Infinity in the context of Hamel dimension

In Islam, God (Arabic: ٱللَّٰه, romanized: Allāh, contraction of ٱلْإِلَٰه al-’ilāh, lit. 'the god', or Arabic: رب, romanized: Rabb, lit. 'lord') is seen as the creator and sustainer of the universe, who lives eternally. God is conceived as a perfect, singular, immortal, omnipotent, and omniscient deity, completely infinite in all of his attributes. Islam further emphasizes that God is most merciful. The Islamic concept of God is variously described as monotheistic, panentheistic, and monistic.

The Islamic concept of tawhid (unification) emphasises that God is absolutely pure and free from association or partnership with other beings, which means attributing the powers and qualities of God to his creation, and vice versa. In Islam, God is never portrayed in any image. The Quran specifically forbids ascribing partners to share his singular sovereignty, as he is considered to be the absolute one without a second, indivisible, and incomparable being, who is similar to nothing, and nothing is comparable to him. Thus, God is absolutely transcendent, unique and utterly other than anything in or of the world as to be beyond all forms of human thought and expression. The briefest and the most comprehensive description of God in the Quran is found in Surat al-Ikhlas.

In mathematics, a series is, roughly speaking, an addition of infinitely many terms, one after the other. The study of series is a major part of calculus and its generalization, mathematical analysis. Series are used in most areas of mathematics, even for studying finite structures in combinatorics through generating functions. The mathematical properties of infinite series make them widely applicable in other quantitative disciplines such as physics, computer science, statistics and finance.

Among the Ancient Greeks, the idea that a potentially infinite summation could produce a finite result was considered paradoxical, most famously in Zeno's paradoxes. Nonetheless, infinite series were applied practically by Ancient Greek mathematicians including Archimedes, for instance in the quadrature of the parabola. The mathematical side of Zeno's paradoxes was resolved using the concept of a limit during the 17th century, especially through the early calculus of Isaac Newton. The resolution was made more rigorous and further improved in the 19th century through the work of Carl Friedrich Gauss and Augustin-Louis Cauchy, among others, answering questions about which of these sums exist via the completeness of the real numbers and whether series terms can be rearranged or not without changing their sums using absolute convergence and conditional convergence of series.

Georg Ferdinand Ludwig Philipp Cantor (/ˈkæntɔːr/ KAN-tor; German: [ˈɡeːɔʁk ˈfɛʁdinant ˈluːtvɪç ˈfiːlɪp ˈkantoːɐ̯]; 3 March [O.S. 19 February] 1845 – 6 January 1918) was a mathematician who played a pivotal role in the creation of set theory, which has become a fundamental theory in mathematics. Cantor established the importance of one-to-one correspondence between the members of two sets, defined infinite and well-ordered sets, and proved that the real numbers are more numerous than the natural numbers. Cantor's method of proof of this theorem implies the existence of an infinity of infinities. He defined the cardinal and ordinal numbers and their arithmetic. Cantor's work is of great philosophical interest, a fact he was well aware of.

Originally, Cantor's theory of transfinite numbers was regarded as counter-intuitive – even shocking. This caused it to encounter resistance from mathematical contemporaries such as Leopold Kronecker and Henri Poincaré and later from Hermann Weyl and L. E. J. Brouwer, while Ludwig Wittgenstein raised philosophical objections; see Controversy over Cantor's theory. Cantor, a devout Lutheran Christian, believed the theory had been communicated to him by God. Some Christian theologians (particularly neo-Scholastics) saw Cantor's work as a challenge to the uniqueness of the absolute infinity in the nature of God – on one occasion equating the theory of transfinite numbers with pantheism – a proposition that Cantor vigorously rejected. Not all theologians were against Cantor's theory; prominent neo-scholastic philosopher Konstantin Gutberlet [de; it] was in favor of it and Cardinal Johann Baptist Franzelin accepted it as a valid theory (after Cantor made some important clarifications).

Everything, every-thing, or every thing, is all that exists; it is an antithesis of nothing, or its complement. It is the totality of things relevant to some subject matter. The universe is everything that exists theoretically, though a multiverse may exist according to theoretical cosmology predictions. It may refer to an anthropocentric worldview, or the sum of human experience, history, and the human condition in general. Every object and entity is a part of everything, including all physical bodies and in some cases all abstract objects.

To describe or know of everything as a spatial consideration in a local environment, such as the world in which humans mostly live, is possible. The detemination of all things in the universe is unknown because of the physics beyond the observed universe and the problem of knowing physics at the range infinite. To know universally everything as a temporal and spatial consideration isn't possible because of the unavailabilty of information at a certain time before the beginning of the universe and because of the problem of eternal causality.

Electric potential (also called the electric field potential, potential drop, the electrostatic potential) is the difference in electric potential energy per unit of electric charge between two points in a static electric field. More precisely, electric potential is the amount of work needed to move a test charge from a reference point to a specific point in a static electric field, normalized to a unit of charge. The test charge used is small enough that disturbance to the field-producing charges is unnoticeable, and its motion across the field is supposed to proceed with negligible acceleration, so as to avoid the test charge acquiring kinetic energy or producing radiation. By definition, the electric potential at the reference point is zero units. Typically, the reference point is earth or a point at infinity, although any point can be used.

