Infinity
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Infinity

Infinity (often denoted by the symbol ) represents something that is boundless or endless or else something that is larger than any real or natural number.[1] Since the time of the ancient Greeks, the nature of infinity was the subject of many discussions among philosophers (see Infinity (philosophy)). In the 17th century, with the introduction of the infinity symbol[2] and the infinitesimal calculus, mathematicians began to work with infinite series and what some mathematicians (including l'Hôpital and Bernoulli[3]) regarded as infinitely small quantities, but infinity continued to be associated with endless processes.[4]As mathematicians struggled with the foundation of the calculus, it remained unclear whether infinity could be considered as a number or magnitude and, if so, how this could be done.[2]

At the end of 19th century, Georg Cantor enlarged the mathematical study of infinity by studying infinite sets and infinite numbers, showing that they can be of various sizes.[2][5] For example, in modern mathematics, a line is commonly viewed as the set of all of its points, and their infinite number (i.e. the cardinality of the line) is larger than the number of integers.[6] In this usage, infinity is a mathematical concept, and infinite mathematical objects can be studied, manipulated, and used just like any other mathematical object.

Thus the mathematical concept of infinity refines and extends the old philosophical concept, in particular by introducing infinitely many different sizes of infinite sets. Among the axioms of Zermelo-Fraenkel set theory, on which most of modern mathematics can be developed, is the axiom of infinity, which guarantees the existence of infinite sets.[2]

The mathematical concept of infinity and the manipulation of infinite sets are used everywhere in mathematics, even in areas such as combinatorics that may seem to have nothing to do with them. For example, Wiles's proof of Fermat's Last Theorem implicitly relies on the existence of very large infinite sets[7] for solving a long-standing problem that is stated in terms of elementary arithmetic.

In physics and cosmology, whether the Universe is finite is an open question; see Universe § Size and regions for more.

History

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

Early Greek

The earliest recorded idea of infinity may be that of Anaximander (c. 610 - c. 546 BC) a pre-Socratic Greek philosopher. He used the word apeiron, which means "unbounded", "indefinite", and perhaps can be translated as "infinite".[2][8]

In accordance with the traditional view of Aristotle, the Hellenistic Greeks generally preferred to distinguish the potential infinity from the actual infinity. It has been argued that they had a "horror of the infinite"[9][10] which would, for example, explain why Euclid (c. 300 BC) did not say that there are an infinity of primes but rather "Prime numbers are more than any assigned multitude of prime numbers."[11] Other scholars, however, disagree that this was characteristic of the perspective of the early Greeks.[12]

Zeno: Achilles and the tortoise

Zeno of Elea (c. 495 - c. 430 BC) did not advance any views concerning the infinite. Nevertheless, his paradoxes,[13] especially "Achilles and the Tortoise", were important contributions in that they made clear the inadequacy of popular conceptions.The paradoxes were described by Bertrand Russell as "immeasurably subtle and profound".[14]

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

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

Etc.

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

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

Finally, in 1821, Augustin-Louis Cauchy provided both a satisfactory definition of a limit and a proof that, for 0 < x < 1,

a + ax + ax2 + ax3 + ax4 + ax5 + · · · = .[15]

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

10 + 0.1 + 0.001 + 0.00001 + · · · =  =  = 10  seconds.

Early Indian

The Jain mathematical text Surya Prajnapti (c. 4th-3rd century BCE) classifies all numbers into three sets: enumerable, innumerable, and infinite. Each of these was further subdivided into three orders:[16]

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

17th century

In the 17th century, European mathematicians started using infinite numbers and infinite expressions in a systematic fashion. In 1655, John Wallis first used the notation for such a number in his De sectionibus conicis,[17] and exploited it in area calculations by dividing the region into infinitesimal strips of width on the order of [18] But in Arithmetica infinitorum (also in 1655), 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."[19]

In 1699, Isaac Newton wrote about equations with an infinite number of terms in his work De analysi per aequationes numero terminorum infinitas.[20]

Mathematics

Hermann Weyl opened a mathematico-philosophic address given in 1930 with:[21]

Mathematics is the science of the infinite.

