999:
5994:
5989:
5979:
5974:
5962:
5959:
5958:
5950:
5949:
5942:
5935:
5927:
5918:
5917:
5915:
5914:
5909:
5903:
5900:
5899:
5897:
5896:
5891:
5886:
5884:Power of three
5881:
5876:
5871:
5866:
5864:Number systems
5861:
5856:
5851:
5846:
5841:
5836:
5831:
5825:
5823:
5819:(alphabetical
5812:
5811:
5808:
5807:
5805:
5804:
5799:
5794:
5789:
5788:
5787:
5782:
5775:Hyperoperation
5771:
5769:
5765:
5764:
5762:
5761:
5756:
5751:
5746:
5740:
5738:
5731:
5725:
5724:
5722:
5721:
5716:
5711:
5706:
5701:
5696:
5691:
5689:Moser's number
5686:
5681:
5676:
5674:Shannon number
5671:
5666:
5661:
5656:
5651:
5646:
5641:
5636:
5631:
5626:
5621:
5616:
5611:
5606:
5601:
5596:
5590:
5588:
5578:
5577:
5570:
5569:
5562:
5555:
5547:
5538:
5537:
5535:
5534:
5529:
5524:
5519:
5514:
5509:
5503:
5501:
5500:Mathematicians
5497:
5496:
5494:
5493:
5488:
5483:
5478:
5472:
5470:
5466:
5465:
5463:
5462:
5457:
5452:
5447:
5441:
5436:
5431:
5426:
5421:
5416:
5411:
5406:
5404:Gimel function
5401:
5399:Epsilon number
5396:
5390:
5385:
5380:
5375:
5370:
5365:
5360:
5355:
5350:
5344:
5342:
5338:
5337:
5335:
5334:
5329:
5324:
5319:
5314:
5308:
5306:
5302:
5301:
5299:
5298:
5293:
5288:
5283:
5278:
5273:
5268:
5263:
5257:
5255:
5251:
5250:
5236:
5235:
5228:
5221:
5213:
5207:
5206:
5200:
5195:
5184:
5166:
5146:
5144:Hotel Infinity
5141:
5116:Grime, James.
5113:
5098:
5079:
5064:
5054:"The Infinite"
5048:
5047:External links
5045:
5043:
5042:
5029:
5023:
5007:
4996:
4978:
4958:
4952:
4934:
4927:
4921:
4904:
4897:
4888:
4837:
4834:
4818:
4812:
4798:
4796:
4793:
4791:
4790:
4785:
4769:
4760:
4755:
4740:
4735:
4722:
4709:
4693:
4688:
4672:
4667:
4654:
4652:(2nd ed.)
4640:
4635:
4622:
4617:
4601:
4599:
4596:
4593:
4592:
4556:
4509:
4490:
4464:
4446:
4420:
4389:
4358:
4334:
4322:
4295:
4286:
4279:
4257:
4242:
4224:
4198:
4188:
4168:
4161:
4141:
4120:
4101:
4089:
4077:
4075:, p. 1143
4065:
4043:
4028:
4004:
3997:
3977:
3971:
3953:
3949:Gemignani 1990
3941:
3922:
3902:
3878:
3874:Swokowski 1983
3865:
3853:
3825:
3814:
3794:
3783:
3763:
3749:
3720:
3709:
3703:, p. 24,
3699:(2 ed.),
3685:
3661:
3636:
3630:
3612:
3598:
3578:
3566:
3552:
3540:
3533:
3513:
3502:
3482:
3454:
3442:
3416:
3409:
3385:
3365:
3358:
3335:
3322:
3302:
3289:10.1086/346308
3259:
3250:
3243:
3217:
3200:
3188:
3169:(3): 359–377.
