Infinity is a term with very distinct, separate meanings which arise in theology, philosophy, mathematics and everyday life. Popular or colloquial usage of the term often does not accord with its more technical meanings. The word infinity comes from Latin : "Infinito", unending. In theology, for example in the work of theologians such as Duns Scotus, the infinite nature of God invokes a sense of being without constraint, rather than a sense of being unlimited in quantity. In philosophy, infinity can be attrib ...

Including:

• Infinity - History
• Infinity - Ancient view of infinity
• Infinity - Views from the Renaissance to modern times
• Infinity - Modern philosophical views
• Infinity - Infinity symbol
• Infinity - Mathematical infinity
• Infinity - Infinity in real analysis
• Infinity - Infinity in complex analysis
• Infinity - Arithmetic properties of infinity
• Infinity - Infinity in set theory
• Infinity - Mathematics without infinity
• Infinity - Use of infinity in common speech
• Infinity - Physical infinity
• Infinity - Infinity in cosmology
• Infinity - Three types of infinities
• Infinity - Infinity in science fiction
• Infinity - Note

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In mathematics, the unit interval is the interval [0,1], that is the set of all real numbers x such that zero is less than or equal to x and x is less than or equal to one. The unit interval plays a fundamental role in homotopy theory, a major branch of topology. It is a metric space, compact, contractible, path connected and locally path connected. As a topological space, it is homeomorphic to the extended real number line. The unit interval is a one-dimensional analytical manifold with boundary {0,1}, carrying a ...

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In mathematics, a complete lattice is a partially ordered set in which all subsets have both a supremum (join) and an infimum (meet). Complete lattices appear in many applications in mathematics and computer science. Being a special instance of lattices, they are studied both in order theory and universal algebra. Complete lattices must not be confused with complete partial orders (cpos), which constitute a strictly more general class of partially ordered sets. More specific complete lattices are complete Boolean ...

Including:

• Complete lattice - Formal definition
• Complete lattice - Complete semilattices
• Complete lattice - Examples
• Complete lattice - Morphisms of complete lattices
• Complete lattice - Free construction and completion
• Complete lattice - Free complete semilattices
• Complete lattice - Free complete lattices
• Complete lattice - Completion
• Complete lattice - Representation
• Complete lattice - Further results
• Complete lattice - Literature

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In analysis the supremum or least upper bound of a set S of real numbers is denoted by sup(S) and is defined to be the smallest real number that is greater than or equal to every number in S. An important property of the real numbers is its completeness: every nonempty set of real numbers that is bounded above has a supremum. If, in addition, we define sup(S) = −∞ when S is empty and sup(S) = +∞ when S is not bounded above, then every set o ...

See also:

Supremum, Supremum - Supremum of a set of real numbers, Supremum - Approximation property, Supremum - Additive property, Supremum - Comparison property, Supremum - Suprema within partially ordered sets, Supremum - Comparison with other order theoretical notions, Supremum - Greatest elements, Supremum - Maximal elements, Supremum - Minimal upper bounds, Supremum - Least-upper-bound property

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In analysis the infimum or greatest lower bound of a set S of real numbers is denoted by inf(S) and is defined to be the biggest real number that is smaller than or equal to every number in S. If no such number exists (because S is not bounded below), then we define inf(S) = −∞. If S is empty, we define inf(S) = ∞ (see extended real number line). An important property of the real numbers is that every set of real numbers has an infimum (any bounded nonempty subset of the real numbers has an infimum in the non-extended real numbers). ...

See also:

Infimum, Infimum - Infima of real numbers, Infimum - Infima in partially ordered sets, Infimum - Least upper bound property

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Infinity - Ancient view of infinity. The earliest known documented knowledge of infinity is presented in the Veda- Yajur Veda which states that "if you remove a part from infinity or add a part to infinity, still what remains is infinity". The Indian Jaina mathematical text Surya Prajinapti (ca. 400 BC) classifies all numbers into three sets: enumerable, innumerable and infinite. It recognises five different types of infinity: infinite in one and two directions, infinite ...