In classical electrostatics, the electrostatic field is a vector quantity expressed as the gradient of the electrostatic potential, which is a scalar quantity denoted by V or occasionally φ, equal to the electric potential energy of any charged particle at any location (measured in joules) divided by the charge of that particle (measured in coulombs). By dividing out the charge on the particle a quotient is obtained that is a property of the electric field itself. In short, an electric potential is the electric potential energy per unit charge.

Zeno's paradoxes are a series of philosophical arguments presented by the ancient Greek philosopher Zeno of Elea (c. 490–430 BC), primarily known through the works of Plato, Aristotle, and later commentators like Simplicius of Cilicia. Zeno devised these paradoxes to support his teacher Parmenides's philosophy of monism, which posits that despite people's sensory experiences, reality is singular and unchanging. The paradoxes famously challenge the notions of plurality (the existence of many things), motion, space, and time by suggesting they lead to logical contradictions.

Zeno's work, primarily known from second-hand accounts since his original texts are lost, comprises forty "paradoxes of plurality," which argue against the coherence of believing in multiple existences, and several arguments against motion and change. Of these, only a few are definitively known today, including the renowned "Achilles Paradox", which illustrates the problematic concept of infinite divisibility in space and time. In this paradox, Zeno argues that a swift runner like Achilles cannot overtake a slower moving tortoise with a head start, because the distance between them can be infinitely subdivided, implying Achilles would require an infinite number of steps to catch the tortoise.

Eternity, also forever, in common parlance, is an infinite amount of time that never ends or the quality, condition or fact of being everlasting or eternal. Classical philosophy, however, defines eternity as what is timeless or exists outside time, whereas sempiternity corresponds to infinite duration.

In mathematics, an infinitesimal number is a non-zero quantity that is closer to 0 than any non-zero real number is. The word infinitesimal comes from a 17th-century Modern Latin coinage infinitesimus, which originally referred to the "infinitieth" item in a sequence.

Infinitesimals do not exist in the standard real number system, but they do exist in other number systems, such as the surreal number system and the hyperreal number system, which can be thought of as the real numbers augmented with both infinitesimal and infinite quantities; the augmentations are the reciprocals of one another.

Mādhava of Sangamagrāma (Mādhavan) (c. 1340 – c. 1425) was an Indian mathematician and astronomer who is considered to be the founder of the Kerala school of astronomy and mathematics in the Late Middle Ages. Madhava made pioneering contributions to the study of infinite series, trigonometry, geometry and algebra. He was the first to use infinite series approximations for a range of trigonometric functions, which has been called the "decisive step onward from the finite procedures of ancient mathematics to treat their limit-passage to infinity".

In set theory, an infinite set is a set that is not a finite set. Infinite sets may be countable or uncountable.

In geometry, an apeirogon (from Ancient Greek ἄπειροv apeiron 'infinite, boundless' and γωνία gonia 'angle') or infinite polygon is a polygon with an infinite number of sides. Apeirogons are the rank 2 case of infinite polytopes. In some literature, the term "apeirogon" may refer only to the regular apeirogon, with an infinite dihedral group of symmetries.

A chord (from the Latin chorda, meaning "catgut or string") of a circle is a straight line segment whose endpoints both lie on a circular arc. If a chord were to be extended infinitely on both directions into a line, the object is a secant line. The perpendicular line passing through the chord's midpoint is called sagitta (Latin for "arrow").

More generally, a chord is a line segment joining two points on any curve, for instance, on an ellipse. A chord that passes through a circle's center point is the circle's diameter.

In probability theory, an experiment or trial (see below) is the mathematical model of any procedure that can be infinitely repeated and has a well-defined set of possible outcomes, known as the sample space. An experiment is said to be random if it has more than one possible outcome, and deterministic if it has only one. A random experiment that has exactly two (mutually exclusive) possible outcomes is known as a Bernoulli trial.

When an experiment is conducted, one (and only one) outcome results— although this outcome may be included in any number of events, all of which would be said to have occurred on that trial. After conducting many trials of the same experiment and pooling the results, an experimenter can begin to assess the empirical probabilities of the various outcomes and events that can occur in the experiment and apply the methods of statistical analysis.

Ein Sof, or Eyn Sof (/eɪn sɒf/, Hebrew: אֵין סוֹף‎ ʾēn sōf; meaning "infinite", lit. '(There is) no end'), in Kabbalah, is understood as God before any self-manifestation in the production of any spiritual realm, probably derived from Solomon ibn Gabirol's (c. 1021 – c. 1070) term, "the Endless One" (שֶׁאֵין לוֹ תִּקְלָה, šeʾēn lo tiqlā). Ein Sof may be translated as "unending", "(there is) no end", or infinity. It was first used by Azriel of Gerona (c. 1160 – c. 1238), who shared the Neoplatonic belief that God can have no desire, thought, word, or action, emphasized by the negation of any attribute.