3149:
3137:
3122:
3102:
3030:
2981:
2980:
2978:
2975:
2973:
2972:
2970:Surreal number
2967:
2962:
2957:
2952:
2947:
2942:
2937:
2932:
2930:Exponentiation
2927:
2922:
2917:
2911:
2909:
2906:
2892:infinite chess
2886:Variations of
2865:
2862:
2772:
2769:
2755:
2752:
2680:Giordano Bruno
2675:
2672:
2643:
2640:
2632:constructivism
2616:
2613:
2609:Koch snowflake
2595:
2592:
2556:
2553:
2506:parallel lines
2449:
2446:
2430:Giuseppe Peano
2303:
2299:
2295:
2290:
2286:
2282:
2278:
2233:
2229:
2224:
2217:
2213:
2208:
2204:
2200:
2164:
2160:
2137:
2121:Main article:
2118:
2115:
2032:
2029:
1995:Ordinal number
1986:
1983:
1979:Keisler (1986)
1930:
1927:
1906:
1881:
1869:complex number
1856:
1853:
1850:
1846:
1842:
1827:Riemann sphere
1797:
1788:point labeled
1774:
1752:
1748:
1744:
1723:
1720:
1717:
1693:
1674:Riemann sphere
1661:
1658:
1613:
1610:
1590:
1587:
1555:
1552:
1532:
1529:
1518:
1517:
1501:
1498:
1495:
1492:
1489:
1486:
1481:
1476:
1473:
1470:
1466:
1455:
1444:
1441:
1417:
1414:
1411:
1408:
1405:
1402:
1397:
1392:
1389:
1386:
1382:
1369:, as follows:
1363:
1362:
1351:
1348:
1328:
1325:
1322:
1319:
1299:
1296:
1293:
1290:
1286:
1283:
1280:
1277:
1272:
1267:
1264:
1260:
1249:
1237:
1234:
1231:
1228:
1208:
1205:
1202:
1199:
1195:
1192:
1189:
1186:
1181:
1176:
1173:
1169:
1158:
1147:
1144:
1124:
1104:
1101:
1098:
1095:
1075:
1072:
1069:
1066:
1062:
1059:
1056:
1053:
1048:
1043:
1039:
1014:
993:
990:
987:
984:
981:
978:
957:
936:
933:
930:
927:
906:
885:
882:
879:
855:
839:
836:
820:
817:
760:
745:Main article:
742:
739:
726:
723:
693:
687:
684:
653:
637:
634:
633:
632:
629:
626:
610:
607:
606:
605:
594:
586:
583:
580:
575:
572:
567:
561:
558:
555:
551:
546:
527:
524:
521:
518:
515:
512:
509:
506:
503:
463:
457:
454:
451:
447:
442:
439:
436:
431:
427:
423:
420:
415:
411:
407:
404:
399:
395:
391:
388:
383:
379:
375:
372:
369:
366:
363:
360:
323:
322:
319:
316:
313:
280:Main article:
277:
274:
213:
210:
191:
188:
90:ancient Greeks
74:
60:natural number
15:
9:
6:
4:
3:
2:
6193:
6182:
6179:
6177:
6174:
6172:
6169:
6167:
6164:
6163:
6161:
6148:
6147:
6141:
6135:
6132:
6130:
6127:
6125:
6122:
6120:
6117:
6115:
6112:
6110:
6107:
6106:
6103:
6100:
6098:
6095:
6092:
6088:
6085:
6083:
6080:
6078:
6075:
6073:
6070:
6068:
6065:
6062:
6058:
6055:
6053:
6050:
6048:
6047:Real analysis
6045:
6044:
6041:
6035:
6032:
6030:
6027:
6025:
6022:
6020:
6017:
6015:
6012:
6010:
6007:
6005:
6002:
5998:
5995:
5993:
5990:
5988:
5985:
5984:
5983:
5980:
5978:
5975:
5973:
5969:
5968:
5964:
5963:
5960:
5956:
5948:
5943:
5941:
5936:
5934:
5929:
5928:
5925:
5913:
5910:
5908:
5905:
5904:
5901:
5895:
5892:
5890:
5887:
5885:
5882:
5880:
5877:
5875:
5872:
5870:
5867:
5865:
5862:
5860:
5857:
5855:
5852:
5850:
5847:
5845:
5844:Infinitesimal
5842:
5840:
5837:
5835:
5832:
5830:
5827:
5826:
5824:
5813:
5803:
5800:
5798:
5795:
5793:
5790:
5786:
5783:
5781:
5778:
5777:
5776:
5773:
5772:
5770:
5766:
5760:
5757:
5755:
5752:
5750:
5747:
5745:
5742:
5741:
5739:
5735:
5732:
5726:
5720:
5717:
5715:
5714:Rayo's number
5712:
5710:
5707:
5705:
5702:
5700:
5697:
5695:
5692:
5690:
5687:
5685:
5682:
5680:
5677:
5675:
5672:
5670:
5667:
5665:
5662:
5660:
5657:
5655:
5652:
5650:
5647:
5645:
5642:
5640:
5637:
5635:
5632:
5630:
5627:
5625:
5622:
5620:
5617:
5615:
5612:
5610:
5607:
5605:
5602:
5600:
5597:
5595:
5592:
5591:
5589:
5579:
5575:
5574:Large numbers
5568:
5563:
5561:
5556:
5554:
5549:
5548:
5545:
5533:
5530:
5528:
5525:
5523:
5520:
5518:
5515:
5513:
5512:David Hilbert
5510:
5508:
5505:
5504:
5502:
5498:
5492:
5489:
5487:
5484:
5482:
5479:
5477:
5474:
5473:
5471:
5467:
5461:
5458:
5456:
5453:
5451:
5448:
5445:
5442:
5440:
5437:
5435:
5432:
5430:
5427:
5425:
5424:Infinitesimal
5422:
5420:
5417:
5415:
5412:
5410:
5409:Hilbert space
5407:
5405:
5402:
5400:
5397:
5394:
5391:
5389:
5386:
5384:
5381:
5379:
5376:
5374:
5371:
5369:
5366:
5364:
5361:
5359:
5356:
5354:
5351:
5349:
5346:
5345:
5343:
5339:
5333:
5330:
5328:
5325:
5323:
5320:
5318:
5315:
5313:
5310:
5309:
5307:
5303:
5297:
5294:
5292:
5289:
5287:
5284:
5282:
5279:
5277:
5274:
5272:
5269:
5267:
5264:
5262:
5259:
5258:
5256:
5252:
5246:
5241:
5234:
5229:
5227:
5222:
5220:
5215:
5214:
5211:
5204:
5201:
5199:
5196:
5193:
5189:
5185:
5182:
5178:
5174:
5171:
5167:
5164:
5163:
5158:
5154:
5151:
5147:
5145:
5142:
5132:on 2017-10-22
5131:
5127:
5123:
5119:
5114:
5111:
5110:
5106:
5103:
5099:
5096:
5092:
5091:
5087:
5084:
5080:
5078:
5074:
5073:
5068:
5065:
5061:
5060:
5055:
5051:
5050:
5039:
5035:
5030:
5026:
5020:
5016:
5012:
5008:
5005:
5001:
4997:
4994:
4990:
4986:
4983:
4979:
4976:
4975:
4970:
4966:
4963:
4959:
4955:
4949:
4945:
4944:
4939:
4935:
4933:
4928:
4924:
4918:
4914:
4913:Penguin Books
4910:
4905:
4902:
4898:
4894:
4889:
4885:
4881:
4876:
4871:
4866:
4861:
4857:
4853:
4849:
4845:
4844:
4838:
4835:
4832:
4828:
4827:
4822:
4819:
4815:
4809:
4805:
4800:
4799:
4788:
4782:
4778:
4774:
4770:
4766:
4761:
4758:
4752:
4748:
4747:
4741:
4738:
4732:
4728:
4723:
4720:
4716:
4712:
4706:
4702:
4698:
4694:
4691:
4685:
4681:
4677:
4676:Kline, Morris
4673:
4670:
4664:
4660:
4655:
4651:
4650:
4645:
4641:
4638:
4632:
4628:
4623:
4620:
4614:
4610:
4609:
4603:
4602:
4582:on 2020-02-26
4578:
4574:
4567:
4560:
4550:
4545:
4536:
4531:
4525:
4521:
4518:
4513:
4506:
4502:
4499:
4494:
4487:(6): 706–718.