Infinity symbol

The infinity symbol (sometimes called the lemniscate) is a mathematical symbol representing the concept of infinity. The symbol is encoded in Unicode at INFINITY (HTML &#8734; · &infin;)[22] and in LaTeX as \infty.[23]

It was introduced in 1655 by John Wallis,[24][25] and since its introduction, it has also been used outside mathematics in modern mysticism[26] and literary symbology.[27]

Calculus

Leibniz, one of the co-inventors of infinitesimal calculus, speculated widely about infinite numbers and their use in mathematics. To Leibniz, both infinitesimals and infinite quantities were ideal entities, not of the same nature as appreciable quantities, but enjoying the same properties in accordance with the Law of Continuity.[28][3]

Real analysis

In real analysis, the symbol , called "infinity", is used to denote an unbounded limit.[29] The notation means that  increases without bound, and means that   decreases without bound. For example, if for every , then[30]

  • means that does not bound a finite area from to
  • means that the area under is infinite.
  • means that the total area under is finite, and is equals to

Infinity can also be used to describe infinite series, as follows:

  • means that the sum of the infinite series converges to some real value
  • means that the sum of the infinite series properly diverges to infinity, in the sense that the partial sums increase without bound.[31]

In addition to defining a limit, infinity can be also used as a value in the extended real number system[1]. Points labeled and can be added to the topological space of the real numbers, producing the two-point compactification of the real numbers. Adding algebraic properties to this gives us the extended real numbers.[32] We can also treat and as the same, leading to the one-point compactification of the real numbers, which is the real projective line.[33]Projective geometry also refers to a line at infinity in plane geometry, a plane at infinity in three-dimensional space, and a hyperplane at infinity for general dimensions, each consisting of points at infinity.[34]

Complex analysis

By stereographic projection, the complex plane can be "wrapped" onto a sphere, with the top point of the sphere corresponding to infinity. This is called the Riemann sphere.

In complex analysis the symbol , called "infinity", denotes an unsigned infinite limit. means that the magnitude  of  grows beyond any assigned value. A point labeled can be added to the complex plane as a topological space giving the one-point compactification of the complex plane.[35] When this is done, the resulting space is a one-dimensional complex manifold, or Riemann surface, called the extended complex plane or the Riemann sphere. Arithmetic operations similar to those given above for the extended real numbers can also be defined, though there is no distinction in the signs (which leads to the one exception that infinity cannot be added to itself). On the other hand, this kind of infinity enables division by zero, namely for any nonzero complex number . In this context, it is often useful to consider meromorphic functions as maps into the Riemann sphere taking the value of at the poles. The domain of a complex-valued function may be extended to include the point at infinity as well. One important example of such functions is the group of Möbius transformations (see Möbius transformation § Overview).

Nonstandard analysis

Infinitesimals (?) and infinites (?) on the hyperreal number line (1/? = ?/1)

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

Set theory

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

A different form of "infinity" are the ordinal and cardinal infinities of set theory--a system of transfinite numbers first developed by Georg Cantor. In this system, the first transfinite cardinal is aleph-null (?0), the cardinality of the set of natural numbers. This modern mathematical conception of the quantitative infinite developed in the late 19th century from works by Cantor, Gottlob Frege, Richard Dedekind and others--using the idea of collections or sets.[2]

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

Cantor defined two kinds of infinite numbers: ordinal numbers and cardinal numbers. Ordinal numbers characterize well-ordered sets, or counting carried on to any stopping point, including points after an infinite number have already been counted. Generalizing finite and (ordinary) infinite sequences which are maps from the positive integers leads to mappings from ordinal numbers to transfinite sequences. Cardinal numbers define the size of sets, meaning how many members they contain, and can be standardized by choosing the first ordinal number of a certain size to represent the cardinal number of that size. The smallest ordinal infinity is that of the positive integers, and any set which has the cardinality of the integers is countably infinite. If a set is too large to be put in one-to-one correspondence with the positive integers, it is called uncountable. Cantor's views prevailed and modern mathematics accepts actual infinity as part of a consistent and coherent theory.[36][37][page needed] Certain extended number systems, such as the hyperreal numbers, incorporate the ordinary (finite) numbers and infinite numbers of different sizes.[]

