Infinity

Infinity (symbol: ) is a concept describing something without any bound, or something 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]

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:[8]

  • 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.[9]

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 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 [10] 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."[11]

In 1699 Isaac Newton wrote about equations with an infinite number of terms in his work De analysi per aequationes numero terminorum infinitas.[12]

Mathematics

Hermann Weyl opened a mathematico-philosophic address given in 1930 with:[13]

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 U+221E INFINITY (HTML ∞ · ∞) and in LaTeX as \infty.

It was introduced in 1655 by John Wallis,[14][15] and, since its introduction, has also been used outside mathematics in modern mysticism[16] and literary symbology.[17]

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.[18][19]

Real analysis

In real analysis, the symbol , called "infinity", is used to denote an unbounded limit.[20] The notation means that x grows without bound, and means that  x decreases without bound. If f(t) ≥ 0 for every t, then[21]

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

Infinity is also used to describe infinite series:

  • means that the sum of the infinite series converges to some real value
  • means that the sum of the infinite series diverges in the specific sense that the partial sums grow without bound.

Infinity can be used not only to define a limit but as a value in the extended real number system. 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.[22] We can also treat and as the same, leading to the one-point compactification of the real numbers, which is the real projective line.[23] 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.[24]

Complex analysis

Riemann sphere1
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 x 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. 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 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 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.

Nonstandard analysis

Números hiperreales
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

Infinity paradoxon - one-to-one correspondence between infinite set and proper subset
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.

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".

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.[25] 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).[26]

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.[27]

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.

Peanocurve
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.[28]

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.[29]

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.[30]

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. 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.

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.

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.

Cosmology

The first published proposal that the universe is infinite came from Thomas Digges in 1576.[31] 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."[32]

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.[33]

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.[34][35][36]

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.[37]

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.[38]

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."[39]

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[40] and J,[41] 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.

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.[42] 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.[43][44]

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. ^ 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
  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. ^ 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.
  9. ^ 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.
  10. ^ Cajori 1993, Sec. 421, Vol. II, p. 44
  11. ^ Cajori 1993, Sec. 435, Vol. II, p. 58
  12. ^ 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
  13. ^ Weyl, Hermann (2012), Peter Pesic, ed., Levels of Infinity / Selected Writings on Mathematics and Philosophy, Dover, p. 17, ISBN 978-0-486-48903-2
  14. ^ 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
  15. ^ 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
  16. ^ 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
  17. ^ 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
  18. ^ Bell, John Lane. "Continuity and Infinitesimals". In Zalta, Edward N. Stanford Encyclopedia of Philosophy.
  19. ^ 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.
  20. ^ Taylor 1955, p. 63
  21. ^ These uses of infinity for integrals and series can be found in any standard calculus text, such as, Swokowski 1983, pp. 468–510
  22. ^ 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
  23. ^ Gemignani 1990, p. 177
  24. ^ Beutelspacher, Albrecht; Rosenbaum, Ute (1998), Projective Geometry / from foundations to applications, Cambridge University Press, p. 27, ISBN 978-0-521-48364-3
  25. ^ Moore, A.W. (1991). The Infinite. Routledge.
  26. ^ Dauben, Joseph (1993). "Georg Cantor and the Battle for Transfinite Set Theory" (PDF). 9th ACMS Conference Proceedings: 4.
  27. ^ Cohen 1963, p. 1143
  28. ^ Sagan 1994, pp. 10–12
  29. ^ Kline 1972, pp. 1197–1198
  30. ^ Doric Lenses Archived 2013-01-24 at the Wayback Machine – Application Note – Axicons – 2. Intensity Distribution. Retrieved 7 April 2014.
  31. ^ 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
  32. ^ 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
  33. ^ 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
  34. ^ "Will the Universe expand forever?". NASA. 24 January 2014. Archived from the original on 1 June 2012. Retrieved 16 March 2015.
  35. ^ "Our universe is Flat". FermiLab/SLAC. 7 April 2015. Archived from the original on 10 April 2015.
  36. ^ Marcus Y. Yoo (2011). "Unexpected connections". Engineering & Science. LXXIV1: 30.
  37. ^ Weeks, Jeffrey (2001). The Shape of Space. CRC Press. ISBN 978-0-8247-0709-5.
  38. ^ Kaku, M. (2006). Parallel worlds. Knopf Doubleday Publishing Group.
  39. ^ Cambridge Dictionary of Philosophy, Second Edition, p. 429
  40. ^ Gosling, James; et. al. (27 July 2012). "4.2.3.". The Java™ Language Specification (Java SE 7 ed.). California: Oracle America, Inc. Archived from the original on 9 June 2012. Retrieved 6 September 2012.
  41. ^ Stokes, Roger (July 2012). "19.2.1". Learning J. Archived from the original on 25 March 2012. Retrieved 6 September 2012.
  42. ^ 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
  43. ^ Infinite chess at the Chess Variant Pages Archived 2017-04-02 at the Wayback Machine An infinite chess scheme.
  44. ^ "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 $ω^4$". arXiv:1510.08155 [math.LO].).