See also:

Infinity, Infinity - History, Infinity - Ancient view of infinity, Infinity - Views from the Renaissance to modern times, Infinity - Modern philosophical views, Infinity - Infinity symbol, Infinity - Mathematical infinity, Infinity - Infinity in real analysis, Infinity - Infinity in complex analysis, Infinity - Arithmetic properties of infinity, Infinity - Infinity in set theory, Infinity - Mathematics without infinity, Infinity - Use of infinity in common speech, Infinity - Physical infinity, Infinity - Infinity in cosmology, Infinity - Three types of infinities, Infinity - Infinity in science fiction, Infinity - Note

Read more here: » Infinity: Encyclopedia II - Infinity - History

Infinity - Ancient view of infinity. The earliest known documented knowledge of infinity is presented in the Hindu Yajur Veda (ca. 1800 BC - 800 BC) which states that "if you remove a part from infinity or add a part to infinity, still what remains is infinity". The Indian Jaina mathematical text Surya Prajinapti (ca. 400 BC) classifies all numbers into three sets: enumerable, innumerable and infinite. It recognises five different types of infinity: infinite in one and tw ...

See also:

Infinity, Infinity - History, Infinity - Ancient view of infinity, Infinity - Views from the Renaissance to modern times, Infinity - Modern philosophical views, Infinity - Infinity symbol, Infinity - Mathematical infinity, Infinity - Infinity in real analysis, Infinity - Infinity in complex analysis, Infinity - Arithmetic properties of infinity, Infinity - Infinity in set theory, Infinity - Mathematics without infinity, Infinity - Use of infinity in common speech, Infinity - Physical infinity, Infinity - Infinity in cosmology, Infinity - Three types of infinities, Infinity - Infinity in science fiction, Infinity - Note

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Suppose x1, x2, ... is a sequence of elements in a topological space T. We say that L∈T is a limit of this sequence and write if and only if for every neighborhood S of L there is an N such that xn∈S for all n>N. If a sequence has a limit, we say the sequence is convergent, and that the sequence converges to the limit. Otherwise, the sequence is ...

See also:

Limit of a sequence, Limit of a sequence - Formal definition, Limit of a sequence - Comments, Limit of a sequence - Examples, Limit of a sequence - Properties, Limit of a sequence - History

Read more here: » Limit of a sequence: Encyclopedia II - Limit of a sequence - Formal definition

The discussion that follows parallels the most common expository approach to the Lebesgue integral. In this approach the theory of integration has two distinct parts: A theory of measurable sets and measures on these sets. A theory of measurable functions and integrals on these functions. Lebesgue integration - Measure theory. Measure theory initially was created to provide a detailed analysis of the notion of length of subsets of the real line and more generally area and volum ...

See also:

Lebesgue integration, Lebesgue integration - Introduction, Lebesgue integration - Construction of the Lebesgue integral, Lebesgue integration - Measure theory, Lebesgue integration - Integration, Lebesgue integration - Intuitive interpretation, Lebesgue integration - Example, Lebesgue integration - Limitations of the Riemann integral, Lebesgue integration - Basic theorems of the Lebesgue integral, Lebesgue integration - Proof techniques, Lebesgue integration - Alternative formulations, Lebesgue integration - Quote

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The most basic of improper integrals are integrals such as: As stated above, this need not be defined as an improper integral, since it can be construed as a Lebesgue integral instead. Nonetheless, for purposes of actually computing this integral, it is more convenient to treat it as an improper integral, i.e., to evaluate it when the upper bound of integration is finite and then take the limit as that bound approaches ∞. The antiderivative of the function being inte ...

See also:

Improper integral, Improper integral - Infinite bounds of integration, Improper integral - Vertical asymptotes at bounds of integration, Improper integral - Cauchy principal values

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An arbitrary point in the projective line P1(K) may be given in homogeneous coordinates by a pair [x1:x2] of points in K which are not both zero. Two such pairs are equal if they differ by an overall (nonzero) factor λ: [x1:x2] = [λx1:λx2]. The line K may be identified with the subset of < ...