Cardinality of the continuum

One of Cantor's most important results was that the cardinality of the continuum is greater than that of the natural numbers ; that is, there are more real numbers R than natural numbers N. Namely, Cantor showed that (see Cantor's diagonal argument or Cantor's first uncountability proof).[38]

The continuum hypothesis states that there is no cardinal number between the cardinality of the reals and the cardinality of the natural numbers, that is, (see Beth one). This hypothesis can neither be proved nor disproved within the widely accepted Zermelo-Fraenkel set theory, even assuming the Axiom of Choice.[39]

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

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

The first of these results is apparent by considering, for instance, the tangent function, which provides a one-to-one correspondence between the interval (-?/2, ?/2) and R (see also Hilbert's paradox of the Grand Hotel). The second result was proved by Cantor in 1878, but only became intuitively apparent in 1890, when Giuseppe Peano introduced the space-filling curves, curved lines that twist and turn enough to fill the whole of any square, or cube, or hypercube, or finite-dimensional space. These curves can be used to define a one-to-one correspondence between the points on one side of a square and the points in the square.[40]

Geometry and topology

Infinite-dimensional spaces are widely used in geometry and topology, particularly as classifying spaces, such as Eilenberg-MacLane spaces. Common examples are the infinite-dimensional complex projective space K(Z,2) and the infinite-dimensional real projective space K(Z/2Z,1).[]

Fractals

The structure of a fractal object is reiterated in its magnifications. Fractals can be magnified indefinitely without losing their structure and becoming "smooth"; they have infinite perimeters--some with infinite, and others with finite surface areas. One such fractal curve with an infinite perimeter and finite surface area is the Koch snowflake.[]

Mathematics without infinity

Leopold Kronecker was skeptical of the notion of infinity and how his fellow mathematicians were using it in the 1870s and 1880s. This skepticism was developed in the philosophy of mathematics called finitism, an extreme form of mathematical philosophy in the general philosophical and mathematical schools of constructivism and intuitionism.[41]

Physics

In physics, approximations of real numbers are used for continuous measurements and natural numbers are used for discrete measurements (i.e. counting). It is therefore assumed by physicists that no measurable quantity could have an infinite value,[] for instance by taking an infinite value in an extended real number system, or by requiring the counting of an infinite number of events. It is, for example, presumed impossible for any type of body to have infinite mass or infinite energy. Concepts of infinite things such as an infinite plane wave exist, but there are no experimental means to generate them.[42]

Cosmology

The first published proposal that the universe is infinite came from Thomas Digges in 1576.[43] Eight years later, in 1584, the Italian philosopher and astronomer Giordano Bruno proposed an unbounded universe in On the Infinite Universe and Worlds: "Innumerable suns exist; innumerable earths revolve around these suns in a manner similar to the way the seven planets revolve around our sun. Living beings inhabit these worlds."[44]

Cosmologists have long sought to discover whether infinity exists in our physical universe: Are there an infinite number of stars? Does the universe have infinite volume? Does space "go on forever"? This is an open question of cosmology. The question of being infinite is logically separate from the question of having boundaries. The two-dimensional surface of the Earth, for example, is finite, yet has no edge. By travelling in a straight line with respect to the Earth's curvature one will eventually return to the exact spot one started from. The universe, at least in principle, might have a similar topology. If so, one might eventually return to one's starting point after travelling in a straight line through the universe for long enough.[45]

The curvature of the universe can be measured through multipole moments in the spectrum of the cosmic background radiation. As to date, analysis of the radiation patterns recorded by the WMAP spacecraft hints that the universe has a flat topology. This would be consistent with an infinite physical universe.[46][47][48]