References

  • Cajori, Florian (1993) [1928 & 1929], A History of Mathematical Notations (Two Volumes Bound as One), Dover, ISBN 978-0-486-67766-8
  • Gemignani, Michael C. (1990), Elementary Topology (2nd ed.), Dover, ISBN 978-0-486-66522-1
  • Keisler, H. Jerome (1986), Elementary Calculus: An Approach Using Infinitesimals (2nd ed.)
  • Maddox, Randall B. (2002), Mathematical Thinking and Writing: A Transition to Abstract Mathematics, Academic Press, ISBN 978-0-12-464976-7
  • Kline, Morris (1972), Mathematical Thought from Ancient to Modern Times, New York: Oxford University Press, pp. 1197–1198, ISBN 978-0-19-506135-2
  • Russell, Bertrand (1996) [1903], The Principles of Mathematics, New York: Norton, ISBN 978-0-393-31404-5, OCLC 247299160
  • Sagan, Hans (1994), Space-Filling Curves, Springer, ISBN 978-1-4612-0871-6
  • Swokowski, Earl W. (1983), Calculus with Analytic Geometry (Alternate ed.), Prindle, Weber & Schmidt, ISBN 978-0-87150-341-1
  • Taylor, Angus E. (1955), Advanced Calculus, Blaisdell Publishing Company
  • Wallace, David Foster (2004), Everything and More: A Compact History of Infinity, Norton, W.W. & Company, Inc., ISBN 978-0-393-32629-1

Sources

External links

Adam Warlock

Adam Warlock, originally known as Him or Adam, is a fictional character appearing in American comic books published by Marvel Comics. The character's earliest appearances were in Fantastic Four #66–67 (cover-dates Sept. 1967 and Oct. 1967) and Thor #165–166 (June–July 1969). He was created by Stan Lee and Jack Kirby, and significantly developed by Roy Thomas and Jim Starlin.

Debuting in the Silver Age of comic books, the character has appeared in over four decades of Marvel publications, and starred in the titles Marvel Premiere and Strange Tales as well as five eponymous volumes and several related limited series. Adam Warlock has been associated with Marvel merchandise including animated television series, and video games.

BioWare

BioWare is a Canadian video game developer based in Edmonton, Alberta. It was founded in May 1995 by newly graduated medical doctors Ray Muzyka and Greg Zeschuk, alongside Trent Oster, Brent Oster, Marcel Zeschuk and Augustine Yip. As of 2007, the company is owned by American publisher Electronic Arts.

BioWare specializes in role-playing video games, and achieved recognition for developing highly praised and successful licensed franchises: Baldur's Gate, Neverwinter Nights, and Star Wars: Knights of the Old Republic. They proceeded to make several other successful games based on original intellectual property: Jade Empire, the Mass Effect series, and the Dragon Age series. In 2011, BioWare launched their first massively multiplayer online role-playing game, Star Wars: The Old Republic.