4486:
4482:
4475:
4468:
4462:
4458:
4455:
4449:
4443:
4439:
4434:
4433:
4424:
4408:
4404:
4400:
4393:
4377:
4373:
4369:
4362:
4348:
4344:
4338:
4331:
4326:
4310:
4306:
4299:
4290:
4282:
4276:
4273:. CRC Press.
4271:
4270:
4261:
4254:. LXXIV1: 30.
4253:
4246:
4238:
4234:
4228:
4212:
4208:
4202:
4196:
4191:
4185:
4181:
4180:
4172:
4164:
4158:
4154:
4153:
4145:
4138:
4134:
4130:
4124:
4117:
4113:
4110:
4105:
4098:
4093:
4086:
4081:
4074:
4069:
4061:
4054:
4047:
4039:
4032:
4018:
4014:
4008:
4000:
3998:9789810203757
3994:
3990:
3989:
3981:
3974:
3968:
3964:
3957:
3950:
3945:
3937:
3933:
3929:
3925:
3919:
3915:
3914:
3906:
3892:
3888:
3882:
3875:
3869:
3862:
3857:
3849:
3848:
3843:
3839:
3835:
3829:
3821:
3817:
3811:
3807:
3806:
3798:
3790:
3786:
3780:
3776:
3775:
3767:
3760:
3756:
3752:
3746:
3742:
3738:
3734:
3730:
3724:
3716:
3712:
3706:
3702:
3698:
3697:
3689:
3675:
3671:
3665:
3651:
3647:
3644:AG, Compart.
3640:
3633:
3627:
3623:
3616:
3610:
3605:
3601:
3595:
3591:
3590:
3582:
3575:
3570:
3562:
3556:
3549:
3544:
3536:
3534:9781602066854
3530:
3526:
3525:
3517:
3509:
3505:
3499:
3495:
3494:
3486:
3470:
3469:
3464:
3458:
3452:, p. 347
3451:
3446:
3430:
3426:
3420:
3412:
3406:
3402:
3399:
3395:
3389:
3381:
3380:
3375:
3369:
3361:
3355:
3348:
3347:
3339:
3325:
3319:
3315:
3314:
3306:
3298:
3294:
3290:
3286:
3282:
3278:
3274:
3272:
3263:
3254:
3246:
3240:
3236:
3232:
3228:
3221:
3213:
3212:
3204:
3197:
3192:
3184:
3180:
3176:
3172:
3168:
3164:
3160:
3153:
3147:, pp. 113–117
3146:
3141:
3133:
3129:
3125:
3119:
3115:
3114:
3106:
3090:
3086:
3082:
3078:
3074:
3070:
3066:
3062:
3058:
3054:
3050:
3049:
3044:
3037:
3035:
3015:
3011:
3004:
2997:
2995:
2993:
2991:
2989:
2987:
2982:
2971:
2968:
2966:
2963:
2961:
2958:
2956:
2955:Infinitesimal
2953:
2951:
2948:
2946:
2943:
2941:
2938:
2936:
2933:
2931:
2928:
2926:
2923:
2921:
2918:
2916:
2913:
2912:
2905:
2902:
2901:George Lakoff
2899:
2895:
2893:
2889:
2884:
2882:
2878:
2874:
2870:
2861:
2859:
2855:
2854:infinite loop
2850:
2848:
2844:
2840:
2836:
2831:
2829:
2825:
2821:
2817:
2813:
2809:
2805:
2801:
2797:
2792:
2790:
2786:
2782:
2778:
2768:
2765:
2761:
2751:
2749:
2745:
2744:cyclic models
2741:
2737:
2732:
2730:
2724:
2722:
2718:
2714:
2709:
2707:
2703:
2699:
2698:go on forever
2695:
2691:
2687:
2685:
2681:
2671:
2669:
2665:
2662:are used for
2661:
2657:
2654:are used for
2653:
2649:
2639:
2637:
2633:
2629:
2625:
2621:
2612:
2610:
2606:
2605:fractal curve
2601:
2591:
2589:
2585:
2580:
2578:
2574:
2570:
2566:
2562:
2561:vector spaces
2552:
2550:
2549:infinite sets
2545:
2543:
2539:
2535:
2531:
2527:
2523:
2518:
2515:
2511:
2507:
2503:
2499:
2495:
2491:
2487:
2482:
2480:
2476:
2472:
2467:
2463:
2459:
2455:
2445:
2443:
2439:
2435:
2431:
2425:
2420:
2417:
2377:
2373:
2369:
2361:
2356:
2352:
2350:
2346:
2342:
2338:
2334:
2332:
2328:
2322:
2317:
2301:
2297:
2293:
2288:
2280:
2267:
2263:
2257:
2253:
2248:
2231:
2222:
2215:
2206:
2202:
2188:
2182:
2162:
2124:
2114:
2112:
2111:
2106:
2101:
2097:
2093:
2089:
2085:
2081:
2076:
2074:
2070:
2066:
2062:
2058:
2054:
2049:
2047:
2043:
2042:Gottlob Frege
2039:
2026:
2022:
2018:
2014:
2010:
2001:
1996:
1992:
1982:
1980:
1976:
1972:
1968:
1964:
1960:
1956:
1952:
1951:infinitesimal
1948:
1944:
1935:
1926:
1924:
1920:
1896:
1892:
1879:
1870:
1851:
1848:
1844:
1840:
1832:
1828:
1824:
1820:
1816:
1812:
1808:
1785:
1772:
1746:
1715:
1707:
1683:
1675:
1671:
1666:
1657:
1655:
1651:
1647:
1643:
1639:
1635:
1631:
1627:
1608:
1585:
1577:
1573:
1569:
1550:
1527:
1515:
1496:
1490:
1484:
1474:
1471:
1468:
1464:
1456:
1442:
1439:
1431:
1415:
1412:
1406:
1400:
1390:
1387:
1384:
1380:
1372:
1371:
1370:
1368:
1349:
1346:
1323:
1317:
1297:
1294:
1291:
1288:
1281:
1275:
1262:
1258:
1250:
1232:
1226:
1203:
1200:
1197:
1190:
1184:
1171:
1167:
1159:
1145:
1142:
1122:
1099:
1093:
1070:
1067:
1064:
1057:
1051:
1046:
1041:
1037:
1029:
1028:
1027:
1025:
1012:
991:
988:
982:
976:
968:
955:
931:
925:
917:
904:
877:
869:
846:, the symbol
845:
844:real analysis
838:Real analysis
835:
833:
829:
825:
816:
814:
810:
805:
799:
778:
774:
748:
737:
733:
731:
722:
720:
719:
714:
709:
707:
691:
682:
671:
670:infinitesimal
667:
643:
630:
627:
624:
623:
622:
620:
616:
592:
589: seconds
584:
581:
578:
573:
570:
565:
559:
556:
553:
549:
544:
525:
522:
519:
516:
513:
510:
507:
504:
501:
494:
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488:
481:
474:
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455:
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429:
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397:
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389:
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364:
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348:
342:
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329:
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320:
317:
314:
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309:
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288:
283:
273:
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197:
187:
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177:
172:
170:
166:
162:
158:
157:combinatorics
154:
150:
145:
143:
139:
135:
131:
127:
126:infinite sets
123:
119:
115:
111:
107:
103:
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65:
61:
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6144:
6133:
5965:
5879:Power of two
5869:Number names
5718:
5604:Ten thousand
5507:Georg Cantor
5481:Möbius plane
5419:Infinite set
5363:Aleph number
5239:
5191:
5180:
5160:
5134:. Retrieved
5130:the original
5121:
5100:
5094:
5081:
5071:
5057:
5037:
5033:
5014:
5011:Rucker, Rudy
5003:
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4972:
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4900:
4892:
4847:
4841:
4824:
4821:D.P. Agrawal
4803:
4776:
4764:
4745:
4729:, Springer,
4726:
4700:
4679:
4658:
4648:
4626:
4607:
4598:Bibliography
4584:. Retrieved
4577:the original
4572:
4559:
4512:
4493:
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4480:
4467:
4431:
4423:
4411:. Retrieved
4407:the original
4402:
4392:
4380:. Retrieved
4371:
4361:
4350:. Retrieved
4346:
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4325:
4313:. Retrieved
4308:
4298:
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4268:
4260:
4251:
4245:
4227:
4215:. Retrieved
4201:
4178:
4171:
4151:
4144:
4139:. p. 88
4128:
4123:
4109:Doric Lenses
4104:
4092:
4080:
4068:
4059:
4046:
4040:. Routledge.