However, the universe could be finite, even if its curvature is flat. An easy way to understand this is to consider two-dimensional examples, such as video games where items that leave one edge of the screen reappear on the other. The topology of such games is toroidal and the geometry is flat. Many possible bounded, flat possibilities also exist for three-dimensional space.[49]

The concept of infinity also extends to the multiverse hypothesis, which, when explained by astrophysicists such as Michio Kaku, posits that there are an infinite number and variety of universes.[50]

Logic

In logic an infinite regress argument is "a distinctively philosophical kind of argument purporting to show that a thesis is defective because it generates an infinite series when either (form A) no such series exists or (form B) were it to exist, the thesis would lack the role (e.g., of justification) that it is supposed to play."[51]

Computing

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

Some programming languages, such as Java[52] and J,[53] allow the programmer an explicit access to the positive and negative infinity values as language constants. These can be used as greatest and least elements, as they compare (respectively) greater than or less than all other values. They have uses as sentinel values in algorithms involving sorting, searching, or windowing.[]

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

In programming, an infinite loop is a loop whose exit condition is never satisfied, thus theoretically executing indefinitely.

Infinite sequences can be represented in the finite memory of a computer as a composite data structure consisting of a few first members of the sequence, and a recursive routine for computing the nth element from the preceding ones. Several techniques can be used for avoiding computing several times the same element of the sequence. One is lazy evaluation. Another one, available in Maple, consists of having routines with a remember option. This option consists of keeping in memory the results of the function that have been computed, and, at each call of the routine, looking if this particular result has been computed, for avoiding to compute it again. For example, the standard definition of the Fibonacci sequence is

Fib(n) == if n = 1 or n = 2 then 1 else Fib(n - 1) + Fib(n - 2).

With a standard implementation, an exponential number of function calls is needed for computing Fib(n), while, with the remember option, only function calls are needed.

Arts, games, and cognitive sciences

Perspective artwork utilizes the concept of vanishing points, roughly corresponding to mathematical points at infinity, located at an infinite distance from the observer. This allows artists to create paintings that realistically render space, distances, and forms.[54] Artist M.C. Escher is specifically known for employing the concept of infinity in his work in this and other ways.[]

Variations of chess played on an unbounded board are called infinite chess.[55][56]

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

The symbol is often used romantically to represent eternal love. Several types of jewelry are fashioned into the infinity shape for this purpose.[]

