2011:
51:
1945:
1676:
2715:. 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
2086:. 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".
2366:
2777:
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
2113:
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
2479:
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
282:
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
4940:
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
1840:. 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
2527:
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.
2124:. 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.
2737:
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
308:
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
841:, 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
346:
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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3169:
614:
277:
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
483:
2613:
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
2914:
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,...>.
2256:
487:
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
1228:
1319:
2325:
719:(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."
265:(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
1521:
2582:, 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
1095:
547:
158:
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
1437:
2697:: "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."
728:
713:
166:, 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
1743:
905:
5123:, 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.
4122:
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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
2158:
2010:
1876:
1013:
1928:
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
2455:, 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.
1633:
1610:
1575:
1552:
1984:. 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
1926:
1817:
1713:
875:
780:
673:
94:
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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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1371:
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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
123:) regarded as infinitely small quantities, but infinity continued to be associated with endless processes. As mathematicians struggled with the foundation of
171:
5183:
4995:
2890:, located at an infinite distance from the observer. This allows artists to create paintings that realistically render space, distances, and forms. Artist
2519:
intersecting "at infinity". Mathematically, points at infinity have the advantage of allowing one to not consider some special cases. For example, in a
3053:
5955:
2633:
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
283:
infinite", underlay all of early Greek philosophy and that
Aristotle's "potential infinity" is an aberration from the general trend of this period.
4386:
5575:
5172:
4984:
5291:
5163:
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2434:
143:, 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
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257:, 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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1964:
quantities. In the second half of the 20th century, it was shown that this treatment could be put on a rigorous footing through various
1660:
5128:
4511:
2719:. If so, one might eventually return to one's starting point after travelling in a straight line through the universe for long enough.
651:
In the 17th century, European mathematicians started using infinite numbers and infinite expressions in a systematic fashion. In 1655,
107:
has been the subject of many discussions among philosophers. In the 17th century, with the introduction of the infinity symbol and the
6191:
5241:
219:
did not define infinity in precise formalism as does modern mathematics, and instead approached infinity as a philosophical concept.
5849:
5486:
4577:"Review of "Where Mathematics comes from: How the Embodied Mind Brings Mathematics Into Being" By George Lakoff and Rafael E. Nunez"
4007:
3543:
1976:. In the latter, infinitesimals are invertible, and their inverses are infinite numbers. The infinities in this sense are part of a
4119:
2817:, allow the programmer an explicit access to the positive and negative infinity values as language constants. These can be used as
2204:
1683:, the complex plane can be "wrapped" onto a sphere, with the top point of the sphere corresponding to infinity. This is called the
6007:
5115:
2378:
4409:
2337:
1173:
159:
6087:
6014:
5342:
5096:
2543:. With the universal use of set theory in mathematics, the point of view has dramatically changed: a line is now considered as
266:
4221:
5997:
1582:
339:
Apparently, Achilles never overtakes the tortoise, since however many steps he completes, the tortoise remains ahead of him.
4316:"Ingenious: Paul J. Steinhardt – The Princeton physicist on what's wrong with inflation theory and his view of the Big Bang"
3614:
2051:. This modern mathematical conception of the quantitative infinite developed in the late 19th century from works by Cantor,
6186:
6002:
5948:
5884:
5764:
5669:
5664:
5659:
5654:
5649:
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5639:
5634:
4553:
Evans, C.D.A; Joel David
Hamkins; Norman Lewis Perlmutter (2015). "A position in infinite chess with game value $ ω^4$ ".
3711:
2262:
1969:
1264:
5769:
5699:
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1531:
In addition to defining a limit, infinity can be also used as a value in the extended real number system. Points labeled
5759:
5388:
3388:
2133:
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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
4243:
1470:
116:
5568:
2810:
2642:
5714:
2734:
spacecraft hints that the universe has a flat topology. This would be consistent with an infinite physical universe.
5208:
5198:
5010:
4140:
In Search of the
Multiverse: Parallel Worlds, Hidden Dimensions, and the Ultimate Quest for the Frontiers of Reality
269:, Euclid "was the first to overcome the horror of the infinite". There is a similar controversy concerning Euclid's
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5859:
5532:
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1825:
1636:
1043:
508:
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5709:
5527:
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36:
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1680:
332:
Step #4: Achilles advances to where the tortoise was at the end of Step #3 while the tortoise goes yet further.
329:
Step #3: Achilles advances to where the tortoise was at the end of Step #2 while the tortoise goes yet further.
326:
Step #2: Achilles advances to where the tortoise was at the end of Step #1 while the tortoise goes yet further.
5496:
4952:
4023:
1933:
1929:
2439:
The second result was proved by Cantor in 1878, but only became intuitively apparent in 1890, when
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3099:
3058:
2955:
2879:
2742:
and the geometry is flat. Many possible bounded, flat possibilities also exist for three-dimensional space.
2512:
686:
232:
128:
4353:
4063:
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6019:
5917:
5812:
5561:
5234:
2950:
2386:
2110:
1953:
1722:
1656:
884:
838:
108:
822:, and since its introduction, it has also been used outside mathematics in modern mysticism and literary
6112:
6107:
6071:
6067:
5992:
5965:
5879:
5869:
5807:
5393:
5301:
5296:
4654:
4378:
3473:
2970:
2708:
2118:. If a set is too large to be put in one-to-one correspondence with the positive integers, it is called
1985:
351:
206:
179:
175:
104:
3236:
Goodman, Nicolas D. (1981). "Reflections on Bishop's philosophy of mathematics". In
Richman, F. (ed.).
5160:
4972:
4836:
3571:
2641:, an extreme form of mathematical philosophy in the general philosophical and mathematical schools of
2488:
that satisfies some property" (singular), where modern mathematicians would generally say "the set of
6139:
6044:
5332:
5286:
4862:
4783:
3883:
These uses of infinity for integrals and series can be found in any standard calculus text, such as,
3054:"Leibniz on the Foundations of the Calculus: The Question of the Reality of Infinitesimal Magnitudes"
2845:
2806:
2272:
2071:
1973:
1905:
1640:
1586:
6134:
6124:
6077:
5754:
5501:
5327:
4587:
2849:
2795:
2787:
2598:
2583:
2500:
2481:
2444:
2370:
2347:
2163:
1716:
1644:
932:
878:
292:
152:
4315:
3024:
2141:
6156:
6039:
5802:
5470:
5454:
4554:
4540:
3189:
3091:
2945:
2712:
2666:
2504:
2359:
2027:
1981:
1847:
625:
270:
140:
5704:
5694:
3533:
2894:
is specifically known for employing the concept of infinity in his work in this and other ways.
2761:, resulting in an infinite variety of universes after each Big Bang event in an infinite cycle.
2373:, showing that there are as many points in a one-dimensional line as in a two-dimensional square
983:
5104:, by Peter Suber. From the St. John's Review, XLIV, 2 (1998) 1–59. The stand-alone appendix to
3225:. Translated by Hardie, R. P.; Gaye, R. K. The Internet Classics Archive. Book 3, Chapters 5–8.
3170:"What Does it Take to Prove Fermat's Last Theorem? Grothendieck and the Logic of Number Theory"
2821:, as they compare (respectively) greater than or less than all other values. They have uses as
2467:, except in the context of processes that could be continued without any limit. For example, a
6181:
6129:
6092:
5444:
5398:
5363:
5271:
5227:
5029:
4958:
4927:
4890:
4818:
4791:
4761:
4741:
4725:
4715:
4694:
4673:
4641:
4623:
4464:
4452:
4448:
4285:
4205:
4194:
4167:
4143:
4003:
3997:
3977:
3928:
3820:
3814:
3789:
3783:
3755:
3739:
3715:
3636:
3604:
3539:
3508:
3502:
3415:
3364:
3328:
3303:
3249:
3138:
3128:
3083:
3075:
2935:
2908:
2887:
2860:, the infinity values may still be accessible and usable as the result of certain operations.
2723:
2630:
2594:
2083:
2066:
as a standard for comparing the size of sets, and to reject the view of
Galileo (derived from
1821:
1798:
1664:
1652:
1615:
1592:
1578:
1557:
1534:
1440:
842:
163:
100:
4847:
Bell, J.L.: Continuity and infinitesimals. Stanford Encyclopedia of philosophy. Revised 2009.
4161:
3922:
3897:
3705:
3619:
2707:: Are there an infinite number of stars? Does the universe have infinite volume? Does space "
1911:
1802:
1698:
860:
765:
658:
79:
6082:
6062:
5724:
5674:
5542:
5537:
5449:
5424:
5403:
5322:
5140:
4880:
4870:
4707:
4508:
3747:
3477:
3322:
3295:
3241:
3181:
3067:
2868:
2834:
2818:
2799:
2774:
2520:
2351:
2074:
where Galileo concludes that positive integers cannot be compared to the subset of positive
2056:
1977:
1841:
1829:
1692:
1648:
1524:
310:
120:
3942:
3769:
178:, very large infinite sets, for solving a long-standing problem that is stated in terms of
6097:
6034:
6029:
6024:
5864:
5383:
5368:
5255:
5187:
5167:
5119:
5100:
4999:
4979:
4534:
4515:
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4126:
3938:
3852:
3765:
2883:
2838:
2524:
2508:
2496:
2468:
2341:
2279:
between the cardinality of the reals and the cardinality of the natural numbers, that is,
2276:
2120:
2094:
2023:
1944:
1833:
1377:
1324:
1233:
1100:
786:) is a mathematical symbol representing the concept of infinity. The symbol is encoded in
757:
609:{\displaystyle ={\frac {10}{1-0.01}}={\frac {10}{0.99}}=10.10101\ldots {\text{ seconds}}.}
253:
112:
74:
55:
32:
5491:
3355:
1748:
323:
Step #1: Achilles runs to the tortoise's starting point while the tortoise walks forward.
5933:
4866:
3999:
Complex Analysis: An Invitation : a Concise Introduction to Complex Function Theory
3746:, Lecture Notes in Computer Science, vol. 417, Berlin: Springer, pp. 146–197,
1446:
1353:
1149:
5894:
5785:
5719:
5684:
5465:
5439:
5414:
5409:
4617:
4484:
3844:
3279:
3220:
2980:
2940:
2902:
2822:
2690:
2670:
2619:
2587:
2516:
2440:
2090:
2048:
2019:
2005:
1965:
1886:
1879:
1837:
1779:
1684:
1129:
1019:
962:
911:
216:
70:
5112:
4885:
4539:
Evans, C.D.A; Joel David Hamkins (2013). "Transfinite game values in infinite chess".
