# Infinity

(Redirected from Infinitely)

Infinity (symbol: ) is a concept describing something without any bound or larger than any natural number. Philosophers have speculated about the nature of the infinite, for example Zeno of Elea, who proposed many paradoxes involving infinity, and Eudoxus of Cnidus, who used the idea of infinitely small quantities in his method of exhaustion. Modern mathematics uses the general concept of infinity in the solution of many practical and theoretical problems, such as in calculus and set theory, and the idea is also used in physics and the other sciences.

In mathematics, "infinity" is often treated as a number (i.e., it counts or measures things: "an infinite number of terms") but it is not the same sort of number as either a natural or a real number.

Georg Cantor formalized many ideas related to infinity and infinite sets during the late 19th and early 20th centuries. In the theory he developed, there are infinite sets of different sizes (called cardinalities).[1] For example, the set of integers is countably infinite, while the infinite set of real numbers is uncountable.[2]

## 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 comes from Anaximander, a pre-Socratic Greek philosopher who lived in Miletus. He used the word apeiron which means infinite or limitless.[3] However, the earliest attestable accounts of mathematical infinity come from Zeno of Elea (born c. 490 BCE), a pre-Socratic Greek philosopher of southern Italy and member of the Eleatic School founded by Parmenides. Aristotle called him the inventor of the dialectic.[4][5] He is best known for his paradoxes,[4] described by Bertrand Russell as "immeasurably subtle and profound".[6]

In accordance with the traditional view of Aristotle, the Hellenistic Greeks generally preferred to distinguish the potential infinity from the actual infinity; for example, instead of saying that there are an infinity of primes, Euclid prefers instead to say that there are more prime numbers than contained in any given collection of prime numbers.[7]

However, recent readings of the Archimedes Palimpsest have found that Archimedes had an understanding about actual infinite quantities. According to Nonlinear Dynamic Systems and Controls, Archimedes has been found to be "the first to rigorously address the science of infinity with infinitely large sets using precise mathematical proofs."[8]

### 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:[9]

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

In this work, two basic types of infinite numbers are distinguished. On both physical and ontological grounds, a distinction was made between asaṃkhyāta ("countless, innumerable") and ananta ("endless, unlimited"), between rigidly bounded and loosely bounded infinities.[10]

### 17th century

European mathematicians started using infinite numbers and expressions in a systematic fashion in the 17th century. In 1655 John Wallis first used the notation ${\displaystyle \infty }$ for such a number in his De sectionibus conicis and exploited it in area calculations by dividing the region into infinitesimal strips of width on the order of ${\displaystyle {\tfrac {1}{\infty }}.}$[11] But in Arithmetica infinitorum (1655 also) he indicates infinite series, infinite products and infinite continued fractions by writing down a few terms or factors and then appending "&c." For example, "1, 6, 12, 18, 24, &c."[12]

In 1699 Isaac Newton wrote about equations with an infinite number of terms in his work De analysi per aequationes numero terminorum infinitas.[13]

## Mathematics

Hermann Weyl opened a mathematico-philosophic address given in 1930 with:[14]

Mathematics is the science of the infinite.

### Infinity symbol

The infinity symbol ${\displaystyle \infty }$ (sometimes called the lemniscate) is a mathematical symbol representing the concept of infinity. The symbol is encoded in Unicode at U+221E infinity (HTML &#8734; · &infin;) and in LaTeX as \infty.

It was introduced in 1655 by John Wallis,[15][16] and, since its introduction, has also been used outside mathematics in modern mysticism[17] and literary symbology.[18]

### 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.[19][20]

#### Real analysis

In real analysis, the symbol ${\displaystyle \infty }$, called "infinity", is used to denote an unbounded limit.[21] The notation ${\displaystyle x\rightarrow \infty }$ means that x grows without bound, and ${\displaystyle x\to -\infty }$ means that  x decreases without bound. If f(t) ≥ 0 for every t, then[22]