See also

Notes

  1. ^ a b "The Definitive Glossary of Higher Mathematical Jargon -- Infinite". Math Vault. 2019-08-01. Retrieved .
  2. ^ a b c d e f Allen, Donale (2003). "The History of Infinity" (PDF). Texas A&M Mathematics. Retrieved .
  3. ^ a b Jesseph, Douglas Michael (1998). "Leibniz on the Foundations of the Calculus: The Question of the Reality of Infinitesimal Magnitudes". Perspectives on Science. 6 (1&2): 6-40. ISSN 1063-6145. OCLC 42413222. Archived from the original on 15 February 2010. Retrieved 2019.
  4. ^ The ontological status of infinitesimals was unclear, but only some mathematicians regarded infinitesimal as a quantity that is smaller (in magnitude) than any positive number. Others viewed it either as an artefact that makes computation easier or as a small quantity that can be made smaller and smaller until the quantity in which it is involved reaches eventually a limit.[]
  5. ^ Gowers, Timothy; Barrow-Green, June; Leader, Imre (2008). The Princeton Companion to Mathematics. Princeton University Press. p. 616. ISBN 978-0-691-11880-2. Archived from the original on 2016-06-03.Extract of page 616 Archived 2016-05-01 at the Wayback Machine
  6. ^ Maddox 2002, pp. 113-117
  7. ^ McLarty, Colin (2010). "What does it take to prove Fermat's Last Theorem? Grothendieck and the logic of number theory". The Bulletin of Symbolic Logic. 16 (3): 359-377. doi:10.2178/bsl/1286284558.
  8. ^ Wallace 2004, p. 44
  9. ^ Nicolas D. Goodman (1981). Richman, F. (ed.). "Reflections on Bishop's philosophy of mathematics". Constructive Mathematics. Lecture Notes in Mathematics. Springer. v. 873.
  10. ^ Maor, p. 3
  11. ^ Heath, Sir Thomas Little; Heiberg, Johan Ludvig (1908). The Thirteen Books of Euclid's Elements. v. 2. The University Press. p. 412. In Euclid's original Elements this is Proposition 20 in Book IX.
  12. ^ Carter, Jason W. (Spring 2012). "Review of Alan Drozdek, In the Beginning Was the Apeiron: Infinity in Greek Philosophy (2008)". Ancient Philosophy. v. 32 (1): 167. doi:10.5840/ancientphil20123219.
  13. ^ "Zeno's Paradoxes". Stanford University. October 15, 2010. Retrieved 2017.
  14. ^ Russell 1996, p. 347
  15. ^ Cauchy, Augustin-Louis (1821). Cours d'Analyse de l'École Royale Polytechnique. Libraires du Roi & de la Bibliothèque du Roi. p. 124. Retrieved 2019.
  16. ^ Ian Stewart (2017). Infinity: a Very Short Introduction. Oxford University Press. p. 117. ISBN 978-0-19-875523-4. Archived from the original on April 3, 2017.
  17. ^ Cajori, Florian (2007). A History of Mathematical Notations. 1. Cosimo, Inc. p. 214. ISBN 9781602066854.CS1 maint: date and year (link)
  18. ^ Cajori 1993, Sec. 421, Vol. II, p. 44
  19. ^ Cajori 1993, Sec. 435, Vol. II, p. 58
  20. ^ Grattan-Guinness, Ivor (2005). Landmark Writings in Western Mathematics 1640-1940. Elsevier. p. 62. ISBN 978-0-08-045744-4. Archived from the original on 2016-06-03.Extract of p. 62
  21. ^ Weyl, Hermann (2012), Peter Pesic (ed.), Levels of Infinity / Selected Writings on Mathematics and Philosophy, Dover, p. 17, ISBN 978-0-486-48903-2
  22. ^ AG, Compart. "Unicode Character "?" (U+221E)". Compart.com. Retrieved .
  23. ^ "List of LaTeX mathematical symbols - OeisWiki". oeis.org. Retrieved .
  24. ^ Scott, Joseph Frederick (1981), The mathematical work of John Wallis, D.D., F.R.S., (1616-1703) (2 ed.), American Mathematical Society, p. 24, ISBN 978-0-8284-0314-6, archived from the original on 2016-05-09
  25. ^ Martin-Löf, Per (1990), "Mathematics of infinity", COLOG-88 (Tallinn, 1988), Lecture Notes in Computer Science, 417, Berlin: Springer, pp. 146-197, doi:10.1007/3-540-52335-9_54, ISBN 978-3-540-52335-2, MR 1064143
  26. ^ O'Flaherty, Wendy Doniger (1986), Dreams, Illusion, and Other Realities, University of Chicago Press, p. 243, ISBN 978-0-226-61855-5, archived from the original on 2016-06-29
