Superreal number
In abstract algebra, the superreal numbers are a class of extensions of the real numbers, introduced by and W. Hugh Woodin as a generalization of the hyperreal numbers and primarily of interest in non
Aevum
In scholastic philosophy, the aevum (also called aeviternity) is the temporal mode of existence experienced by angels and by the saints in heaven. In some ways, it is a state that logically lies betwe
Countable set
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is countable if there exists an injective
Directed infinity
A directed infinity is a type of infinity in the complex plane that has a defined complex argument θ but an infinite absolute value r. For example, the limit of 1/x where x is a positive real number a
Extended real number line
In mathematics, the affinely extended real number system is obtained from the real number system by adding two infinity elements: and where the infinities are treated as actual numbers. It is useful i
Hilbert cube
In mathematics, the Hilbert cube, named after David Hilbert, is a topological space that provides an instructive example of some ideas in topology. Furthermore, many interesting topological spaces can
Infinity mirror
The infinity mirror (also sometimes called an infinite mirror) is a configuration of two or more parallel or nearly parallel mirrors, creating a series of smaller and smaller reflections that appear t
Nonstandard analysis
The history of calculus is fraught with philosophical debates about the meaning and logical validity of fluxions or infinitesimal numbers. The standard way to resolve these debates is to define the op
Continuum (set theory)
In the mathematical field of set theory, the continuum means the real numbers, or the corresponding (infinite) cardinal number, denoted by . Georg Cantor proved that the cardinality is larger than the
Projectively extended real line
In real analysis, the projectively extended real line (also called the one-point compactification of the real line), is the extension of the set of the real numbers, by a point denoted ∞. It is thus t
Axiom of infinity
In axiomatic set theory and the branches of mathematics and philosophy that use it, the axiom of infinity is one of the axioms of Zermelo–Fraenkel set theory. It guarantees the existence of at least o
Beth number
In mathematics, particularly in set theory, the beth numbers are a certain sequence of infinite cardinal numbers (also known as transfinite numbers), conventionally written , where is the second Hebre
Ideal point
In hyperbolic geometry, an ideal point, omega point or point at infinity is a well-defined point outside the hyperbolic plane or space.Given a line l and a point P not on l, right- and left-limiting p
Supernatural number
In mathematics, the supernatural numbers, sometimes called generalized natural numbers or Steinitz numbers, are a generalization of the natural numbers. They were used by Ernst Steinitz in 1910 as a p
Hyperinteger
In nonstandard analysis, a hyperinteger n is a hyperreal number that is equal to its own integer part. A hyperinteger may be either finite or infinite. A finite hyperinteger is an ordinary integer. An
Plane at infinity
In projective geometry, a plane at infinity is the hyperplane at infinity of a three dimensional projective space or to any plane contained in the hyperplane at infinity of any projective space of hig
Surreal number
In mathematics, the surreal number system is a totally ordered proper class containing the real numbers as well as infinite and infinitesimal numbers, respectively larger or smaller in absolute value
Infinity (philosophy)
In philosophy and theology, infinity is explored in articles under headings such as the Absolute, God, and Zeno's paradoxes. In Greek philosophy, for example in Anaximander, 'the Boundless' is the ori
Circular points at infinity
In projective geometry, the circular points at infinity (also called cyclic points or isotropic points) are two special points at infinity in the complex projective plane that are contained in the com
Temporal finitism
Temporal finitism is the doctrine that time is finite in the past. The philosophy of Aristotle, expressed in such works as his Physics, held that although space was finite, with only void existing bey
Beyond Infinity (mathematics book)
Beyond Infinity : An Expedition to the Outer Limits of Mathematics is a popular mathematics book by Eugenia Cheng centered on concepts of infinity. It was published by Basic Books and (with a slightly
Hilbert's paradox of the Grand Hotel
Hilbert's paradox of the Grand Hotel (colloquial: Infinite Hotel Paradox or Hilbert's Hotel) is a thought experiment which illustrates a counterintuitive property of infinite sets. It is demonstrated
Line at infinity
In geometry and topology, the line at infinity is a projective line that is added to the real (affine) plane in order to give closure to, and remove the exceptional cases from, the incidence propertie
Uncountable set
In mathematics, an uncountable set (or uncountably infinite set) is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal numb
Heh (god)
Ḥeḥ (ḥḥ, also Huh, Hah, Hauh, Huah, and Hehu) was the personification of infinity or eternity in the Ogdoad in ancient Egyptian religion. His name originally meant "flood", referring to the watery cha
Paradoxes of the Infinite
Paradoxes of the Infinite (German title: Paradoxien des Unendlichen) is a mathematical work by Bernard Bolzano on the theory of sets. It was published by a friend and student, , in 1851, three years a
Point at infinity
In geometry, a point at infinity or ideal point is an idealized limiting point at the "end" of each line. In the case of an affine plane (including the Euclidean plane), there is one ideal point for e
Cardinality of the continuum
In set theory, the cardinality of the continuum is the cardinality or "size" of the set of real numbers , sometimes called the continuum. It is an infinite cardinal number and is denoted by (lowercase
Division by zero
In mathematics, division by zero is division where the divisor (denominator) is zero. Such a division can be formally expressed as , where a is the dividend (numerator). In ordinary arithmetic, the ex
Ad infinitum
Ad infinitum is a Latin phrase meaning "to infinity" or "forevermore".