Buzz Lightyear

Buzz Lightyear is a fictional character in the Toy Story franchise. He is a toy Space Ranger superhero according to the movies and an action figure in the franchise. Along with Sheriff Woody, he is one of the two lead characters in all three Toy Story movies. He also appeared in the movie Buzz Lightyear of Star Command: The Adventure Begins and the television series spin-off Buzz Lightyear of Star Command.

He is voiced by Tim Allen in the Toy Story films, a few video games, and the Buzz Lightyear movie, Patrick Warburton in the TV series, and by Pat Fraley for the video games and the Buzz Lightyear attractions in Disney theme parks.

Call of Duty

Call of Duty is a first-person shooter video game franchise. Starting out in 2003, it first focused on games set in World War II and was based primarily on Microsoft Windows. Over time, the series has seen games set in modern times, the midst of the Cold War, futuristic worlds, and outer space. Infinity Ward was the series' first developer, with Treyarch later becoming the second, creating a two-team development cycle. Sledgehammer Games later became the third developer in the cycle. Activision has served as the publisher for the series since its creation. Several spin-offs and handheld versions of titles have also been made by other developers. The most recent title, Call of Duty: Black Ops 4, was released on October 12, 2018.

The series originally focused on the World War II setting, with Infinity Ward developing the first (2003) and second (2005) titles in the series and Treyarch developing the third (2006). Call of Duty 4: Modern Warfare (2007) introduced a new, modern setting, and proved to be the breakthrough title for the series, creating the Modern Warfare sub-series. The game's legacy also influenced the creation of a remastered version, released in 2016. Two other entries, Modern Warfare 2 (2009) and 3 (2011), were made. Infinity Ward have also developed two games outside of the Modern Warfare sub-series, Ghosts (2013) and Infinite Warfare (2016). Treyarch made one last World War II-based game, World at War (2008), before releasing Black Ops (2010) and subsequently creating the Black Ops sub-series. Three other entries, Black Ops II (2012), III (2015), and 4 (2018), were made. Sledgehammer Games, who were co-developers for Modern Warfare 3, have also developed two titles, Advanced Warfare (2014) and WWII (2017).

As of February 2016, the series has sold over 250 million copies. Sales of all Call of Duty games topped US$15 billion. Other products in the franchise include a line of action figures designed by Plan-B Toys, a card game created by Upper Deck Company, Mega bloks sets by Mega Brands, and a comic book mini-series published by WildStorm Productions.

Countable set

In mathematics, a countable set is a set with the same cardinality (number of elements) as some subset of the set of natural numbers. A countable set is either a finite set or a countably infinite set. Whether finite or infinite, the elements of a countable set can always be counted one at a time and, although the counting may never finish, every element of the set is associated with a unique natural number.

Some authors use countable set to mean countably infinite alone. To avoid this ambiguity, the term at most countable may be used when finite sets are included and countably infinite, enumerable, or denumerable otherwise.

Georg Cantor introduced the term countable set, contrasting sets that are countable with those that are uncountable (i.e., nonenumerable or nondenumerable). Today, countable sets form the foundation of a branch of mathematics called discrete mathematics.

Disney Infinity

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As Avalanche Software was closed down on May 11, 2016, Disney announced that the franchise would be retired. However, "Gold Editions" of the three games in the series were released for Microsoft Windows (via Steam) on December 9, 2016, which contain all the existing released figures and playset contents up to and including the third game's Zootopia figures.

Gamora

Gamora Zen Whoberi Ben Titan () is a fictional character appearing in American comic books published by Marvel Comics. Created by writer/artist Jim Starlin, the character first appeared in Strange Tales #180 (June 1975). Gamora is the adopted daughter of Thanos, and the last of her species. Her powers include superhuman strength and agility and an accelerated healing factor. She also is an elite combatant, being able to beat most of the opponents in the galaxy. She is a member of the group known as the Infinity Watch. The character played a role in the 2007 crossover comic book event "Annihilation: Conquest", and became a member of the titular team in its spin-off comic, Guardians of the Galaxy.