  27. ^ Toker, Leona (1989), Nabokov: The Mystery of Literary Structures, Cornell University Press, p. 159, ISBN 978-0-8014-2211-9, archived from the original on 2016-05-09
  28. ^ Bell, John Lane. "Continuity and Infinitesimals". In Zalta, Edward N. (ed.). Stanford Encyclopedia of Philosophy.
  29. ^ Taylor 1955, p. 63
  30. ^ These uses of infinity for integrals and series can be found in any standard calculus text, such as, Swokowski 1983, pp. 468-510
  31. ^ "Properly Divergent Sequences - Mathonline". mathonline.wikidot.com. Retrieved .
  32. ^ Aliprantis, Charalambos D.; Burkinshaw, Owen (1998), Principles of Real Analysis (3rd ed.), San Diego, CA: Academic Press, Inc., p. 29, ISBN 978-0-12-050257-8, MR 1669668, archived from the original on 2015-05-15
  33. ^ Gemignani 1990, p. 177
  34. ^ Beutelspacher, Albrecht; Rosenbaum, Ute (1998), Projective Geometry / from foundations to applications, Cambridge University Press, p. 27, ISBN 978-0-521-48364-3
  35. ^ Weisstein, Eric W. "Extended Complex Plane". mathworld.wolfram.com. Retrieved .
  36. ^ "Infinity". math.dartmouth.edu. Retrieved .
  37. ^ Moore, A.W. (1991). The Infinite. Routledge.
  38. ^ Dauben, Joseph (1993). "Georg Cantor and the Battle for Transfinite Set Theory" (PDF). 9th ACMS Conference Proceedings: 4.
  39. ^ Cohen 1963, p. 1143
  40. ^ Sagan 1994, pp. 10-12
  41. ^ Kline 1972, pp. 1197-1198
  42. ^ Doric Lenses Archived 2013-01-24 at the Wayback Machine - Application Note - Axicons - 2. Intensity Distribution. Retrieved 7 April 2014.
  43. ^ John Gribbin (2009), In Search of the Multiverse: Parallel Worlds, Hidden Dimensions, and the Ultimate Quest for the Frontiers of Reality, ISBN 978-0-470-61352-8. p. 88
  44. ^ Brake, Mark (2013). Alien Life Imagined: Communicating the Science and Culture of Astrobiology. Physics Today. 67 (illustrated ed.). Cambridge University Press. p. 63. Bibcode:2014PhT....67f..49S. doi:10.1063/PT.3.2420. ISBN 978-0-521-49129-7.Extract of p. 63
  45. ^ Koupelis, Theo; Kuhn, Karl F. (2007). In Quest of the Universe (illustrated ed.). Jones & Bartlett Learning. p. 553. ISBN 978-0-7637-4387-1.Extract of p. 553
  46. ^ "Will the Universe expand forever?". NASA. 24 January 2014. Archived from the original on 1 June 2012. Retrieved 2015.
  47. ^ "Our universe is Flat". FermiLab/SLAC. 7 April 2015. Archived from the original on 10 April 2015.
  48. ^ Marcus Y. Yoo (2011). "Unexpected connections". Engineering & Science. LXXIV1: 30.
  49. ^ Weeks, Jeffrey (2001). The Shape of Space. CRC Press. ISBN 978-0-8247-0709-5.
  50. ^ Kaku, M. (2006). Parallel worlds. Knopf Doubleday Publishing Group.
  51. ^ Cambridge Dictionary of Philosophy, Second Edition, p. 429
  52. ^ Gosling, James; et al. (27 July 2012). "4.2.3.". The Java(TM) Language Specification (Java SE 7 ed.). California: Oracle America, Inc. Archived from the original on 9 June 2012. Retrieved 2012.
  53. ^ Stokes, Roger (July 2012). "19.2.1". Learning J. Archived from the original on 25 March 2012. Retrieved 2012.
  54. ^ Kline, Morris (1985). Mathematics for the nonmathematician. Courier Dover Publications. p. 229. ISBN 978-0-486-24823-3. Archived from the original on 2016-05-16., Section 10-7, p. 229 Archived 2016-05-16 at the Wayback Machine
  55. ^ Infinite chess at the Chess Variant Pages Archived 2017-04-02 at the Wayback Machine An infinite chess scheme.
  56. ^ "Infinite Chess, PBS Infinite Series" Archived 2017-04-07 at the Wayback Machine PBS Infinite Series,with academic sources by J. Hamkins (infinite chess: Evans, C.D.A; Joel David Hamkins (2013). "Transfinite game values in infinite chess". arXiv:1302.4377 [math.LO]. and Evans, C.D.A; Joel David Hamkins; Norman Lewis Perlmutter (2015). "A position in infinite chess with game value $?^4$". arXiv:1510.08155 [math.LO].).

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