3680:
628:
text Surya Prajnapti (c. 4th–3rd century BCE) classifies all numbers into three sets:
241:, which means "unbounded", "indefinite", and perhaps can be translated as "infinite".
6170:
6101:
6057:
5854:
5629:
5609:
5584:
5522:
5434:
5419:
4923:
4755:
4441:
4417:
3095:
2965:
2911:
2864:
2615:
2358:, but also that this is equal to the number of points on a plane and, indeed, in any
2115:
2079:
2075:
2052:
1961:
854:
680:
632:, innumerable, and infinite. Each of these was further subdivided into three orders:
167:
5093:
3193:
5889:
5517:
5429:
5373:
5043:
Singh, Navjyoti (1988). "Jaina Theory of Actual Infinity and Transfinite Numbers".
4831:
4686:
4278:
4217:
3435:
2960:
2930:
2754:
2646:
2571:
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2355:
2098:
2031:
1957:
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297:
212:
136:
132:
50:
1828:
of the complex plane. When this is done, the resulting space is a one-dimensional
4188:
4129:– Application Note – Axicons – 2. Intensity Distribution. Retrieved 7 April 2014.
3848:
3598:
3363:. Translated by Fitzpatrick, Richard. Lulu.com. p. 6 (Book I, Postulate 5).
2790:
standard (IEEE 754) specifies a positive and a negative infinity value (and also
2138:
One of Cantor's most important results was that the cardinality of the continuum
5899:
5839:
5378:
5216:(compilation of articles about infinity in physics, mathematics, and philosophy)
5209:
The Mystery Of The Aleph: Mathematics, the Kabbalah, and the Search for Infinity
5136:
5021:
4815:
The Mystery of the Aleph: Mathematics, the Kabbalah, and the Search for Infinity
2891:
2750:
2662:
2475:, with the proviso that one can extend it as far as one wants; but extending it
2331:
2001:
819:
652:
258:
228:
144:
4854:
Proceedings of the National Academy of Sciences of the United States of America
2677:
measurements (i.e., counting). Concepts of infinite things such as an infinite
639:
Innumerable: nearly innumerable, truly innumerable, and innumerably innumerable
155:
can be studied, manipulated, and used just like any other mathematical object.
5987:
5904:
5689:
5337:
4942:
4658:
4527:
2746:
2678:
2532:
2447:, curved lines that twist and turn enough to fill the whole of any square, or
2035:
1948:
Infinitesimals (ε) and infinities (ω) on the hyperreal number line (1/ε = ω/1)
1675:
783:
629:
4729:
3751:
3307:
3185:
3142:
3079:
5795:
5790:
5624:
5460:
5077:
3656:
3127:. Imre Leader, Princeton University. Princeton: Princeton University Press.
3087:
2975:
2857:
2826:
2700:
2674:
2452:
2114:
positive integers, and any set which has the cardinality of the integers is
823:
261:
Greeks had a "horror of the infinite" which would, for example, explain why
244:
190:
4894:
4875:
3240:. Lecture Notes in Mathematics. Vol. 873. Springer. pp. 135–145.
2753:, posits that there are an infinite number and variety of universes. Also,
342:
Zeno was not attempting to make a point about infinity. As a member of the
4537:
PBS Infinite Series, with academic sources by J. Hamkins (infinite chess:
4163:
Alien Life Imagined: Communicating the Science and Culture of Astrobiology
3324:
The origins of science; an inquiry into the foundations of Western thought
3122:
2539:, points and lines were viewed as distinct entities, and a point could be
478:{\displaystyle a+ax+ax^{2}+ax^{3}+ax^{4}+ax^{5}+\cdots ={\frac {a}{1-x}}.}
127:, it remained unclear whether infinity could be considered as a number or
17:
5982:
5977:
5358:
5108:, below. A concise introduction to Cantor's mathematics of infinite sets.
4948:
3071:
2925:
2758:
2716:
2704:
2638:
2575:
2464:
2377:
The first of these results is apparent by considering, for instance, the
2336:
This hypothesis cannot be proved or disproved within the widely accepted
2106:
2102:
354:
provided both a satisfactory definition of a limit and a proof that, for
316:
194:
124:
43:
5213:
5619:
5276:
4670:
Mathematical Thinking and Writing: A Transition to Abstract Mathematics
3245:
2844:
In languages that do not have greatest and least elements but do allow
2830:
2658:
2610:
2365:
2014:
One-to-one correspondence between an infinite set and its proper subset
1527:
to infinity, in the sense that the partial sums increase without bound.
787:
343:
237:
186:
148:
2871:
whose exit condition is never satisfied, thus executing indefinitely.
1585:
of the real numbers. Adding algebraic properties to this gives us the
5874:
5679:
5614:
2703:
have long sought to discover whether infinity exists in our physical
2067:
262:
211:
Ancient cultures had various ideas about the nature of infinity. The
4910:
Jain, L.C. (1973). "Set theory in the Jaina school of mathematics",
4851:
Cohen, Paul (1963), "The Independence of the Continuum Hypothesis",
4193:(illustrated ed.). Jones & Bartlett Learning. p. 553.
3633:
Levels of Infinity / Selected Writings on Mathematics and Philosophy
2463:
Until the end of the 19th century, infinity was rarely discussed in
2070:) that the whole cannot be the same size as the part. (However, see
679:, and exploited it in area calculations by dividing the region into
131:
and, if so, how this could be done. At the end of the 19th century,
4559:
4304:
Kaku, M. (2006). Parallel worlds. Knopf Doubleday Publishing Group.
3299:
726:
wrote about equations with an infinite number of terms in his work
5064:
4545:
2898:
2770:
2739:
2364:
2009:
1674:
808:
151:. In this usage, infinity is a mathematical concept, and infinite
59:
49:
5026:
Infinity and the Mind: The Science and Philosophy of the Infinite
3927:(3rd ed.), San Diego, CA: Academic Press, Inc., p. 29,
2369:
The first three steps of a fractal construction whose limit is a
5553:
4841:
4166:(illustrated ed.). Cambridge University Press. p. 63.
2731:
2448:
227:
The earliest recorded idea of infinity in Greece may be that of
5937:
5557:
5223:
5219:
3707:
The mathematical work of John Wallis, D.D., F.R.S., (1616–1703)
3482:. Libraires du Roi & de la Bibliothèque du Roi. p. 124
2495:
One of the rare exceptions of a mathematical concept involving
162:, on which most of modern mathematics can be developed, is the
5604:
5087:
4619:
A History of Mathematical Notations (Two Volumes Bound as One)
2791:
2730:. To date, analysis of the radiation patterns recorded by the
642:
Infinite: nearly infinite, truly infinite, infinitely infinite
3327:. Internet Archive. London, Allen and Unwin. pp. 1–241.
2749:
hypothesis, which, when explained by astrophysicists such as
2681:
exist, but there are no experimental means to generate them.
2251:{\displaystyle \mathbf {c} =2^{\aleph _{0}}>{\aleph _{0}}}
1980:; there is no equivalence between them as with the Cantorian
743:
opened a mathematico-philosophic address given in 1930 with:
69:
is something which is boundless, endless, or larger than any
2350:
can be used to show not only that the number of points in a
170:
that may seem to have nothing to do with them. For example,
4920:
The Crest of the Peacock: Non-European Roots of Mathematics
2597:
of infinite dimension. In particular, this is the case of
1223:{\displaystyle \int _{-\infty }^{\infty }f(t)\,dt=\infty }
4693:, New York: Oxford University Press, pp. 1197–1198,
2062:
Dedekind's approach was essentially to adopt the idea of
4064:"Georg Cantor and the Battle for Transfinite Set Theory"
135:
enlarged the mathematical study of infinity by studying
4385:(Java SE 7 ed.). California: Oracle America, Inc.
3974:
Projective Geometry / from foundations to applications
2722:
The curvature of the universe can be measured through
729:
De analysi per aequationes numero terminorum infinitas
691:
4660:
Elementary Calculus: An Approach Using Infinitesimals
3921:
Aliprantis, Charalambos D.; Burkinshaw, Owen (1998),
3310:– via The University of Chicago Press Journals.
2285:
2207:
2166:
2144:
1914:
1889:
1850:
1805:
1782:
1751:
1725:
1701:
1618:
1595:
1560:
1537:
1473:
1449:
1389:
1356:
1327:
1267:
1236:
1176:
1152:
1132:
1103:
1046:
1022:
986:
965:
935:
914:
887:
863:
768:
689:
661:
554:
511:
502:. Achilles does overtake the tortoise; it takes him
368:
82:
4760:(Alternate ed.), Prindle, Weber & Schmidt,
3393:. Vol. v. 1. The University Press. p. 212.
2034:. In this system, the first transfinite cardinal is
1908:
as maps into the Riemann sphere taking the value of
1314:{\displaystyle \int _{-\infty }^{\infty }f(t)\,dt=a}
877:, called "infinity", is used to denote an unbounded
5825:
5778:
5747:
5738:
5591:
5510:
5479:
5351:
5315:
5264:
2590:are generally vector spaces of infinite dimension.
2320:{\displaystyle \mathbf {c} =\aleph _{1}=\beth _{1}}
1523:means that the sum of the infinite series properly
35:. For other uses of "Infinity" and "Infinite", see
5094:A Crash Course in the Mathematics of Infinite Sets
4991:O'Connor, John J. and Edmund F. Robertson (2000).
4971:O'Connor, John J. and Edmund F. Robertson (1998).
4788:Everything and More: A Compact History of Infinity
4440:
4277:
3600:Landmark Writings in Western Mathematics 1640-1940
2618:with an infinite perimeter and finite area is the
2319:
2250:
2181:
2152:
2059:and others—using the idea of collections or sets.
1920:
1904:. In this context, it is often useful to consider
1895:
1870:
1811:
1788:
1767:
1737:
1715:, called "infinity", denotes an unsigned infinite
1707:
1627:
1604:
1569:
1546:
1515:
1458:
1431:
1365:
1342:
1313:
1251:
1222:
1161:
1138:
1118:
1089:
1028:
1007:
971:
950:
920:
899:
869:
774:
707:
667:
608:
541:
477:
319:races a tortoise, giving the latter a head start.