See also:

Projective line, Projective line - Homogeneous coordinates, Projective line - Examples, Projective line - Real projective line, Projective line - Complex projective line: the Riemann sphere, Projective line - For a finite field, Projective line - Symmetry group, Projective line - As algebraic curve

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Some authors use the extended real number line and allow the distance function d to attain the value ∞. Such a metric is called an extended metric. Every extended metric can be rescaled to a finite metric (using d'(x, y) = d(x, y) / (1 + d(x, y)) or d''(x, y) = min(1, d(x, y))) and the two concepts of metric space are therefore equiva ...

See also:

Metric mathematics, Metric mathematics - Definition, Metric mathematics - Notes, Metric mathematics - Examples, Metric mathematics - Equivalence of metrics, Metric mathematics - Relation of norms and metrics, Metric mathematics - Related concepts and alternative axiom systems

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Real number - Completeness. The main reason for introducing the reals is that the reals contain all limits. More technically, the reals are complete (in the sense of metric spaces or uniform spaces, which is a different sense than the Dedekind completeness of the order in the previous section). This means the following: A sequence (xn) of real numbers is called a Cauchy sequence if for any ε > 0 there exists an integer N (possibly depending on ε) such t ...

See also:

Real number, Real number - History, Real number - Definition, Real number - Construction from the rational numbers, Real number - Axiomatic approach, Real number - Properties, Real number - Completeness, Real number - The complete ordered field, Real number - Advanced properties, Real number - Generalizations and extensions

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Various inequivalent definitions of Kleene algebras and related structures have been given in the literature. See [1] for a survey. Here we will give the definition that seems to be the most common nowadays. A Kleene algebra is a set A together with two binary operations + : A × A → A and · : A × A → A and one function * : A → A, written as a + b, ab and a* respectively, so that the following axioms are satisfied.

See also:

Kleene algebra, Kleene algebra - Definition, Kleene algebra - Examples, Kleene algebra - Properties, Kleene algebra - History

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Complete lattice - Free complete semilattices. As usual, the construction of free objects depends on the chosen class of morphisms. Let us first consider functions that preserve all joins (i.e. lower adjoints of Galois connections), since this case is simpler than the situation for complete homomorphisms. Using the aforementioned terminology, this could be called a free complete join-semilattice. Using the standard definition from universal algebra, a free complete lattice over a generating set S ...

See also:

Complete lattice, Complete lattice - Formal definition, Complete lattice - Complete semilattices, Complete lattice - Examples, Complete lattice - Morphisms of complete lattices, Complete lattice - Free construction and completion, Complete lattice - Free complete semilattices, Complete lattice - Free complete lattices, Complete lattice - Completion, Complete lattice - Representation, Complete lattice - Further results, Complete lattice - Literature

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Limit of a function - Functions on metric spaces. Suppose f : (M,dM) -> (N,dN) is a map between two metric spaces, p is a limit point of M and L∈N. We say that "the limit of f at p is L" and write if and only if for every ε > 0 there exists a δ > 0 such that for all x∈M with 0 < dM(x, p) < δ, we have dN(f(x), L) < ε. See also:

Limit of a function, Limit of a function - History, Limit of a function - Formal definition, Limit of a function - Functions on metric spaces, Limit of a function - Real-valued functions, Limit of a function - Complex-valued functions, Limit of a function - Limit of a function of more than one variable, Limit of a function - Examples, Limit of a function - Real-valued functions, Limit of a function - Functions on metric spaces, Limit of a function - Properties

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Infinity - Ancient view of infinity. The earliest known documented knowledge of infinity is presented in the Hindu Yajur Veda (ca. 1800 BC - 800 BC) which states that "if you remove a part from infinity or add a part to infinity, still what remains is infinity". The Indian Jaina mathematical text Surya Prajinapti (ca. 400 BC) classifies all numbers into three sets: enumerable, innumerable and infinite. It recognises five different types of infinity: infinite in one and tw ...