• ${\displaystyle \int _{a}^{b}f(t)\,dt=\infty }$ means that f(t) does not bound a finite area from ${\displaystyle a}$ to ${\displaystyle b}$
• ${\displaystyle \int _{-\infty }^{\infty }f(t)\,dt=\infty }$ means that the area under f(t) is infinite.
• ${\displaystyle \int _{-\infty }^{\infty }f(t)\,dt=a}$ means that the total area under f(t) is finite, and equals ${\displaystyle a}$

Infinity is also used to describe infinite series:

• ${\displaystyle \sum _{i=0}^{\infty }f(i)=a}$ means that the sum of the infinite series converges to some real value ${\displaystyle a}$.
• ${\displaystyle \sum _{i=0}^{\infty }f(i)=\infty }$ means that the sum of the infinite series diverges in the specific sense that the partial sums grow without bound.[citation needed]

Infinity can be used not only to define a limit but as a value in the extended real number system. Points labeled ${\displaystyle +\infty }$ and ${\displaystyle -\infty }$ 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.[23] We can also treat ${\displaystyle +\infty }$ and ${\displaystyle -\infty }$ as the same, leading to the one-point compactification of the real numbers, which is the real projective line.[24] 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.[25]

#### 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 ${\displaystyle \infty }$, called "infinity", denotes an unsigned infinite limit. ${\displaystyle x\rightarrow \infty }$ means that the magnitude ${\displaystyle |x|}$ of x grows beyond any assigned value. A point labeled ${\displaystyle \infty }$ can be added to the complex plane as a topological space giving the one-point compactification of the complex plane. 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 (therefore one exception is that infinity cannot be added to itself). On the other hand, this kind of infinity enables division by zero, namely ${\displaystyle z/0=\infty }$ for any nonzero complex number z. In this context it is often useful to consider meromorphic functions as maps into the Riemann sphere taking the value of ${\displaystyle \infty }$ at the poles. The domain of a complex-valued function may be extended to include the point at infinity as well. One important example of such functions is the group of Möbius transformations.[citation needed]

### Nonstandard analysis

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

The original formulation of infinitesimal calculus by Isaac Newton and Gottfried Leibniz used infinitesimal quantities. In the twentieth 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, 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 infinite set and proper subset

A different form of "infinity" are the ordinal and cardinal infinities of set theory. Georg Cantor developed a system of transfinite numbers, in which 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 nineteenth century from work by Cantor, Gottlob Frege, Richard Dedekind and others, using the idea of collections, or sets.[citation needed]

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 (which derived from Euclid) that the whole cannot be the same size as the part (however, see Galileo's paradox where he concludes that positive integers which are squares and all positive integers are the same size). 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 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".[citation needed]

Cantor defined two kinds of infinite numbers: ordinal numbers and cardinal numbers. Ordinal numbers may be identified with well-ordered sets, or counting carried on to any stopping point, including points after an infinite number have already been counted. Generalizing finite and the ordinary infinite sequences which are maps from the positive integers leads to mappings from ordinal numbers, and 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.[26][page needed] Certain extended number systems, such as the hyperreal numbers, incorporate the ordinary (finite) numbers and infinite numbers of different sizes.[citation needed]

#### Cardinality of the continuum

One of Cantor's most important results was that the cardinality of the continuum ${\displaystyle \mathbf {c} }$ is greater than that of the natural numbers ${\displaystyle {\aleph _{0}}}$; that is, there are more real numbers R than natural numbers N. Namely, Cantor showed that ${\displaystyle \mathbf {c} =2^{\aleph _{0}}>{\aleph _{0}}}$ (see Cantor's diagonal argument or Cantor's first uncountability proof).[27]

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, ${\displaystyle \mathbf {c} =\aleph _{1}=\beth _{1}}$ (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.[28]

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 that this is equal to the number of points on a plane and, indeed, in any finite-dimensional space.[citation needed]

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 (−π/2, π/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.[29]

### 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).[citation needed]

### 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.[citation needed]

### 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.[30]

## 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,[citation needed] 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.[31]