Gamora has been featured in a variety of associated Marvel merchandise. Zoe Saldana plays the character in the Marvel Cinematic Universe, starting with the 2014 film Guardians of the Galaxy. She reprised her role in the sequel Guardians of the Galaxy Vol. 2 and Avengers: Infinity War, and will appear in Avengers: Endgame.

Infinity-Man

Infinity-Man is a fictional character appearing in DC Comics.

Infinity Gems

The Infinity Gems (originally referred to as Soul Gems and later as Infinity Stones) are six gems appearing in Marvel Comics. The six gems are the Mind, Soul, Space, Power, Time and Reality Gems. In later storylines, crossovers and other media, a seventh gem has also been included. The Gems have been used by various characters in the Marvel Universe.

The Gems play a prominent role in the first three phases of the Marvel Cinematic Universe, where they are referred to as the Infinity Stones.

Infinity Ward

Infinity Ward, Inc. is an American video game developer. They developed the video game Call of Duty, along with six other installments in the Call of Duty series. Vince Zampella, Grant Collier, and Jason West established Infinity Ward in 2002 after working at 2015, Inc. previously. All of the 22 original team members of Infinity Ward came from the team that had worked on Medal of Honor: Allied Assault while at 2015, Inc. Activision helped fund Infinity Ward in its early days, buying up 30 percent of the company. The studio's first game, World War II shooter Call of Duty, was released on the PC in 2003. The day after the game was released, Activision bought the rest of Infinity Ward, signing employees to long term contracts. Infinity Ward went on to make Call of Duty 2, Call of Duty 4: Modern Warfare, Call of Duty: Modern Warfare 2, Call of Duty: Modern Warfare 3, Call of Duty: Ghosts and most recently Call of Duty: Infinite Warfare.

Co-founder Collier left the company in early 2009 to join parent company Activision. In 2010, West and Zampella were fired by Activision for "breaches of contract and insubordination", they soon founded a game studio called Respawn Entertainment. On May 3, 2014, Neversoft was merged into Infinity Ward.

Infinity symbol

The infinity symbol ∞ (sometimes called the lemniscate) is a mathematical symbol representing the concept of infinity.

List of Marvel Cinematic Universe films

The Marvel Cinematic Universe (MCU) films are an American series of superhero films based on characters that appear in publications by Marvel Comics. The MCU is the shared universe in which all of the films are set. The films have been in production since 2007, and in that time Marvel Studios has produced and released 20 films, with 11 more in various stages of production. It is the highest-grossing film franchise of all time, having grossed over $17.5 billion at the global box office.

Kevin Feige has produced every film in the series, alongside Avi Arad for the first two releases, Gale Anne Hurd for The Incredible Hulk, Amy Pascal for the Spider-Man films, and Stephen Broussard for Ant-Man and the Wasp. The films are written and directed by a variety of individuals and feature large, often ensemble, casts. Many of the actors, including Robert Downey Jr., Chris Evans, Chris Hemsworth, Samuel L. Jackson, and Scarlett Johansson signed contracts to star in numerous films.

The first film in the series was Iron Man (2008), which was distributed by Paramount Pictures. Paramount also distributed Iron Man 2 (2010), Thor (2011) and Captain America: The First Avenger (2011), while Universal Pictures distributed The Incredible Hulk (2008). Walt Disney Studios Motion Pictures began distributing the films with the 2012 crossover film The Avengers, which concluded Phase One of the franchise. Phase Two includes Iron Man 3 (2013), Thor: The Dark World (2013), Captain America: The Winter Soldier (2014), Guardians of the Galaxy (2014), Avengers: Age of Ultron (2015), and Ant-Man (2015).