88:
5179:John J. O'Connor and Edmund F. Robertson (2000).
5159:John J. O'Connor and Edmund F. Robertson (1998).
4691:Mathematical Thought from Ancient to Modern Times
1516:{\displaystyle \sum _{i=0}^{\infty }f(i)=\infty }
293:Zeno's paradoxes § Achilles and the tortoise
4261:Marcus Y. Yoo (2011). "Unexpected connections".
3972:Beutelspacher, Albrecht; Rosenbaum, Ute (1998),
2558:In particular, in modern mathematics, lines are
3681:"List of LaTeX mathematical symbols - OeisWiki"
3479:Cours d'Analyse de l'École Royale Polytechnique
745:
275:
58:, it seems that there is a boundless amount of
2089:Cantor defined two kinds of infinite numbers:
5949:
5569:
5235:
2593:In topology, some constructions can generate
1581:of the real numbers, producing the two-point
1090:{\displaystyle \int _{a}^{b}f(t)\,dt=\infty }
636:Enumerable: lowest, intermediate, and highest
8:
4377:Gosling, James; et al. (27 July 2012).
3788:, University of Chicago Press, p. 243,
3414:. Stuttgart, Germany: Franz Steiner Verlag.
3121:Gowers, Timothy; Barrow-Green, June (2008).
2794:values). These are defined as the result of
2745:The concept of infinity also extends to the
2555:(however, the latter phrase is still used).
2160:is greater than that of the natural numbers
542:{\displaystyle 10+0.1+0.001+0.00001+\cdots }
4943:http://www.math.wisc.edu/~keisler/calc.html
3898:"Properly Divergent Sequences - Mathonline"
3816:Nabokov: The Mystery of Literary Structures
1836:, called the extended complex plane or the
1432:{\displaystyle \sum _{i=0}^{\infty }f(i)=a}
747:Mathematics is the science of the infinite.
5956:
5942:
5934:
5744:
5576:
5562:
5554:
5242:
5228:
5220:
4837:Ancient Jaina Mathematics: an Introduction
3976:, Cambridge University Press, p. 27,
1439:means that the sum of the infinite series
195:whether the universe is spatially infinite
147:of the line) is larger than the number of
4884:
4874:
4558:
4544:
4509:Infinite chess at the Chess Variant Pages
3959:
3884:
3819:, Cornell University Press, p. 159,
3631:Weyl, Hermann (2012), Peter Pesic (ed.),
3538:. Vol. 1. Cosimo, Inc. p. 214.
3047:
3045:
2311:
2298:
2286:
2284:
2241:
2236:
2225:
2220:
2208:
2206:
2172:
2167:
2165:
2145:
2143:
2082:parts; this notion of infinity is called
1913:
1888:
1854:
1849:
1804:
1781:
1760:
1752:
1750:
1724:
1700:
1617:
1594:
1559:
1536:
1489:
1478:
1472:
1448:
1405:
1394:
1388:
1355:
1326:
1298:
1280:
1272:
1266:
1235:
1207:
1189:
1181:
1175:
1151:
1131:
1102:
1074:
1056:
1051:
1045:
1021:
985:
980:decreases without bound. For example, if
964:
934:
913:
886:
862:
767:
690:
688:
660:
598:
579:
558:
553:
510:
454:
439:
423:
407:
391:
367:
81:
3507:. Oxford University Press. p. 117.
2901:played on an unbounded board are called
2886:, roughly corresponding to mathematical
2354:is equal to the number of points in any
1943:
115:and what some mathematicians (including
5203:MacTutor History of Mathematics archive
5192:MacTutor History of Mathematics archive
5173:MacTutor History of Mathematics archive
5161:'Georg Ferdinand Ludwig Philipp Cantor'
5015:MacTutor History of Mathematics archive
5004:MacTutor History of Mathematics archive
4985:MacTutor History of Mathematics archive
4973:'Georg Ferdinand Ludwig Philipp Cantor'
4575:Elglaly, Yasmine Nader; Quek, Francis.
4485:"The Mathematical Side of M. C. Escher"
3460:
3390:The Thirteen Books of Euclid's Elements
3282:The Thirteen Books of Euclid's Elements
3206:
3196:– via Cambridge University Press.
3052:Jesseph, Douglas Michael (1998-05-01).
2993:
2189:; that is, there are more real numbers
1989:
1820:can be added to the complex plane as a
313:as "immeasurably subtle and profound".
4447:. Courier Dover Publications. p.
4354:"Infinity and NaN (The GNU C Library)"
4187:Koupelis, Theo; Kuhn, Karl F. (2007).
3871:
3584:
3558:
3155:
3124:The Princeton companion to mathematics
2018:A different form of "infinity" is the
1376:Infinity can also be used to describe
708:{\displaystyle {\tfrac {1}{\infty }}.}
174:implicitly relies on the existence of
172:Wiles's proof of Fermat's Last Theorem
4528:"Infinite Chess, PBS Infinite Series"
4107:
4095:
4083:
3996:Rao, Murali; Stetkær, Henrik (1991).
3785:Dreams, Illusion, and Other Realities
2852:, it is possible for a programmer to
2711:"? This is still an open question of
2026:infinities of set theory—a system of
54:Due to the constant light reflection
7:
5292:Hilbert's paradox of the Grand Hotel
4912:Indian Journal of History of Science
4790:, Norton, W.W. & Company, Inc.,
4443:Mathematics for the nonmathematician
3007:
3005:
3003:
3001:
2999:
2997:
2802:, and other exceptional operations.
2435:Hilbert's paradox of the Grand Hotel
1738:{\displaystyle x\rightarrow \infty }
900:{\displaystyle x\rightarrow \infty }
235:Greek philosopher. He used the word
111:, mathematicians began to work with
5129:"Infinity is bigger than you think"
5070:Internet Encyclopedia of Philosophy
4314:McKee, Maggie (25 September 2014).
4250:from the original on 10 April 2015.
4218:"Will the Universe expand forever?"
3858:Stanford Encyclopedia of Philosophy
3742:(1990), "Mathematics of infinity",
3535:A History of Mathematical Notations
3521:from the original on April 3, 2017.
3504:Infinity: a Very Short Introduction
3168:McLarty, Colin (15 January 2014) .
2875:Arts, games, and cognitive sciences
2695:On the Infinite Universe and Worlds
1797:grows beyond any assigned value. A
4341:Cambridge Dictionary of Philosophy
3782:O'Flaherty, Wendy Doniger (1986),
3321:Hutten, Ernest Hirschlaff (1962).
2693:proposed an unbounded universe in
2492:that have the property" (plural).
2295:
2238:
2222:
2169:
1915:
1865:
1806:
1732:
1702:
1655:in three-dimensional space, and a
1639:of the real numbers, which is the
1622:
1599:
1564:
1541:
1510:
1490:
1406:
1281:
1276:
1217:
1190:
1185:
1126:does not bound a finite area from
1084:
945:
894:
864:
769:
696:
662:
83:
25:
5850:Indefinite and fictitious numbers
5487:Differential geometry of surfaces
4904:Exact Sciences from Jaina Sources
4002:. World Scientific. p. 113.
2267:Cantor's first set theory article
2105:which are maps from the positive
2047:), the cardinality of the set of
5282:Controversy over Cantor's theory
4389:from the original on 9 June 2012
4322:. No. 17. NautilusThink Inc
4224:from the original on 1 June 2012
3704:Scott, Joseph Frederick (1981),
3657:"Unicode Character "∞" (U+221E)"
3617:from the original on 2016-06-03.
3387:; Heiberg, Johan Ludvig (1908).
2287:
2209:
2146:
1934:Möbius transformation § Overview
1321:means that the total area under
683:strips of width on the order of
105:philosophical nature of infinity
6088:Least-squares spectral analysis
6015:Fundamental theorem of calculus
5343:Synthetic differential geometry
4757:Calculus with Analytic Geometry
4483:Schattschneider, Doris (2010).
4383:The Java Language Specification
4246:. FermiLab/SLAC. 7 April 2015.
4071:9th ACMS Conference Proceedings
3949:from the original on 2015-05-15
3849:"Continuity and Infinitesimals"
3833:from the original on 2016-05-09
3802:from the original on 2016-06-29
3728:from the original on 2016-05-09
3597:Grattan-Guinness, Ivor (2005).
2097:. Ordinal numbers characterize
287:Zeno: Achilles and the tortoise
267:infinitude of the prime numbers
5045:Journal of the Asiatic Society
5028:. Princeton University Press.
4957:. Princeton University Press.
4778:, Blaisdell Publishing Company
4636:Gemignani, Michael C. (1990),
3411:: Infinity in Greek Philosophy
3174:The Bulletin of Symbolic Logic
1761:
1753:
1745:means that the magnitude
1729:
1504:
1498:
1420:
1414:
1337:
1331:
1295:
1289:
1246:
1240:
1204:
1198:
1113:
1107:
1071:
1065:
996:
990:
939:
891:
62:and repetition inside of them.
1:
5765:Conway chained arrow notation
4712:The Principles of Mathematics
4343:, Second Edition, p. 429
3712:American Mathematical Society
3357:Euclid's Elements of Geometry
3278:Sarton, George (March 1928).
2201:. Namely, Cantor showed that
2182:{\displaystyle {\aleph _{0}}}
1970:smooth infinitesimal analysis
951:{\displaystyle x\to -\infty }
929:increases without bound, and
837:, one of the co-inventors of
818:It was introduced in 1655 by
305:
301:
73:. It is often denoted by the
5389:Cardinality of the continuum
2882:artwork uses the concept of
2757:posit an infinite amount of
2626:Mathematics without infinity
2547:, and one says that a point
2153:{\displaystyle \mathbf {c} }
2134:Cardinality of the continuum
2128:Cardinality of the continuum
1952:The original formulation of
1635:as the same, leading to the
4754:Swokowski, Earl W. (1983),
4668:Maddox, Randall B. (2002),
4408:Stokes, Roger (July 2012).