### Theoretical applications of physical infinity

The practice of refusing infinite values for measurable quantities does not come from a priori or ideological motivations, but rather from more methodological and pragmatic motivations[disputed ][citation needed]. One of the needs of any physical and scientific theory is to give usable formulas that correspond to or at least approximate reality. As an example, if any object of infinite gravitational mass were to exist, any usage of the formula to calculate the gravitational force would lead to an infinite result, which would be of no benefit since the result would be always the same regardless of the position and the mass of the other object. The formula would be useful neither to compute the force between two objects of finite mass nor to compute their motions. If an infinite mass object were to exist, any object of finite mass would be attracted with infinite force (and hence acceleration) by the infinite mass object, which is not what we can observe in reality. Sometimes infinite result of a physical quantity may mean that the theory being used to compute the result may be approaching the point where it fails. This may help to indicate the limitations of a theory.[citation needed]

This point of view does not mean that infinity cannot be used in physics. For convenience's sake, calculations, equations, theories and approximations often use infinite series, unbounded functions, etc., and may involve infinite quantities. Physicists however require that the end result be physically meaningful. In quantum field theory infinities arise which need to be interpreted in such a way as to lead to a physically meaningful result, a process called renormalization.[citation needed]

However, there are some theoretical circumstances where the end result is infinity. One example is the singularity in the description of black holes. Some solutions of the equations of the general theory of relativity allow for finite mass distributions of zero size, and thus infinite density. This is an example of what is called a mathematical singularity, or a point where a physical theory breaks down. This does not necessarily mean that physical infinities exist; it may mean simply that the theory is incapable of describing the situation properly. Two other examples occur in inverse-square force laws of the gravitational force equation of Newtonian gravity and Coulomb's law of electrostatics. At r=0 these equations evaluate to infinities.[citation needed]

### Cosmology

The first published proposal that the universe is infinite came from Thomas Digges in 1576.[32] 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."[33]

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.[34]

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.[35][36][37]

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.[38]

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.[39]

## 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."[40]

## 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.[citation needed]

Some programming languages, such as Java[41] and J,[42] 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.[citation needed]

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.[citation 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.[43] Artist M. C. Escher is specifically known for employing the concept of infinity in his work in this and other ways.[citation needed]

Variations of chess played on an unbounded board are called infinite chess.[44][45]

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,...>.[citation needed]