4038:The Infinite
4037:
4031:
4020:. Retrieved
4016:
4007:
3987:
3980:
3962:
3956:
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3912:
3905:
3894:. Retrieved
3890:
3881:
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3673:
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3649:
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3581:
3569:
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3473:. Retrieved
3467:
3457:
3450:Russell 1996
3445:
3433:. Retrieved
3428:
3419:
3401:
3397:
3394:
3388:
3378:
3368:
3345:
3338:
3327:. Retrieved
3312:
3305:
3283:(1): 60–62.
3280:
3276:
3270:
3262:
3253:
3226:
3220:
3210:
3203:
3198:, p. 44
3196:Wallace 2004
3191:
3166:
3162:
3152:
3140:
3112:
3105:
3093:. Retrieved
3089:the original
3052:
3046:
3021:. Retrieved
3014:the original
3009:
2950:Infinite set
2920:Aleph number
2896:
2885:
2867:
2851:
2842:
2832:
2793:
2774:
2757:
2733:
2725:
2710:
2690:Cosmologists
2688:
2683:
2677:
2652:real numbers
2645:
2636:intuitionism
2618:
2597:
2581:
2558:
2548:
2546:
2541:
2537:
2533:
2529:
2519:
2483:
2478:
2474:
2465:
2462:line segment
2451:
2427:
2415:
2374:between the
2365:
2335:
2324:
2259:
2186:
2180:
2126:
2108:
2088:well-ordered
2077:
2050:
2021:Georg Cantor
2006:
1971:transfinites
1957:, including
1947:Isaac Newton
1940:
1872:
1765:
1679:
1648:for general
1519:
1364:
1248:is infinite.
1005:
948:
897:
841:
822:
806:
750:
735:
730:Hermann Weyl
728:
716:
713:Isaac Newton
710:
705:
665:
639:
636:17th century
612:
609:Early Indian
486:
482:= 10 seconds
479:
475:
346:
338:
330:
327:
324:
304:
287:Zeno of Elea
285:
270:
265:
241:
237:
232:
225:
222:pre-Socratic
215:
199:
173:
146:
122:Georg Cantor
87:
55:
54:
5972:Integration
5889:Power of 10
5829:Busy beaver
5634:Quintillion
5629:Quadrillion
5368:Beth number
5126:Brady Haran
5122:Numberphile
5072:In Our Time
4413:6 September
4382:6 September
4347:www.gnu.org
3861:Taylor 1955
3650:Compart.com
3574:Cajori 1993
3548:Cajori 1993
3475:October 12,
3208:Aristotle.