3924:Principles of Real Analysis
3284:. Thomas L. Heath, Heiberg"
2819:greatest and least elements
2728:cosmic background radiation
2381:function, which provides a
2338:Zermelo–Fraenkel set theory
2332:Beth number § Beth one
1960:and Gottfried Leibniz used
1871:{\displaystyle z/0=\infty }
1350:is finite, and is equal to
160:Zermelo–Fraenkel set theory
6208:
5860:Largest known prime number
5352:Formalizations of infinity
5214:Dictionary of the Infinite
4918:Joseph, George G. (2000).
4817:. New York: Pocket Books.
3587:, Sec. 435, Vol. II, p. 58
3561:, Sec. 421, Vol. II, p. 44
2432:
2329:
2263:Cantor's diagonal argument
2260:
2131:
1999:
1826:one-point compactification
1637:one-point compactification
1230:means that the area under
1008:{\displaystyle f(t)\geq 0}
755:
290:
204:
29:
6192:Philosophy of mathematics
6153:
6053:
5972:
5913:
5845:Extended real number line
5760:Knuth's up-arrow notation
5528:Gottfried Wilhelm Leibniz
4774:Taylor, Angus E. (1955),
4616:Cajori, Florian (1993) ,
4518:An infinite chess scheme.
4465:Section 10-7, p. 229
4263:Engineering & Science
4220:. NASA. 24 January 2014.
3572:"Arithmetica Infinitorum"
3110:– via Project MUSE.
3021:Texas A&M Mathematics
3014:"The History of Infinity"
2635:philosophy of mathematics
2537:foundation of mathematics
2471:was what is now called a
2383:one-to-one correspondence
2064:one-to-one correspondence
675:for such a number in his
37:Infinity (disambiguation)
5770:Steinhaus–Moser notation
4190:In Quest of the Universe
3752:10.1007/3-540-52335-9_54
3744:COLOG-88 (Tallinn, 1988)
3603:. Elsevier. p. 62.
3532:Cajori, Florian (2007).
3407:In the Beginning Was the
3385:Heath, Sir Thomas Little
3238:Constructive Mathematics
2788:IEEE floating-point
2574:that occur in classical
2275:states that there is no
1681:stereographic projection
1628:{\displaystyle -\infty }
1605:{\displaystyle +\infty }
1570:{\displaystyle -\infty }
1547:{\displaystyle +\infty }
655:first used the notation
273:, sometimes translated:
56:between opposing mirrors
42:Not to be confused with
5533:August Ferdinand Möbius
5316:Branches of mathematics
5307:Paradoxes of set theory
4813:Aczel, Amir D. (2001).
4640:(2nd ed.), Dover,
4276:Weeks, Jeffrey (2001).
4062:Dauben, Joseph (1993).
3059:Perspectives on Science
2956:Infinite monkey theorem
2726:in the spectrum of the
1921:{\displaystyle \infty }
1812:{\displaystyle \infty }
1708:{\displaystyle \infty }
870:{\displaystyle \infty }
775:{\displaystyle \infty }
717:Arithmetica infinitorum
668:{\displaystyle \infty }
247:(350 BC) distinguished
231:(c. 610 – c. 546 BC) a
89:{\displaystyle \infty }
6020:Calculus of variations
5993:Differential equations
5813:Fast-growing hierarchy
4954:To Infinity and Beyond
4876:10.1073/pnas.50.6.1143
4439:Kline, Morris (1985).
4244:"Our universe is Flat"
3902:mathonline.wikidot.com
3474:Cauchy, Augustin-Louis
3404:Drozdek, Adam (2008).
3186:10.2178/bsl/1286284558
3012:Allen, Donald (2003).
2951:Names of large numbers
2374:
2321:
2252:
2183:
2154:
2015:
1988:is fully developed in
1954:infinitesimal calculus
1949:
1930:Möbius transformations
1922:
1897:
1872:
1813:
1790:
1769:
1739:
1709:
1688:
1657:hyperplane at infinity
1629:
1606:
1571:
1548:
1517:
1494:
1460:
1433:
1410:
1367:
1344:
1315:
1253:
1224:
1163:
1140:
1120:
1091:
1030:
1009:
973:
952:
922:
901:
871:
839:infinitesimal calculus
782:(sometimes called the
776:
749:
709:
677:De sectionibus conicis
669:
610:
543:
479:
280:
176:Grothendieck universes
109:infinitesimal calculus
90:
63:
6113:Representation theory
6072:quaternionic analysis
6068:Hypercomplex analysis
5966:mathematical analysis
5870:Long and short scales
5808:Grzegorczyk hierarchy
5497:Möbius transformation
5394:Dedekind-infinite set
5302:Paradoxes of infinity
5297:Infinity (philosophy)
5009:Pearce, Ian. (2002).
4784:Wallace, David Foster
4138:John Gribbin (2009),
3813:Toker, Leona (1989),
3635:, Dover, p. 17,
2971:Paradoxes of infinity
2807:programming languages
2578:have always a finite
2545:the set of its points
2368:
2330:Further information:
2322:
2261:Further information:
2253:
2195:than natural numbers
2184:
2155:
2013:
1986:non-standard calculus
1947:
1923:
1906:meromorphic functions
1898:
1873:
1814:
1791:
1770:
1740:
1710:
1678:
1663:, each consisting of
1651:in plane geometry, a
1630:
1607:
1587:extended real numbers
1572:
1549:
1518:
1474:
1461:
1434:
1390:
1368:
1345:
1316:
1254:
1225:
1164:
1141:
1121:
1092:
1031:
1010:
974:
953:
923:
902:
872:
777:
710:
670:
611:
544:
480:
352:Augustin-Louis Cauchy
207:Infinity (philosophy)
205:Further information:
197:is an open question.
180:elementary arithmetic
99:From the time of the
91:
53:
6187:Mathematical objects
6045:Table of derivatives
5333:Nonstandard analysis
5113:Infinite Reflections
5106:Infinite Reflections
4738:Space-Filling Curves
4736:Sagan, Hans (1994),
4714:, New York: Norton,
4160:Brake, Mark (2013).
4110:, pp. 1197–1198
4047:Moore, A.W. (1991).
3501:Ian Stewart (2017).
3072:10.1162/posc_a_00543
2850:relational operators
2661:, approximations of
2599:iterated loop spaces
2553:is located on a line
2445:space-filling curves
2356:segment of that line
2340:, even assuming the
2283:
2273:continuum hypothesis
2205:
2164:
2142:
1974:nonstandard analysis
1940:Nonstandard analysis
1912:
1887:
1848:
1803:
1780:
1749:
1723:
1699:
1641:real projective line
1616:
1593:
1589:. We can also treat
1577:can be added to the
1558:
1535:
1471:
1447:
1387:
1354:
1343:{\displaystyle f(t)}
1325:
1265:
1252:{\displaystyle f(t)}
1234:
1174:
1150:
1130:
1119:{\displaystyle f(t)}
1101:
1044:
1020:
984:
963:
933:
912:
885:
861:
766:
762:The infinity symbol
687:
659:
552:
509:
366:
153:mathematical objects
80:
31:For the symbol, see
27:Mathematical concept
6125:Continuous function
6078:Functional analysis
5885:Orders of magnitude
5755:Scientific notation
5502:Riemannian manifold
5471:Transfinite numbers
5328:Internal set theory
5197:Ian Pearce (2002).
5181:'Jaina mathematics'
4993:'Jaina mathematics'
4902:Jain, L.C. (1982).
4867:1963PNAS...50.1143C
4842:Infinity Foundation
4638:Elementary Topology
3440:Stanford University
2909:Cognitive scientist
2863:In programming, an
2796:arithmetic overflow
2778:supposed to play."
2609:The structure of a
2584:functional analysis
2501:projective geometry
2371:space-filling curve
2348:Cardinal arithmetic
2030:first developed by
2028:transfinite numbers
1768:{\displaystyle |x|}
1645:Projective geometry
1443:to some real value
1285:
1194:
1061:
6157:Mathematics portal
6040:Lists of integrals
5803:Ackermann function
5455:Sphere at infinity
5406:(Complex infinity)
5186:2008-12-20 at the
5166:2006-09-16 at the
5118:2009-11-05 at the
5099:2010-02-27 at the
4998:2008-12-20 at the
4978:2006-09-16 at the
4672:, Academic Press,
4655:Keisler, H. Jerome
4533:2017-04-07 at the
4514:2017-04-02 at the
4492:Notices of the AMS
4470:2016-05-16 at the
4280:The Shape of Space
4125:2013-01-24 at the
4028:math.dartmouth.edu
3887:, pp. 468–510
3442:. October 15, 2010
3436:"Zeno's Paradoxes"
3246:10.1007/BFb0090732
3102:on 11 January 2012
2946:Indeterminate form
2888:points at infinity
2595:topological spaces
2566:Infinite dimension
2531:Before the use of
2515:effect that shows
2505:points at infinity
2375:
2360:finite-dimensional
2317:
2248:
2179:
2150:
2116:countably infinite
2016:
1950:
1918:
1893:
1868:
1809:
1786:
1765:
1735:
1705:
1689:
1665:points at infinity
1625:
1602:
1567:
1544:
1513:
1459:{\displaystyle a.}
1456:
1429:
1366:{\displaystyle a.}
1363:
1340:
1311:
1268:
1249:
1220:
1177:
1162:{\displaystyle b.}
1159:
1136:
1116:
1087:
1047:
1026:
1005:
969:
948:
918:
897:
867:
772:
705:
700:
665:
606:
539:
475:
350:Finally, in 1821,
271:parallel postulate
249:potential infinity
86:
64:
6177:Concepts in logic
6164:
6163:
6130:Special functions
6093:Harmonic analysis
5931:
5930:
5821:
5820:
5551:
5550:
5445:Point at infinity
5425:Hyperreal numbers
5399:Directed infinity
5364:Absolute infinite
5287:Galileo's paradox
5272:Ananta (infinite)
5035:978-0-691-00172-2
4964:978-0-691-02511-7
4933:978-0-14-027778-4
4824:978-0-7434-2299-4
4797:978-0-393-32629-1
4776:Advanced Calculus
4767:978-0-87150-341-1
4747:978-1-4612-0871-6
4721:978-0-393-31404-5
4708:Russell, Bertrand
4700:978-0-19-506135-2
4679:978-0-12-464976-7
4647:978-0-486-66522-1
4629:978-0-486-67766-8
4458:978-0-486-24823-3
4291:978-0-8247-0709-5
4206:Extract of p. 553
4200:978-0-7637-4387-1
4173:978-0-521-49129-7
4148:978-0-470-61352-8
3983:978-0-521-48364-3
3934:978-0-12-050257-8
3826:978-0-8014-2211-9
3795:978-0-226-61855-5
3761:978-3-540-52335-2
3721:978-0-8284-0314-6
3642:978-0-486-48903-2
3610:978-0-08-045744-4
3514:978-0-19-875523-4
3421:978-3-515-09258-6
3370:978-0-6151-7984-1
3334:978-0-04-946007-2
3255:978-3-540-10850-4
3134:978-1-4008-3039-8
3066:(1&2): 6–40.