Infinite monkey theorem
The infinite monkey theorem states that a monkey hitting keys at random on a typewriter keyboard for an infinite amount of time will almost surely type any given text, such as the complete works of Wi
Division by infinity
In mathematics, division by infinity is division where the divisor (denominator) is infinity. In ordinary arithmetic, this does not have a well-defined meaning, since infinity is a mathematical concep
Infinity symbol
The infinity symbol is a mathematical symbol representing the concept of infinity. This symbol is also called a lemniscate, after the lemniscate curves of a similar shape studied in algebraic geometry
Actual infinity
In the philosophy of mathematics, the abstraction of actual infinity involves the acceptance (if the axiom of infinity is included) of infinite entities as given, actual and completed objects. These m
Pascal's wager
Pascal's wager is a philosophical argument presented by the seventeenth-century French mathematician, philosopher, physicist and theologian Blaise Pascal (1623–1662). It posits that human beings wager
Where Mathematics Comes From
Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being (hereinafter WMCF) is a book by George Lakoff, a cognitive linguist, and Rafael E. Núñez, a psychologist. Published in
Law of continuity
The law of continuity is a heuristic principle introduced by Gottfried Leibniz based on earlier work by Nicholas of Cusa and Johannes Kepler. It is the principle that "whatever succeeds for the finite
Levi-Civita field
In mathematics, the Levi-Civita field, named after Tullio Levi-Civita, is a non-Archimedean ordered field; i.e., a system of numbers containing infinite and infinitesimal quantities. Each member can b
Continuum hypothesis
In mathematics, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets. It states that there is no set whose cardinality is strictly between that of the in
Hyperreal number
In mathematics, the system of hyperreal numbers is a way of treating infinite and infinitesimal (infinitely small but non-zero) quantities. The hyperreals, or nonstandard reals, *R, are an extension o
Infinity
Infinity is that which is boundless, endless, or larger than any natural number. It is often denoted by the infinity symbol . Since the time of the ancient Greeks, the philosophical nature of infinity
Playing with Infinity
Playing with Infinity: Mathematical Explorations and Excursions is a book in popular mathematics by Hungarian mathematician Rózsa Péter, published in German in 1955 and in English in 1961.
Aleph number
In mathematics, particularly in set theory, the aleph numbers are a sequence of numbers used to represent the cardinality (or size) of infinite sets that can be well-ordered. They were introduced by t
Ultrafinitism
In the philosophy of mathematics, ultrafinitism (also known as ultraintuitionism, strict formalism, strict finitism, actualism, predicativism, and strong finitism) is a form of finitism and intuitioni
Infinitism
Infinitism is the view that knowledge may be justified by an infinite chain of reasons. It belongs to epistemology, the branch of philosophy that considers the possibility, nature, and means of knowle
Infinitesimal
In mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. The word infinitesimal comes from a 17th-century Modern Latin coinage
Irresistible force paradox
The irresistible force paradox (also unstoppable force paradox or shield and spear paradox), is a classic paradox formulated as "What happens when an unstoppable force meets an immovable object?" The
Hyperplane at infinity
In geometry, any hyperplane H of a projective space P may be taken as a hyperplane at infinity. Then the set complement P ∖ H is called an affine space. For instance, if (x1, ..., xn, xn+1) are homoge
Eternity
Eternity, in common parlance, means infinite time that never ends or the quality, condition, or fact of being everlasting or eternal. Classical philosophy, however, defines eternity as what is timeles
John Wallis
John Wallis (/ˈwɒlɪs/; Latin: Wallisius; 3 December [O.S. 23 November] 1616 – 8 November [O.S. 28 October] 1703) was an English clergyman and mathematician who is given partial credit for the developm
Absolute Infinite
The Absolute Infinite (symbol: Ω) is an extension of the idea of infinity proposed by mathematician Georg Cantor. It can be thought of as a number that is bigger than any other conceivable or inconcei
Apeirogon
In geometry, an apeirogon (from Ancient Greek ἄπειρος apeiros 'infinite, boundless', and γωνία gonia 'angle') or infinite polygon is a generalized polygon with a countably infinite number of sides. Ap
Semi-infinite
In mathematics, semi-infinite objects are objects which are infinite or unbounded in some but not all possible ways.
Cantor's diagonal argument
In set theory, Cantor's diagonal argument, also called the diagonalisation argument, the diagonal slash argument, the anti-diagonal argument, the diagonal method, and Cantor's diagonalization proof, w
Finitism
Finitism is a philosophy of mathematics that accepts the existence only of finite mathematical objects. It is best understood in comparison to the mainstream philosophy of mathematics where infinite m
Nonstandard calculus
In mathematics, nonstandard calculus is the modern application of infinitesimals, in the sense of nonstandard analysis, to infinitesimal calculus. It provides a rigorous justification for some argumen