Captain America: Civil War (2016) is the first film in the franchise's Phase Three, and is followed by Doctor Strange (2016), Guardians of the Galaxy Vol. 2 (2017), Spider-Man: Homecoming (2017), Thor: Ragnarok (2017), Black Panther (2018), Avengers: Infinity War (2018), and Ant-Man and the Wasp (2018), with Captain Marvel (2019) and Avengers: Endgame (2019) still scheduled for the phase. Spider-Man: Far From Home has also been scheduled for 2019, beginning Phase Four. Two untitled films are scheduled for 2020, three for 2021, and three for 2022. Sony Pictures distributes the Spider-Man films, which they continue to own, finance, and have final creative control over.

Respawn Entertainment

Respawn Entertainment, LLC is an American video game development studio founded by Jason West and Vince Zampella. West and Zampella previously co-founded Infinity Ward and created the Call of Duty franchise, where they were responsible for its development until 2010. Respawn was acquired by Electronic Arts on December 1, 2017.

Robert Jordan

James Oliver Rigney Jr. (October 17, 1948 – September 16, 2007), better known by his pen name Robert Jordan, was an American author of epic fantasy. He is best known for the Wheel of Time series, which comprises 14 books and a prequel novel. He is one of several writers to have written original Conan the Barbarian novels; his are highly acclaimed to this day. Rigney also wrote historical fiction under his pseudonym Reagan O'Neal, a western as Jackson O'Reilly, and dance criticism as Chang Lung. Additionally, he ghostwrote an "international thriller" that is still believed to have been written by someone else.

Thanos

Thanos (UK: , US: ) is a fictional supervillain appearing in American comic books published by Marvel Comics. The character, created by writer/artist Jim Starlin, first appeared in The Invincible Iron Man #55 (cover dated February 1973). Thanos is one of the most powerful villains in the Marvel Universe and has clashed with many heroes including the Avengers, the Guardians of the Galaxy, the Fantastic Four, and the X-Men.

The character appears in the Marvel Cinematic Universe, portrayed by Damion Poitier in The Avengers (2012), and by Josh Brolin in Guardians of the Galaxy (2014), Avengers: Age of Ultron (2015), Avengers: Infinity War (2018), and Avengers: Endgame (2019) through voice and motion capture. The character has also appeared in various comic adaptations, including animated television series, arcade, and video games.

The Infinity Gauntlet

The Infinity Gauntlet is an American comic book published by Marvel Comics. The story, written by Jim Starlin and pencilled by George Pérez and Ron Lim, was first serialized as a six-issue limited series from July to December 1991. As the main piece of a crossover event, some plot elements were featured in tie-in issues of other Marvel publications. Since its initial publication, the series has been reprinted in various formats and editions.

The roots of the series date to concepts developed in comics Starlin wrote and drew for Marvel in the 1970s, primarily Thanos and the Infinity Gems. Starlin returned to Marvel in 1990 as the writer for Silver Surfer volume 3 beginning with issue #34, assisted by Lim on pencils. Their storyline developed through the next sixteen issues and the two-issue spin-off limited series Thanos Quest before concluding in The Infinity Gauntlet. Fan-favorite artist Pérez drew the first three issues and eight pages of issue four before his busy schedule and unhappiness with the story led to him being replaced by Lim.

At the start of The Infinity Gauntlet, the alien nihilist Thanos has collected the six Infinity Gems and attached them to his gauntlet. With their combined power, he becomes like a god and sets out to win the affection of Mistress Death, the living embodiment of death in the Marvel Universe. When Thanos uses his powers to kill half of the living beings in the universe, Adam Warlock leads Earth's remaining heroes against him. After the Infinity Gauntlet is stolen by Thanos' villainous granddaughter Nebula, Thanos aids the remaining heroes in defeating her. Warlock ultimately obtains the Infinity Gauntlet and uses its power to undo the death and destruction caused by Thanos.

The series was a top seller for Marvel during publication and was followed by two immediate sequels, The Infinity War (1992) and The Infinity Crusade (1993). The story's events continued to be referenced in-story by comics for decades. The Infinity Gauntlet remained popular among fans, warranting multiple reprint editions and merchandise. Themes and plot elements have been repeatedly adapted into video games, animated cartoons, and film.

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