3030:on August 1, 2020
2724:multipole moments
2669:measurements and
2631:Leopold Kronecker
2549:belongs to a line
2541:located on a line
2511:for modeling the
2507:are added to the
2084:Dedekind infinite
2072:Galileo's paradox
1896:{\displaystyle z}
1822:topological space
1789:{\displaystyle x}
1719:. The expression
1653:plane at infinity
1647:also refers to a
1579:topological space
1139:{\displaystyle a}
1029:{\displaystyle t}
972:{\displaystyle x}
921:{\displaystyle x}
843:Law of continuity
835:Gottfried Leibniz
699:
626:Jain mathematical
601:
587:
574:
470:
164:axiom of infinity
16:(Redirected from
6199:
6083:Fourier analysis
6063:Complex analysis
5964:Major topics in
5958:
5951:
5944:
5935:
5745:
5675:Eddington number
5620:Hundred thousand
5578:
5571:
5564:
5555:
5543:Abraham Robinson
5538:Bernhard Riemann
5457:(Kleinian group)
5450:Regular cardinal
5404:Division by zero
5384:Cardinal numbers
5323:Complex analysis
5258:
5244:
5237:
5230:
5221:
5151:
5149:
5148:
5139:. Archived from
5074:
5052:
5039:
4968:
4937:
4922:(2nd ed.).
4907:
4897:
4888:
4878:
4861:(6): 1143–1148,
4828:
4800:
4779:
4770:
4750:
4732:
4703:
4682:
4664:
4650:
4632:
4602:
4601:
4599:
4598:
4592:
4586:. Archived from
4581:
4572:
4566:
4564:
4562:
4550:
4548:
4525:
4519:
4506:
4500:
4499:
4489:
4480:
4474:
4462:
4446:
4436:
4430:
4429:
4427:
4425:
4420:on 25 March 2012
4416:. Archived from
4405:
4399:
4398:
4396:
4394:
4374:
4368:
4367:
4365:
4364:
4350:
4344:
4338:
4332:
4331:
4329:
4327:
4311:
4305:
4302:
4296:
4295:
4283:
4273:
4267:
4266:
4258:
4252:
4251:
4240:
4234:
4233:
4231:
4229:
4214:
4208:
4204:
4184:
4178:
4177:
4157:
4151:
4136:
4130:
4117:
4111:
4105:
4099:
4098:, pp. 10–12
4093:
4087:
4081:
4075:
4074:
4068:
4059:
4053:
4052:
4044:
4038:
4037:
4035:
4034:
4020:
4014:
4013:
3993:
3987:
3986:
3969:
3963:
3957:
3951:
3950:
3918:
3912:
3911:
3909:
3908:
3894:
3888:
3881:
3875:
3869:
3863:
3862:
3853:Zalta, Edward N.
3841:
3835:
3834:
3810:
3804:
3803:
3779:
3773:
3772:
3736:
3730:
3729:
3701:
3695:
3694:
3692:
3691:
3677:
3671:
3670:
3668:
3667:
3652:
3646:
3645:
3628:
3622:
3620:Extract of p. 62
3618:
3594:
3588:
3582:
3576:
3575:
3568:
3562:
3556:
3550:
3549:
3529:
3523:
3522:
3498:
3492:
3491:
3489:
3487:
3470:
3464:
3458:
3452:
3451:
3449:
3447:
3432:
3426:
3425:
3401:
3395:
3394:
3381:
3375:
3374:
3362:
3354:Euclid (2008) .
3351:
3345:
3344:
3342:
3341:
3318:
3312:
3311:
3275:
3269:
3266:
3260:
3259:
3233:
3227:
3226:
3216:
3210:
3204:
3198:
3197:
3165:
3159:
3153:
3147:
3146:
3118:
3112:
3111:
3109:
3107:
3098:. Archived from
3049:
3040:
3039:
3037:
3035:
3029:
3023:. Archived from
3018:
3009:
2884:vanishing points
2800:division by zero
2775:infinite regress
2521:projective plane
2480:expression "the
2429:
2423:
2422:
2420:
2419:
2416:
2413:
2406:
2404:
2403:
2400:
2397:
2352:real number line
2326:
2324:
2323:
2318:
2316:
2315:
2303:
2302:
2290:
2257:
2255:
2254:
2249:
2247:
2246:
2245:
2232:
2231:
2230:
2229:
2212:
2200:
2194:
2188:
2186:
2185:
2180:
2178:
2177:
2176:
2159:
2157:
2156:
2151:
2149:
2095:cardinal numbers
2057:Richard Dedekind
2046:
1927:
1925:
1924:
1919:
1902:
1900:
1899:
1894:
1878:for any nonzero
1877:
1875:
1874:
1869:
1858:
1842:division by zero
1830:complex manifold
1818:
1816:
1815:
1810:
1795:
1793:
1792:
1787:
1774:
1772:
1771:
1766:
1764:
1756:
1744:
1742:
1741:
1736:
1714:
1712:
1711:
1706:
1693:complex analysis
1671:Complex analysis
1649:line at infinity
1634:
1632:
1631:
1626:
1611:
1609:
1608:
1603:
1583:compactification
1576:
1574:
1573:
1568:
1553:
1551:
1550:
1545:
1522:
1520:
1519:
1514:
1493:
1488:
1465:
1463:
1462:
1457:
1438:
1436:
1435:
1430:
1409:
1404:
1372:
1370:
1369:
1364:
1349:
1347:
1346:
1341:
1320:
1318:
1317:
1312:
1284:
1279:
1258:
1256:
1255:
1250:
1229:
1227:
1226:
1221:
1193:
1188:
1168:
1166:
1165:
1160:
1145:
1143:
1142:
1137:
1125:
1123:
1122:
1117:
1096:
1094:
1093:
1088:
1060:
1055:
1035:
1033:
1032:
1027:
1014:
1012:
1011:
1006:
978:
976:
975:
970:
958:means that
957:
955:
954:
949:
927:
925:
924:
919:
907:means that
906:
904:
903:
898:
876:
874:
873:
868:
814:
806:
802:
799:
796:
794:
781:
779:
778:
773:
714:
712:
711:
706:
701:
692:
674:
672:
671:
666:
615:
613:
612:
607:
602:
599:
588:
580:
575:
573:
559:
548:
546:
545:
540:
501:
494:
484:
482:
481:
476:
471:
469:
455:
444:
443:
428:
427:
412:
411:
396:
395:
361:
311:Bertrand Russell
307:
303:
141:infinite numbers
95:
93:
92:
87:
47:
40:
21:
6207:
6206:
6202:
6201:
6200:
6198:
6197:
6196:
6167:
6166:
6165:
6160:
6149:
6098:P-adic analysis
6049:
6035:Matrix calculus
6030:Tensor calculus
6025:Vector calculus
5988:Differentiation
5968:
5962:
5932:
5927:
5909:
5865:List of numbers
5833:
5831:
5829:
5827:
5817:
5774:
5740:
5734:
5705:Graham's number
5695:Skewes's number
5597:
5595:
5593:
5587:
5582:
5552:
5547:
5506:
5475:
5466:Surreal numbers
5440:Ordinal numbers
5369:Actual infinity
5347:
5311:
5260:
5254:
5248:
5188:Wayback Machine
5168:Wayback Machine
5146:
5144:
5126:
5120:Wayback Machine
5101:Wayback Machine
5063:
5060:
5055:
5042:
5036:
5020:
5000:Wayback Machine
4980:Wayback Machine
4965:
4947:
4934:
4917:
4901:
4850:
4825:
4812:
4808:
4803:
4798:
4782:
4773:
4768:
4753:
4748:
4735:
4722:
4706:
4701:
4685:
4680:
4667:
4653:
4648:
4635:
4630:
4615:
4611:
4606:
4605:
4596:
4594:
4590:
4579:
4574:
4573:
4569:
4552:
4538:
4535:Wayback Machine
4526:
4522:
4516:Wayback Machine
4507:
4503:
4487:
4482:
4481:
4477:
4472:Wayback Machine
4459:
4438:
4437:
4433:
4423:
4421:
4407:
4406:
4402:
4392:
4390:
4376:
4375:
4371:
4362:
4360:
4352:
4351:
4347:
4339:
4335:
4325:
4323:
4313:
4312:
4308:
4303:
4299:
4292:
4275:
4274:
4270:
4260:
4259:
4255:
4242:
4241:
4237:
4227:
4225:
4216:
4215:
4211:
4201:
4186:
4185:
4181:
4174:
4159:
4158:
4154:
4137:
4133:
4127:Wayback Machine
4118:
4114:
4106:
4102:
4094:
4090:
4082:
4078:
4066:
4061:
4060:
4056:
4046:
4045:
4041:
4032:
4030:
4022:
4021:
4017:
4010:
3995:
3994:
3990:
3984:
3971:
3970:
3966:
3958:
3954:
3935:
3920:
3919:
3915:
3906:
3904:
3896:
3895:
3891:
3882:
3878:
3870:
3866:
3845:Bell, John Lane
3843:
3842:
3838:
3827:
3812:
3811:
3807:
3796:
3781:
3780:
3776:
3762:
3740:Martin-Löf, Per
3738:
3737:
3733:
3722:
3703:
3702:
3698:
3689:
3687:
3679:
3678:
3674:
3665:
3663:
3654:
3653:
3649:
3643:
3630:
3629:
3625:
3611:
3596:
3595:
3591:
3583:
3579:
3570:
3569:
3565:
3557:
3553:
3546:
3531:
3530:
3526:
3515:
3500:
3499:
3495:
3485:
3483:
3472:
3471:
3467:
3459:
3455:
3445:
3443:
3434:
3433:
3429:
3422:
3403:
3402:
3398:
3383:
3382:
3378:
3371:
3360:
3353:
3352:
3348:
3339:
3337:
3335:
3320:
3319:
3315:
3277:
3276:
3272:
3267:
3263:
3256:
3235:
3234:
3230:
3218:
3217:
3213:
3205:
3201:
3167:
3166:
3162:
3154:
3150:
3135:
3120:
3119:
3115:
3105:
3103:
3051:
3050:
3043:
3033:
3031:
3027:
3016:
3011:
3010:
2995:
2990:
2985:
2921:
2877:
2823:sentinel values
2784:
2767:
2687:
2671:natural numbers
2655:
2628:
2607:
2588:function spaces
2568:
2523:, two distinct
2509:Euclidean space
2497:actual infinity
2461:
2443:introduced the
2437:
2425:
2417:
2414:
2411:
2410:
2408:
2401:
2398:
2395:
2394:
2392:
2390:
2342:Axiom of Choice
2334:
2307:
2294:
2281:
2280:
2277:cardinal number
2269:
2237:
2221:
2216:
2203:
2202:
2196:
2190:
2168:
2162:
2161:
2140:
2139:
2136:
2130:
2091:ordinal numbers
2076:square integers
2049:natural numbers
2045:
2042:
2039:
2008:
2000:Main articles:
1998:
1978:hyperreal field
1966:logical systems
1942:
1910:
1909:
1885:
1884:
1846:
1845:
1834:Riemann surface
1801:
1800:
1778:
1777:
1747:
1746:
1721:
1720:
1697:
1696:
1673:
1614:
1613:
1591:
1590:
1556:
1555:
1533:
1532:
1469:
1468:
1445:
1444:
1385:
1384:
1378:infinite series
1352:
1351:
1323:
1322:
1263:
1262:
1232:
1231:
1172:
1171:
1148:
1147:
1128:
1127:
1099:
1098:
1042:
1041:
1018:
1017:
1015:for every
982:
981:
961:
960:
931:
930:
910:
909:
883:
882:
881:. The notation
859:
858:
851:
832:
812:
804:
800:
797:
792:
791:
764:
763:
760:
758:Infinity symbol
754:
738:
685:
684:
657:
656:
649:
622:
563:
550:
549:
507:
506:
496:
489:
459:
435:
419:
403:
387:
364:
363:
355:
295:
289:
254:actual infinity
225:
213:ancient Indians
209:
203:
113:infinite series
78:
77:
75:infinity symbol
48:
41:
33:Infinity symbol
30:
28:
23:
22:
15:
12:
11:
5:
6205:
6203:
6195:
6194:
6189:
6184:
6179:
6169:
6168:
6162:
6161:
6154:
6151:
6150:
6148:
6147:
6142:
6137:
6132:
6127:
6122:
6116:
6115:
6110:
6108:Measure theory
6105:
6102:P-adic numbers
6095:
6090:
6085:
6080:
6075:
6065:
6060:
6054:
6051:
6050:
6048:
6047:
6042:
6037:
6032:
6027:
6022:
6017:
6012:
6011:
6010:
6005:
6000:
5990:
5985:
5973:
5970:
5969:
5963:
5961:
5960:
5953:
5946:
5938:
5929:
5928:
5926:
5925:
5920:
5914:
5911:
5910:
5908:
5907:
5902:
5897:
5895:Power of three
5892:
5887:
5882:
5877:
5875:Number systems
5872:
5867:
5862:
5857:
5852:
5847:
5842:
5836:
5834:
5830:(alphabetical
5823:
5822:
5819:
5818:
5816:
5815:
5810:
5805:
5800:
5799:
5798:
5793:
5786:Hyperoperation
5782:
5780:
5776:
5775:
5773:
5772:
5767:
5762:
5757:
5751:
5749:
5742:
5736:
5735:
5733:
5732:
5727:
5722:
5717:
5712:
5707:
5702:
5700:Moser's number
5697:
5692:
5687:
5685:Shannon number
5682:
5677:
5672:
5667:
5662:
5657:
5652:
5647:
5642:
5637:
5632:
5627:
5622:
5617:
5612:
5607:
5601:
5599:
5589:
5588:
5583:
5581:
5580:
5573:
5566:
5558:
5549:
5548:
5546:
5545:
5540:
5535:
5530:
5525:
5520:
5514:
5512:
5511:Mathematicians
5508:
5507:
5505:
5504:
5499:
5494:
5489:
5483:
5481:
5477:
5476:
5474:
5473:
5468:
5463:
5458:
5452:
5447:
5442:
5437:
5432:
5427:
5422:
5417:
5415:Gimel function
5412:
5410:Epsilon number
5407:
5401:
5396:
5391:
5386:
5381:
5376:
5371:
5366:
5361:
5355:
5353:
5349:
5348:
5346:
5345:
5340:
5335:
5330:
5325:
5319:
5317:
5313:
5312:
5310:
5309:
5304:
5299:
5294:
5289:
5284:
5279:
5274:
5268:
5266:
5262:
5261:
5249:
5247:
5246:
5239:
5232:
5224:
5218:
5217:
5211:
5206:
5195:
5177:
5157:
5155:Hotel Infinity
5152:
5127:Grime, James.
5124:
5109:
5090:
5075:
5065:"The Infinite"
5059:
5058:External links
5056:
5054:
5053:
5040:
5034:
5018:
5007:
4989:
4969:
4963:
4945:
4938:
4932:
4915:
4908:
4899:
4848:
4845:
4829:
4823:
4809:
4807:
4804:
4802:
4801:
4796:
4780:
4771:
4766:
4751:
4746:
4733:
4720:
4704:
4699:
4683:
4678:
4665:
4663:(2nd ed.)
4651:
4646:
4633:
4628:
4612:
4610:
4607:
4604:
4603:
4567:
4520:
4501:
4475:
4457:
4431:
4400:
4369:
4345:
4333:
4306:
4297:
4290:
4268:
4253:
4235:
4209:
4199:
4179:
4172:
4152:
4131:
4112:
4100:
4088:
4086:, p. 1143
4076:
4054:
4039:
4015:
4008:
3988:
3982:
3964:
3960:Gemignani 1990
3952:
3933:
3913:
3889:
3885:Swokowski 1983
3876:
3864:
3836:
3825:
3805:
3794:
3774:
3760:
3731:
3720:
3714:, p. 24,
3710:(2 ed.),
3696:
3672:
3647:
3641:
3623:
3609:
3589:
3577:
3563:
3551:
3544:
3524:
3513:
3493:
3465:
3453:
3427:
3420:
3396:
3376:
3369:
3346:
3333:
3313:
3300:10.1086/346308
3270:
3261:
3254:
3228:
3211:
3199:
3180:(3): 359–377.
3160:
3148:
3133:
3113:
3041:
2992:
2991:
2989:
2986:
2984:
2983:
2981:Surreal number
2978:
2973:
2968:
2963:
2958:
2953:
2948:
2943:
2941:Exponentiation
2938:
2933:
2928:
2922:
2920:
2917:
2903:infinite chess
2897:Variations of
2876:
2873:
2783:
2780:
2766:
2763:
2691:Giordano Bruno
2686:
2683:
2654:
2651:
2643:constructivism
2627:
2624:
2620:Koch snowflake
2606:
2603:
2567:
2564:
2517:parallel lines
2460:
2457:
2441:Giuseppe Peano
2314:
2310:
2306:
2301:
2297:
2293:
2289:
2244:
2240:
2235:
2228:
2224:
2219:
2215:
2211:
2175:
2171:
2148:
2132:Main article:
2129:
2126:
2043:
2040:
2006:Ordinal number
1997:
1994:
1990:Keisler (1986)
1941:
1938:
1917:
1892:
1880:complex number
1867:
1864:
1861:
1857:
1853:
1838:Riemann sphere
1808:
1799:point labeled
1785:
1763:
1759:
1755:
1734:
1731:
1728:
1704:
1685:Riemann sphere
1672:
1669:
1624:
1621:
1601:
1598:
1566:
1563:
1543:
1540:
1529:
1528:
1512:
1509:
1506:
1503:
1500:
1497:
1492:
1487:
1484:
1481:
1477:
1466:
1455:
1452:
1428:
1425:
1422:
1419:
1416:
1413:
1408:
1403:
1400:
1397:
1393:
1380:, as follows:
1374:
1373:
1362:
1359:
1339:
1336:
1333:
1330:
1310:
1307:
1304:
1301:
1297:
1294:
1291:
1288:
1283:
1278:
1275:
1271:
1260:
1248:
1245:
1242:
1239:
1219:
1216:
1213:
1210:
1206:
1203:
1200:
1197:
1192:
1187:
1184:
1180:
1169:
1158:
1155:
1135:
1115:
1112:
1109:
1106:
1086:
1083:
1080:
1077:
1073:
1070:
1067:
1064:
1059:
1054:
1050:
1025:
1004:
1001:
998:
995:
992:
989:
968:
947:
944:
941:
938:
917:
896:
893:
890:
866:
850:
847:
831:
828:
771:
756:Main article:
753:
750:
737:
734:
704:
698:
695:
664:
648:
645:
644:
643:
640:
637:
621:
618:
617:
616:
605:
597:
594:
591:
586:
583:
578:
572:
569:
566:
562:
557:
538:
535:
532:
529:
526:
523:
520:
517:
514:
474:
468:
465:
462:
458:
453:
450:
447:
442:
438:
434:
431:
426:
422:
418:
415:
410:
406:
402:
399:
394:
390:
386:
383:
380:
377:
374:
371:
334:
333:
330:
327:
324:
291:Main article:
288:
285:
224:
221:
202:
199:
101:ancient Greeks
85:
71:natural number
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
6204:
6193:
6190:
6188:
6185:
6183:
6180:
6178:
6175:
6174:
6172:
6159:
6158:
6152:
6146:
6143:
6141:
6138:
6136:
6133:
6131:
6128:
6126:
6123:
6121:
6118:
6117:
6114:
6111:
6109:
6106:
6103:
6099:
6096:
6094:
6091:
6089:
6086:
6084:
6081:
6079:
6076:
6073:
6069:
6066:
6064:
6061:
6059:
6058:Real analysis
6056:
6055:
6052:
6046:
6043:
6041:
6038:
6036:
6033:
6031:
6028:
6026:
6023:
6021:
6018:
6016:
6013:
6009:
6006:
6004:
6001:
5999:
5996:
5995:
5994:
5991:
5989:
5986:
5984:
5980:
5979:
5975:
5974:
5971:
5967:
5959:
5954:
5952:
5947:
5945:
5940:
5939:
5936:
5924:
5921:
5919:
5916:
5915:
5912:
5906:
5903:
5901:
5898:
5896:
5893:
5891:
5888:
5886:
5883:
5881:
5878:
5876:
5873:
5871:
5868:
5866:
5863:
5861:
5858:
5856:
5855:Infinitesimal
5853:
5851:
5848:
5846:
5843:
5841:
5838:
5837:
5835:
5824:
5814:
5811:
5809:
5806:
5804:
5801:
5797:
5794:
5792:
5789:
5788:
5787:
5784:
5783:
5781:
5777:
5771:
5768:
5766:
5763:
5761:
5758:
5756:
5753:
5752:
5750:
5746:
5743:
5737:
5731:
5728:
5726:
5725:Rayo's number
5723:
5721:
5718:
5716:
5713:
5711:
5708:
5706:
5703:
5701:
5698:
5696:
5693:
5691:
5688:
5686:
5683:
5681:
5678:
5676:
5673:
5671:
5668:
5666:
5663:
5661:
5658:
5656:
5653:
5651:
5648:
5646:
5643:
5641:
5638:
5636:
5633:
5631:
5628:
5626:
5623:
5621:
5618:
5616:
5613:
5611:
5608:
5606:
5603:
5602:
5600:
5590:
5586:
5585:Large numbers
5579:
5574:
5572:
5567:
5565:
5560:
5559:
5556:
5544:
5541:
5539:
5536:
5534:
5531:
5529:
5526:
5524:
5523:David Hilbert
5521:
5519:
5516:
5515:
5513:
5509:
5503:
5500:
5498:
5495:
5493:
5490:
5488:
5485:
5484:
5482:
5478:
5472:
5469:
5467:
5464:
5462:
5459:
5456:
5453:
5451:
5448:
5446:
5443:
5441:
5438:
5436:
5435:Infinitesimal
5433:
5431:
5428:
5426:
5423:
5421:
5420:Hilbert space
5418:
5416:
5413:
5411:
5408:
5405:
5402:
5400:
5397:
5395:
5392:
5390:
5387:
5385:
5382:
5380:
5377:
5375:
5372:
5370:
5367:
5365:
5362:
5360:
5357:
5356:
5354:
5350:
5344:
5341:
5339:
5336:
5334:
5331:
5329:
5326:
5324:
5321:
5320:
5318:
5314:
5308:
5305:
5303:
5300:
5298:
5295:
5293:
5290:
5288:
5285:
5283:
5280:
5278:
5275:
5273:
5270:
5269:
5267:
5263:
5257:
5252:
5245:
5240:
5238:
5233:
5231:
5226:
5225:
5222:
5215:
5212:
5210:
5207:
5204:
5200:
5196:
5193:
5189:
5185:
5182:
5178:
5175:
5174:
5169:
5165:
5162:
5158:
5156:
5153:
5143:on 2017-10-22
5142:
5138:
5134:
5130:
5125:
5122:
5121:
5117:
5114:
5110:
5107:
5103:
5102:
5098:
5095:
5091:
5089:
5085:
5084:
5079:
5076:
5072:
5071:
5066:
5062:
5061:
5057:
5050:
5046:
5041:
5037:
5031:
5027:
5023:
5019:
5016:
5012:
5008:
5005:
5001:
4997:
4994:
4990:
4987:
4986:
4981:
4977:
4974:
4970:
4966:
4960:
4956:
4955:
4950:
4946:
4944:
4939:
4935:
4929:
4925:
4924:Penguin Books
4921:
4916:
4913:
4909:
4905:
4900:
4896:
4892:
4887:
4882:
4877:
4872:
4868:
4864:
4860:
4856:
4855:
4849:
4846:
4843:
4839:
4838:
4833:
4830:
4826:
4820:
4816:
4811:
4810:
4805:
4799:
4793:
4789:
4785:
4781:
4777:
4772:
4769:
4763:
4759:
4758:
4752:
4749:
4743:
4739:
4734:
4731:
4727:
4723:
4717:
4713:
4709:
4705:
4702:
4696:
4692:
4688:
4687:Kline, Morris
4684:
4681:
4675:
4671:
4666:
4662:
4661:
4656:
4652:
4649:
4643:
4639:
4634:
4631:
4625:
4621:
4620:
4614:
4613:
4608:
4593:on 2020-02-26
4589:
4585:
4578:
4571:
4568:
4561:
4556:
4547:
4542:
4536:
4532:
4529:
4524:
4521:
4517:
4513:
4510:
4505:
4502:
4498:(6): 706–718.
4497:
4493:
4486:
4479:
4476:
4473:
4469:
4466:
4460:
4454:
4450:
4445:
4444:
4435:
4432:
4419:
4415:
4411:
4404:
4401:
4388:
4384:
4380:
4373:
4370:
4359:
4355:
4349:
4346:
4342:
4337:
4334:
4321:
4317:
4310:
4307:
4301:
4298:
4293:
4287:
4284:. CRC Press.
4282:
4281:
4272:
4269:
4265:. LXXIV1: 30.
4264:
4257:
4254:
4249:
4245:
4239:
4236:
4223:
4219:
4213:
4210:
4207:
4202:
4196:
4192:
4191:
4183:
4180:
4175:
4169:
4165:
4164:
4156:
4153:
4149:
4145:
4141:
4135:
4132:
4128:
4124:
4121:
4116:
4113:
4109:
4104:
4101:
4097:
4092:
4089:
4085:
4080:
4077:
4072:
4065:
4058:
4055:
4050:
4043:
4040:
4029:
4025:
4019:
4016:
4011:
4009:9789810203757
4005:
4001:
4000:
3992:
3989:
3985:
3979:
3975:
3968:
3965:
3961:
3956:
3953:
3948:
3944:
3940:
3936:
3930:
3926:
3925:
3917:
3914:
3903:
3899:
3893:
3890:
3886:
3880:
3877:
3873:
3868:
3865:
3860:
3859:
3854:
3850:
3846:
3840:
3837:
3832:
3828:
3822:
3818:
3817:
3809:
3806:
3801:
3797:
3791:
3787:
3786:
3778:
3775:
3771:
3767:
3763:
3757:
3753:
3749:
3745:
3741:
3735:
3732:
3727:
3723:
3717:
3713:
3709:
3708:
3700:
3697:
3686:
3682:
3676:
3673:
3662:
3658:
3655:AG, Compart.
3651:
3648:
3644:
3638:
3634:
3627:
3624:
3621:
3616:
3612:
3606:
3602:
3601:
3593:
3590:
3586:
3581:
3578:
3573:
3567:
3564:
3560:
3555:
3552:
3547:
3545:9781602066854
3541:
3537:
3536:
3528:
3525:
3520:
3516:
3510:
3506:
3505:
3497:
3494:
3481:
3480:
3475:
3469:
3466:
3463:, p. 347
3462:
3457:
3454:
3441:
3437:
3431:
3428:
3423:
3417:
3413:
3410:
3406:
3400:
3397:
3392:
3391:
3386:
3380:
3377:
3372:
3366:
3359:
3358:
3350:
3347:
3336:
3330:
3326:
3325:
3317:
3314:
3309:
3305:
3301:
3297:
3293:
3289:
3285:
3283:
3274:
3271:
3265:
3262:
3257:
3251:
3247:
3243:
3239:
3232:
3229:
3224:
3223:
3215:
3212:
3208:
3203:
3200:
3195:
3191:
3187:
3183:
3179:
3175:
3171:
3164:
3161:
3158:, pp. 113–117
3157:
3152:
3149:
3144:
3140:
3136:
3130:
3126:
3125:
3117:
3114:
3101:
3097:
3093:
3089:
3085:
3081:
3077:
3073:
3069:
3065:
3061:
3060:
3055:
3048:
3046:
3042:
3026:
3022:
3015:
3008:
3006:
3004:
3002:
3000:
2998:
2994:
2987:
2982:
2979:
2977:
2974:
2972:
2969:
2967:
2966:Infinitesimal
2964:
2962:
2959:
2957:
2954:
2952:
2949:
2947:
2944:
2942:
2939:
2937:
2934:
2932:
2929:
2927:
2924:
2923:
2918:
2916:
2913:
2912:George Lakoff
2910:
2906:
2904:
2900:
2895:
2893:
2889:
2885:
2881:
2874:
2872:
2870:
2866:
2865:infinite loop
2861:
2859:
2855:
2851:
2847:
2842:
2840:
2836:
2832:
2828:
2824:
2820:
2816:
2812:
2808:
2803:
2801:
2797:
2793:
2789:
2781:
2779:
2776:
2772:
2764:
2762:
2760:
2756:
2755:cyclic models
2752:
2748:
2743:
2741:
2735:
2733:
2729:
2725:
2720:
2718:
2714:
2710:
2709:go on forever
2706:
2702:
2698:
2696:
2692:
2684:
2682:
2680:
2676:
2673:are used for
2672:
2668:
2665:are used for
2664:
2660:
2652:
2650:
2648:
2644:
2640:
2636:
2632:
2625:
2623:
2621:
2617:
2616:fractal curve
2612:
2604:
2602:
2600:
2596:
2591:
2589:
2585:
2581:
2577:
2573:
2572:vector spaces
2565:
2563:
2561:
2560:infinite sets
2556:
2554:
2550:
2546:
2542:
2538:
2534:
2529:
2526:
2522:
2518:
2514:
2510:
2506:
2502:
2498:
2493:
2491:
2487:
2483:
2478:
2474:
2470:
2466:
2458:
2456:
2454:
2450:
2446:
2442:
2436:
2431:
2428:
2388:
2384:
2380:
2372:
2367:
2363:
2361:
2357:
2353:
2349:
2345:
2343:
2339:
2333:
2328:
2312:
2308:
2304:
2299:
2291:
2278:
2274:
2268:
2264:
2259:
2242:
2233:
2226:
2217:
2213:
2199:
2193:
2173:
2135:
2127:
2125:
2123:
2122:
2117:
2112:
2108:
2104:
2100:
2096:
2092:
2087:
2085:
2081:
2077:
2073:
2069:
2065:
2060:
2058:
2054:
2053:Gottlob Frege
2050:
2037:
2033:
2029:
2025:
2021:
2012:
2007:
2003:
1995:
1993:
1991:
1987:
1983:
1979:
1975:
1971:
1967:
1963:
1962:infinitesimal
1959:
1955:
1946:
1939:
1937:
1935:
1931:
1907:
1903:
1890:
1881:
1862:
1859:
1855:
1851:
1843:
1839:
1835:
1831:
1827:
1823:
1819:
1796:
1783:
1757:
1726:
1718:
1694:
1686:
1682:
1677:
1670:
1668:
1666:
1662:
1658:
1654:
1650:
1646:
1642:
1638:
1619:
1596:
1588:
1584:
1580:
1561:
1538:
1526:
1507:
1501:
1495:
1485:
1482:
1479:
1475:
1467:
1453:
1450:
1442:
1426:
1423:
1417:
1411:
1401:
1398:
1395:
1391:
1383:
1382:
1381:
1379:
1360:
1357:
1334:
1328:
1308:
1305:
1302:
1299:
1292:
1286:
1273:
1269:
1261:
1243:
1237:
1214:
1211:
1208:
1201:
1195:
1182:
1178:
1170:
1156:
1153:
1133:
1110:
1104:
1081:
1078:
1075:
1068:
1062:
1057:
1052:
1048:
1040:
1039:
1038:
1036:
1023:
1002:
999:
993:
987:
979:
966:
942:
936:
928:
915:
888:
880:
857:, the symbol
856:
855:real analysis
849:Real analysis
848:
846:
844:
840:
836:
829:
827:
825:
821:
816:
810:
789:
785:
759:
751:
748:
744:
742:
735:
733:
731:
730:
725:
720:
718:
702:
693:
682:
681:infinitesimal
678:
654:
646:
641:
638:
635:
634:
633:
631:
627:
619:
603:
600: seconds
595:
592:
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220:
218:
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188:
183:
181:
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173:
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168:combinatorics
165:
161:
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138:
137:infinite sets
134:
130:
126:
122:
118:
114:
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5976:
5890:Power of two
5880:Number names
5729:
5615:Ten thousand
5518:Georg Cantor
5492:Möbius plane
5430:Infinite set
5374:Aleph number
5250:
5202:
5191:
5171:
5145:. Retrieved
5141:the original
5132:
5111:
5105:
5092:
5082:
5068:
5048:
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5022:Rucker, Rudy
5014:
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4911:
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4852:
4835:
4832:D.P. Agrawal
4814:
4787:
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4756:
4740:, Springer,
4737:
4711:
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4669:
4659:
4637:
4618:
4609:Bibliography
4595:. Retrieved
4588:the original
4583:
4570:
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4504:
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4491:
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4434:
4422:. Retrieved
4418:the original
4413:
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4391:. Retrieved
4382:
4372:
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4357:
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4340:
4336:
4324:. Retrieved
4319:
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4300:
4279:
4271:
4262:
4256:
4238:
4226:. Retrieved
4212:
4189:
4182:
4162:
4155:
4150:. p. 88
4139:
4134:
4120:Doric Lenses
4115:
4103:
4091:
4079:
4070:
4057:
4051:. Routledge.
4049:The Infinite
4048:
4042:
4031:. Retrieved
4027:
4018:
3998:
3991:
3973:
3967:
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3916:
3905:. Retrieved
3901:
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3867:
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3684:
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3664:. Retrieved
3660:
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3632:
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3592:
3580:
3566:
3554:
3534:
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3503:
3496:
3484:. Retrieved
3478:
3468:
3461:Russell 1996
3456:
3444:. Retrieved
3439:
3430:
3412:
3408:
3405:
3399:
3389:
3379:
3356:
3349:
3338:. Retrieved
3323:
3316:
3294:(1): 60–62.
3291:
3287:
3281:
3273:
3264:
3237:
3231:
3221:
3214:
3209:, p. 44
3207:Wallace 2004
3202:
3177:
3173:
3163:
3151:
3123:
3116:
3104:. Retrieved
3100:the original
3063:
3057:
3032:. Retrieved
3025:the original
3020:
2961:Infinite set
2931:Aleph number
2907:
2896:
2878:
2862:
2853:
2843:
2804:
2785:
2768:
2744:
2736:
2721:
2701:Cosmologists
2699:
2694:
2688:
2663:real numbers
2656:
2647:intuitionism
2629:
2608:
2592:
2569:
2559:
2557:
2552:
2548:
2544:
2540:
2530:
2494:
2489:
2485:
2476:
2473:line segment
2462:
2438:
2426:
2385:between the
2376:
2346:
2335:
2270:
2197:
2191:
2137:
2119:
2099:well-ordered
2088:
2061:
2032:Georg Cantor
2017:
1982:transfinites
1968:, including
1958:Isaac Newton
1951:
1883:
1776:
1690:
1659:for general
1530:
1375:
1259:is infinite.
1016:
959:
908:
852:
833:
817:
761:
746:
741:Hermann Weyl
739:
727:
724:Isaac Newton
721:
716:
676:
650:
647:17th century
623:
620:Early Indian
497:
493:= 10 seconds
490:
486:
357:
349:
341:
338:
335:
315:
298:Zeno of Elea
296:
281:
276:
252:
248:
243:
236:
233:pre-Socratic
226:
210:
184:
157:
133:Georg Cantor
98:
66:
65:
5983:Integration
5900:Power of 10
5840:Busy beaver
5645:Quintillion
5640:Quadrillion
5379:Beth number
5137:Brady Haran
5133:Numberphile
5083:In Our Time
4424:6 September
4393:6 September
4358:www.gnu.org
3872:Taylor 1955
3661:Compart.com
3585:Cajori 1993
3559:Cajori 1993
3486:October 12,
3219:Aristotle.
3156:Maddox 2002
2892:M.C. Escher
2880:Perspective
2846:overloading
2751:Michio Kaku
2551:instead of
2513:perspective
2121:uncountable
2002:Cardinality
1824:giving the
1695:the symbol
1097:means that
820:John Wallis
805:∞
736:Mathematics
653:John Wallis
259:Hellenistic
229:Anaximander
223:Early Greek
145:cardinality
6171:Categories
6008:stochastic
5905:Sagan Unit
5739:Expression
5690:Googolplex
5655:Septillion
5650:Sextillion
5596:numerical
5480:Geometries
5338:Set theory
5147:2013-04-06
4597:2021-03-25
4560:1510.08155
4414:Learning J
4363:2021-03-15
4108:Kline 1972
4096:Sagan 1994
4084:Cohen 1963
4033:2019-11-16
4024:"Infinity"
3907:2019-11-15
3690:2019-11-15
3666:2019-11-15
3340:2020-01-09
3268:Maor, p. 3
3106:1 November
2988:References
2829:involving
2827:algorithms
2809:, such as
2792:indefinite
2747:multiverse
2679:plane wave
2667:continuous
2533:set theory
2490:the points
2477:infinitely
2433:See also:
2036:aleph-null
1996:Set theory
1661:dimensions
784:lemniscate
630:enumerable
18:Infinitely
6120:Functions
5796:Pentation
5791:Tetration
5779:Operators
5748:Notations
5670:Decillion
5665:Nonillion
5660:Octillion
5592:Examples
5461:Supertask
5199:'Jainism'
5011:'Jainism'
4730:247299160
4710:(1996) ,
4622:, Dover,
4546:1302.4377
3308:0021-1753
3143:659590835
3096:118227996
3080:1063-6145
2976:Supertask
2858:data type
2839:windowing
2835:searching
2782:Computing
2759:Big Bangs
2713:cosmology
2685:Cosmology
2580:dimension
2453:hypercube
2309:ℶ
2296:ℵ
2239:ℵ
2223:ℵ
2170:ℵ
2109:leads to
2103:sequences
1916:∞
1866:∞
1844:, namely
1807:∞
1733:∞
1730:→
1703:∞
1623:∞
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1600:∞
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1476:∑
1441:converges
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1392:∑
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1270:∫
1218:∞
1191:∞
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940:→
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807:) and in
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697:∞
663:∞
596:…
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537:⋯
464:−
449:⋯
245:Aristotle
191:cosmology
129:magnitude
121:Bernoulli
117:l'Hôpital
84:∞
6182:Infinity
6145:Infinity
5998:ordinary
5978:Calculus
5828:articles
5826:Related
5730:Infinity
5635:Trillion
5610:Thousand
5359:0.999...
5251:Infinity
5184:Archived
5164:Archived
5116:Archived
5097:Archived
5078:Infinity
5024:(1995).
4996:Archived
4976:Archived
4951:(1991).
4949:Eli Maor
4895:16578557
4834:(2000).
4786:(2004),
4689:(1972),
4657:(1986),
4584:CHI 2009
4531:Archived
4512:Archived
4468:Archived
4410:"19.2.1"
4387:Archived
4379:"4.2.3."
4326:31 March
4320:Nautilus
4248:Archived
4228:16 March
4222:Archived
4123:Archived
3962:, p. 177
3947:archived
3831:archived
3800:archived
3726:archived
3685:oeis.org
3615:Archived
3519:Archived
3476:(1821).
3446:April 3,
3194:13475845
3088:42413222
2926:0.999...
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2740:toroidal
2717:topology
2705:universe
2675:discrete
2639:finitism
2605:Fractals
2576:geometry
2535:for the
2503:, where
2465:geometry
2459:Geometry
2387:interval
2111:mappings
2107:integers
2024:cardinal
1775:of
1525:diverges
830:Calculus
801:INFINITY
798:∞
593:10.10101
344:Eleatics
317:Achilles
215:and the
149:integers
125:calculus
67:Infinity
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6003:partial
5923:History
5741:methods
5715:SSCG(3)
5710:TREE(3)
5630:Billion
5625:Million
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5265:History
5086:at the
4863:Bibcode
4806:Sources
3943:1669668
3874:, p. 63
3855:(ed.).
3770:1064143
3409:Apeiron
3222:Physics
3034:Nov 15,
2831:sorting
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263:Euclid
217:Greeks
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1932:(see
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