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

## Notes

1. ^ Gowers, Timothy; Barrow-Green, June; Leader, Imre (2008). The Princeton Companion to Mathematics. Princeton University Press. p. 616. ISBN 0-691-11880-9. Archived from the original on 2016-06-03. Extract of page 616 Archived 2016-05-01 at the Wayback Machine.
2. ^ Maddox 2002, pp. 113 –117
3. ^ Wallace 2004, p. 44
4. ^ a b "Zeno's Paradoxes". Stanford University. October 15, 2010. Retrieved April 3, 2017.
5. ^ "Zeno of Elea". Stanford University. January 5, 2017. Retrieved April 3, 2017.
6. ^ Russell 1996, p. 347
7. ^ Euclid. Euclid's Elements, Book IX, Proposition 20.
8. ^ Wassim M. Haddad; VijaySekhar Chellaboina (February 17, 2008). Nonlinear Dynamical Systems and Control: A Lyapunov-Based Approach. Princeton University Press. p. xxv. ISBN 0-691-13329-8. Archived from the original on April 4, 2017.
9. ^ Ian Stewart (March 23, 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.
10. ^ Dutta, Bidyarthi (December 2015). "Ranganathan's elucidation of subject in the light of 'Infinity (∞)'". Annals of Library and Information Studies. 62: 255–264. Retrieved 16 May 2017.
11. ^ Cajori 1993, Sec. 421, Vol. II, pg. 44
12. ^ Cajori 1993, Sec. 435, Vol. II, pg. 58
13. ^ 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 page 62
14. ^ Weyl, Hermann (2012), Peter Pesic, ed., Levels of Infinity / Selected Writings on Mathematics and Philosophy, Dover, p. 17, ISBN 978-0-486-48903-2
15. ^ 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 0-8284-0314-7, archived from the original on 2016-05-09
16. ^ 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
17. ^ O'Flaherty, Wendy Doniger (1986), Dreams, Illusion, and Other Realities, University of Chicago Press, p. 243, ISBN 9780226618555, archived from the original on 2016-06-29
18. ^ Toker, Leona (1989), Nabokov: The Mystery of Literary Structures, Cornell University Press, p. 159, ISBN 9780801422119, archived from the original on 2016-05-09
19. ^
20. ^ 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 16 February 2010.
21. ^ Taylor 1955, p. 63
22. ^ These uses of infinity for integrals and series can be found in any standard calculus text, such as, Swokowski 1983, pp. 468–510
23. ^ Aliprantis, Charalambos D.; Burkinshaw, Owen (1998), Principles of Real Analysis (3rd ed.), San Diego, CA: Academic Press, Inc., p. 29, ISBN 0-12-050257-7, MR 1669668, archived from the original on 2015-05-15
24. ^ Gemignani 1990, p. 177
25. ^ Beutelspacher, Albrecht; Rosenbaum, Ute (1998), Projective Geometry / from foundations to applications, Cambridge University Press, p. 27, ISBN 978-0-521-48364-3
26. ^ Moore, A. W. (1991). The Infinite. Routledge.
27. ^ Dauben, Joseph (1993). "Georg Cantor and the Battle for Transfinite Set Theory" (PDF). 9th ACMS Conference Proceedings: 4.
28. ^ Cohen 1963, p. 1143
29. ^ Sagan 1994, pp. 10–12
30. ^ Kline 1972, pp. 1197–1198
31. ^ Doric Lenses Archived 2013-01-24 at the Wayback Machine. – Application Note – Axicons – 2. Intensity Distribution. Retrieved 7 April 2014.
32. ^ John Gribbin (2009), In Search of the Multiverse: Parallel Worlds, Hidden Dimensions, and the Ultimate Quest for the Frontiers of Reality, ISBN 9780470613528. p. 88
33. ^ Brake, Mark (2013). Alien Life Imagined: Communicating the Science and Culture of Astrobiology (illustrated ed.). Cambridge University Press. p. 63. ISBN 978-0-521-49129-7. Extract of page 63
34. ^ 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 page 553
35. ^ "Will the Universe expand forever?". NASA. 24 January 2014. Archived from the original on 1 June 2012. Retrieved 16 March 2015.
36. ^ "Our universe is Flat". FermiLab/SLAC. 7 April 2015. Archived from the original on 10 April 2015.
37. ^ Marcus Y. Yoo (2011). "Unexpected connections". Engineering & Science. Caltech. LXXIV1: 30.
38. ^ Weeks, Jeffrey (December 12, 2001). The Shape of Space. CRC Press. ISBN 978-0824707095.
39. ^ Kaku, M. (2006). Parallel worlds. Knopf Doubleday Publishing Group.
40. ^ Cambridge Dictionary of Philosophy, Second Edition, p. 429
41. ^ Gosling, James; et. al. (27 July 2012). "4.2.3.". The Java™ Language Specification (Java SE 7 ed.). California, U.S.A.: Oracle America, Inc. Archived from the original on 9 June 2012. Retrieved 6 September 2012.
42. ^ Stokes, Roger (July 2012). "19.2.1". Learning J. Archived from the original on 25 March 2012. Retrieved 6 September 2012.
43. ^ Kline, Morris (1985). Mathematics for the nonmathematician. Courier Dover Publications. p. 229. ISBN 0-486-24823-2. Archived from the original on 2016-05-16., Section 10-7, p. 229 Archived 2016-05-16 at the Wayback Machine.
44. ^ Infinite chess at the Chess Variant Pages Archived 2017-04-02 at the Wayback Machine. An infinite chess scheme.
45. ^ "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: [math.LO]. and Evans, C. D. A; Joel David Hamkins; Norman Lewis Perlmutter (2015). "A position in infinite chess with game value $ω^4$". arXiv: [math.LO].).