3145:Maddox 2002
2881:M.C. Escher
2869:Perspective
2835:overloading
2740:Michio Kaku
2540:instead of
2502:perspective
2110:uncountable
1991:Cardinality
1813:giving the
1684:the symbol
1086:means that
809:John Wallis
794:∞
725:Mathematics
642:John Wallis
248:Hellenistic
218:Anaximander
212:Early Greek
134:cardinality
6160:Categories
5997:stochastic
5894:Sagan Unit
5728:Expression
5679:Googolplex
5644:Septillion
5639:Sextillion
5585:numerical
5469:Geometries
5327:Set theory
5136:2013-04-06
4586:2021-03-25
4549:1510.08155
4403:Learning J
4352:2021-03-15
4097:Kline 1972
4085:Sagan 1994
4073:Cohen 1963
4022:2019-11-16
4013:"Infinity"
3896:2019-11-15
3679:2019-11-15
3655:2019-11-15
3329:2020-01-09
3257:Maor, p. 3
3095:1 November
2977:References
2818:involving
2816:algorithms
2798:, such as
2781:indefinite
2736:multiverse
2668:plane wave
2656:continuous
2522:set theory
2479:the points
2466:infinitely
2422:See also:
2025:aleph-null
1985:Set theory
1650:dimensions
773:lemniscate
619:enumerable
6109:Functions
5785:Pentation
5780:Tetration
5768:Operators
5737:Notations
5659:Decillion
5654:Nonillion
5649:Octillion
5581:Examples
5450:Supertask
5188:'Jainism'
5000:'Jainism'
4719:247299160
4699:(1996) ,
4611:, Dover,
4535:1302.4377
3297:0021-1753
3132:659590835
3085:118227996
3069:1063-6145
2965:Supertask
2847:data type
2828:windowing
2824:searching
2771:Computing
2748:Big Bangs
2702:cosmology
2674:Cosmology
2569:dimension
2442:hypercube
2298:ℶ
2285:ℵ
2228:ℵ
2212:ℵ
2159:ℵ
2098:leads to
2092:sequences
1905:∞
1855:∞
1833:, namely
1796:∞
1722:∞
1719:→
1692:∞
1612:∞
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1500:∞
1480:∞
1465:∑
1430:converges
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1381:∑
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759:∞
711:In 1699,
686:∞
652:∞
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526:⋯
453:−
438:⋯
234:Aristotle
180:cosmology
118:magnitude
110:Bernoulli
106:l'Hôpital
73:∞
6171:Infinity
6134:Infinity
5987:ordinary
5967:Calculus
5817:articles
5815:Related
5719:Infinity
5624:Trillion
5599:Thousand
5348:0.999...
5240:Infinity
5173:Archived
5153:Archived
5105:Archived
5086:Archived
5067:Infinity
5013:(1995).
4985:Archived
4965:Archived
4940:(1991).
4938:Eli Maor
4884:16578557
4823:(2000).
4775:(2004),
4678:(1972),
4646:(1986),
4573:CHI 2009
4520:Archived
4501:Archived
4457:Archived
4399:"19.2.1"
4376:Archived
4368:"4.2.3."
4315:31 March
4309:Nautilus
4237:Archived
4217:16 March
4211:Archived
4112:Archived
3951:, p. 177
3936:archived
3820:archived
3789:archived
3715:archived
3674:oeis.org
3604:Archived
3508:Archived
3465:(1821).
3435:April 3,
3183:13475845
3077:42413222
2915:0.999...
2908:See also
2729:toroidal
2706:topology
2694:universe
2664:discrete
2628:finitism
2594:Fractals
2565:geometry
2524:for the
2492:, where
2454:geometry
2448:Geometry
2376:interval
2100:mappings
2096:integers
2013:cardinal
1764:of
1514:diverges
819:Calculus
790:INFINITY
787:∞
582:10.10101
333:Eleatics
306:Achilles
204:and the
138:integers
114:calculus
56:Infinity
33:Infiniti
5992:partial
5912:History
5730:methods
5704:SSCG(3)
5699:TREE(3)
5619:Billion
5614:Million
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5254:History
5075:at the
4852:Bibcode
4795:Sources
3932:1669668
3863:, p. 63
3844:(ed.).
3759:1064143
3398:Apeiron
3211:Physics
3023:Nov 15,
2820:sorting
2648:physics
2642:Physics
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