See also:

Infinity, Infinity - History, Infinity - Ancient view of infinity, Infinity - Views from the Renaissance to modern times, Infinity - Modern philosophical views, Infinity - Infinity symbol, Infinity - Mathematical infinity, Infinity - Infinity in real analysis, Infinity - Infinity in complex analysis, Infinity - Arithmetic properties of infinity, Infinity - Infinity in set theory, Infinity - Mathematics without infinity, Infinity - Use of infinity in common speech, Infinity - Physical infinity, Infinity - Infinity in cosmology, Infinity - Three types of infinities, Infinity - Infinity in science fiction, Infinity - Note

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Infinity - Infinity in real analysis. In real analysis, the symbol , called "infinity", denotes an unbounded limit. means that x grows beyond any assigned value, and means x is eventually less than any assigned value. Points labeled and can be added to the real numbers as a topological space, producing the two-point compactification of the real numbers. Adding algebraic properties to this gives us the extended real numbers. We can also treat and as the same, leading to the one-point compact ...

See also:

Infinity, Infinity - History, Infinity - Ancient view of infinity, Infinity - Views from the Renaissance to modern times, Infinity - Modern philosophical views, Infinity - Infinity symbol, Infinity - Mathematical infinity, Infinity - Infinity in real analysis, Infinity - Infinity in complex analysis, Infinity - Arithmetic properties of infinity, Infinity - Infinity in set theory, Infinity - Mathematics without infinity, Infinity - Use of infinity in common speech, Infinity - Physical infinity, Infinity - Infinity in cosmology, Infinity - Three types of infinities, Infinity - Infinity in science fiction, Infinity - Note

Read more here: » Infinity: Encyclopedia II - Mathematical infinity

In common parlance, infinity is often used in a hyperbolic sense. For example, "The movie was infinitely boring, but we had to wait forever to get tickets." In video games, infinite lives and infinite ammo refer to a never-ending supply of lives and ammunition. An infinite loop in computer programming is a conditional loop construction whose condition always evaluates to true. In theory, as long as there is no external interaction, the loop will continue to run for all time. In practice however, most programming loops co ...

See also:

Infinity, Infinity - History, Infinity - Ancient view of infinity, Infinity - Views from the Renaissance to modern times, Infinity - Modern philosophical views, Infinity - Infinity symbol, Infinity - Mathematical infinity, Infinity - Infinity in real analysis, Infinity - Infinity in complex analysis, Infinity - Arithmetic properties of infinity, Infinity - Infinity in set theory, Infinity - Mathematics without infinity, Infinity - Use of infinity in common speech, Infinity - Physical infinity, Infinity - Infinity in cosmology, Infinity - Three types of infinities, Infinity - Infinity in science fiction, Infinity - Note

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The Hitchhiker's Guide to the Galaxy contains the following definition of infinity: "Bigger than the biggest thing ever and then some, much bigger than that, in fact really amazingly immense, a totally stunning size, real 'Wow, that's big!' time. Infinity is just so big that by comparison, bigness itself looks really titchy. Gigantic multiplied by colossal multiplied by staggeringly huge is the s ...

See also:

Infinity, Infinity - History, Infinity - Ancient view of infinity, Infinity - Views from the Renaissance to modern times, Infinity - Modern philosophical views, Infinity - Infinity symbol, Infinity - Mathematical infinity, Infinity - Infinity in real analysis, Infinity - Infinity in complex analysis, Infinity - Arithmetic properties of infinity, Infinity - Infinity in set theory, Infinity - Mathematics without infinity, Infinity - Use of infinity in common speech, Infinity - Physical infinity, Infinity - Infinity in cosmology, Infinity - Three types of infinities, Infinity - Infinity in science fiction, Infinity - Note

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