This is the origin of the Ohr Ein Sof or "Infinite Light" of paradoxical divine self-knowledge, nullified within the Ein Sof before creation. In Lurianic Kabbalah, the first act of creation, the tzimtzum or self-withdrawal of the divine to create a space, takes place from there.

A gravitational singularity, spacetime singularity, or simply singularity, is a theoretical condition in which gravity is predicted to be so intense that spacetime itself would break down catastrophically. As such, a singularity is by definition no longer part of the regular spacetime and cannot be determined by "where" or "when". Gravitational singularities exist at a junction between general relativity and quantum mechanics; therefore, the properties of the singularity cannot be described without an established theory of quantum gravity. Trying to find a complete and precise definition of singularities in the theory of general relativity, the best theory of gravity available, remains a difficult problem. A singularity in general relativity can be defined by the scalar invariant curvature becoming infinite or, better, by a geodesic being incomplete.

General relativity predicts that any object collapsing beyond its Schwarzschild radius would form a black hole, inside which a singularity will form. A black hole singularity is, however, covered by an event horizon, so it is never in the causal past of any outside observer, and at no time can it be objectively said to have formed. General relativity also predicts that the initial state of the universe, at the beginning of the Big Bang, was a singularity of infinite density and temperature. However, classical gravitational theories are not expected to be accurate under these conditions, and a quantum description is likely needed. For example, quantum mechanics does not permit particles to inhabit a space smaller than their Compton wavelengths.

In mathematics, cardinality is an intrinsic property of sets, roughly meaning the number of individual objects they contain, which may be infinite. The concept is understood through one-to-one correspondences between sets. That is, if their objects can be paired such that each object has a pair, and no object is paired more than once.

The basic concepts of cardinality go back as early as the 6th century BCE, and there are several close encounters with it throughout history, however, the results were generally dismissed as paradoxical. It is considered to have been first introduced formally to mathematics by Georg Cantor at the turn of the 20th century. Cantor's theory of cardinality was then formalized, popularized, and explored by many influential mathematicians of the time, and has since become a fundamental concept of mathematics.

Jorge Francisco Isidoro Luis Borges (/ˈbɔːrhɛs/ BOR-hess; Spanish: [ˈxoɾxe ˈlwis ˈboɾxes] ; 24 August 1899 – 14 June 1986) was an Argentine short-story writer, essayist, poet and translator regarded as a key figure in Spanish-language and international literature. His best-known works, Ficciones (transl. Fictions) and El Aleph (transl. The Aleph), published in the 1940s, are collections of short stories exploring motifs such as dreams, labyrinths, chance, infinity, archives, mirrors, fictional writers, and mythology. Borges's works have contributed to philosophical literature and the fantasy genre, and have had a major influence on the magical realist movement in 20th century Latin American literature.

Born in Buenos Aires, Borges later moved with his family to Switzerland in 1914, where he studied at the Collège de Genève. The family travelled widely in Europe, including Spain. On his return to Argentina in 1921, Borges began publishing his poems and essays in surrealist literary journals. He also worked as a librarian and public lecturer. In 1955, he was appointed director of the National Public Library and professor of English Literature at the University of Buenos Aires. He became completely blind by the age of 55. Scholars have suggested that his progressive blindness helped him to create innovative literary symbols through imagination. By the 1960s, his work was translated and published widely in the United States and Europe. Borges himself was fluent in several languages.

In mathematics, transfinite numbers or infinite numbers are numbers that are "infinite" in the sense that they are larger than all finite numbers. These include the transfinite cardinals, which are cardinal numbers used to quantify the size of infinite sets, and the transfinite ordinals, which are ordinal numbers used to provide an ordering of infinite sets. The term transfinite was coined in 1895 by Georg Cantor, who wished to avoid some of the implications of the word infinite. In particular he believed that "truly infinite" is a perfect and thus divine quality and so refused to attribute this term to mathematical constructs comprehensible by humans. Few contemporary writers share these qualms; it is now accepted usage to refer to transfinite cardinals and ordinals as infinite numbers. Nevertheless, the term transfinite also remains in use.

Notable work on transfinite numbers was done by Wacław Sierpiński: Leçons sur les nombres transfinis (1928 book) much expanded into Cardinal and Ordinal Numbers (1958, 2nd ed. 1965).

In mathematics, a real interval is the set of all real numbers lying between two fixed endpoints with no "gaps". Each endpoint is either a real number or positive or negative infinity, indicating the interval extends without a bound. A real interval can contain neither endpoint, either endpoint, or both endpoints, excluding any endpoint which is infinite.

For example, the set of real numbers consisting of 0, 1, and all numbers in between is an interval, denoted [0, 1] and called the unit interval; the set of all positive real numbers is an interval, denoted (0, ∞); the set of all real numbers is an interval, denoted (−∞, ∞); and any single real number a is an interval, denoted [a, a].