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axioms
A Wisdom Archive on axioms

axioms A selection of articles related to axioms

axioms, Axiom, Axiom - Etymology, Axiom - Mathematics, Axiom - <span id=role>Role in mathematical logic</span>, Axiom - Further discussion, Axiom - Logical axioms, Axiom - Non-logical axioms, Axiomatic system, Peano axioms, Axiom of choice, Axiom of countability, Axiomatic set theory, Parallel postulate, Continuum hypothesis, Axiomatization, List of axioms

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ARTICLES RELATED TO axioms

axioms: Encyclopedia II - List of axioms - Zermelo-Frankel axioms

These are the de facto standard axioms for contemporary mathematics Axiom of extensionality Axiom of empty set Axiom of pairing Axiom of union Axiom of infinity Axiom schema of replacement Axiom of power set Axiom of regularity Axiom of separation Axiom schema of specification See also Zermelo set theory. ...

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List of axioms, List of axioms - Zermelo-Frankel axioms, List of axioms - Axiom of choice, List of axioms - Equivalents of AC, List of axioms - Weaker than AC, List of axioms - Alternates incompatible with AC, List of axioms - Other axioms of mathematical logic, List of axioms - Geometry, List of axioms - Other axioms

Read more here: » List of axioms: Encyclopedia II - List of axioms - Zermelo-Frankel axioms

axioms: Encyclopedia II - Cissoid of Diocles - Delian problem

The cissoid of Diocles is named after the Greek geometer Diocles who used it in 180 B.C. to solve the Delian problem: how much must the length of a cube be increased in order to double the volume of the cube? Diocles' solution is correct, except that the solution involves an intersection of a line with a construction of a cissoid of Diocles, which cannot be accomplished by means of the simple but strict Greek rules of compass and straightedge constructio ...

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Cissoid of Diocles, Cissoid of Diocles - Construction, Cissoid of Diocles - Delian problem, Cissoid of Diocles - Roulette, Cissoid of Diocles - The cissoid of Diocles as a pedal curve, Cissoid of Diocles - Inversion

Read more here: » Cissoid of Diocles: Encyclopedia II - Cissoid of Diocles - Delian problem

axioms: Encyclopedia II - Physical law - Description

Several general properties of physical laws have been identified (see Davies (1992) and Feynman (1965) as noted, although each of the characterizations is not necessarily original to them). Physical laws are: true. By definition, there have never been repeatable contradicting observations. universal. They appear to apply everywhere in the universe. (Davies) simple. They are typically expressed in terms of a single mathematical equation. (Davies) absolute. Nothing in the universe appears to affect them. ...

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Physical law, Physical law - Description, Physical law - Examples, Physical law - Laws as approximations, Physical law - Necessity origin and existence, Physical law - History and religious influence, Physical law - Significance and renown of discoverers, Physical law - Other fields

Read more here: » Physical law: Encyclopedia II - Physical law - Description

axioms: Encyclopedia II - Charles Peirce - Peirce's philosophy

It is not sufficiently recognized that Peirce’s career was that of a scientist, not a philosopher; and that during his lifetime he was known and valued chiefly as a scientist, only secondly as a logician, and scarcely at all as a philosopher. Even his work in philosophy and logic will not be understood until this fact becomes a standing premise of Peircian studies. (Max Fisch, in Moore and Robin 1964: 486). Upon this first, and in one sense sole, rule of reason, that in order to learn you must desire to learn, and in so desiring not ...

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Charles Peirce, Charles Peirce - Life, Charles Peirce - Reception, Charles Peirce - Works, Charles Peirce - Major publications, Charles Peirce - Peirce's philosophy, Charles Peirce - Pragmatism, Charles Peirce - Scholastic realism, Charles Peirce - Formal perspective, Charles Peirce - Dynamics of representation, Charles Peirce - Normative sciences, Charles Peirce - Parallels with Leibniz, Charles Peirce - Bibliography, Charles Peirce - Primary literature, Charles Peirce - Secondary literature

Read more here: » Charles Peirce: Encyclopedia II - Charles Peirce - Peirce's philosophy

axioms: Encyclopedia II - Inner product space - Norms on inner product spaces

Inner product spaces have a naturally defined norm This is well defined by the nonnegativity axiom of the definition of inner product space. The norm is thought of as the length of the vector x. Directly from the axioms, we can prove the following: Cauchy-Schwarz inequality: for x, y elements of V with equality if and only if x and y are linearly dependent. This is one of the most important inequal ...

See also:

Inner product space, Inner product space - Definitions, Inner product space - Examples, Inner product space - Norms on inner product spaces, Inner product space - Orthonormal sequences, Inner product space - Operators on inner product spaces, Inner product space - Degenerate inner products

Read more here: » Inner product space: Encyclopedia II - Inner product space - Norms on inner product spaces

axioms: Encyclopedia II - Vector space - Formal definition

A vector space over a field F (such as the field of real numbers or the field of complex numbers) is a set V together with two operations, vector addition, denoted v + w, where v, w ∈ V, mapping V × V (see Cartesian product) back into V, and scalar multiplication, denoted a v, where a ∈ F and v ∈ V, ...

See also:

Vector space, Vector space - Formal definition, Vector space - Elementary properties, Vector space - Examples, Vector space - Subspaces and bases, Vector space - Linear transformations, Vector space - Generalizations and additional structures

Read more here: » Vector space: Encyclopedia II - Vector space - Formal definition

axioms: Encyclopedia II - Imre Lakatos - Research programs

Lakatos' contribution to the philosophy of science was an attempt to resolve the perceived conflict between Popper's Falsificationism and the revolutionary structure of science described by Kuhn. Popper's theory implied that scientists should give up a theory as soon as they encounter any falsifying evidence, immediately replacing it with increasingly 'bold and powerful' new hypotheses. However, Kuhn described science as consisting of periods of normal science in which scientists continue to hold their theories in the face of anomalies, int ...

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Imre Lakatos, Imre Lakatos - Life, Imre Lakatos - Proofs and refutations, Imre Lakatos - Research programs, Imre Lakatos - Notes, Imre Lakatos - Selected works, Imre Lakatos - Further Information

Read more here: » Imre Lakatos: Encyclopedia II - Imre Lakatos - Research programs

axioms: Encyclopedia II - Kuratowski closure axioms - Definition

A topological space (X,cl) is a set X with a function called the closure operator where is the power set of X. The closure operator has to satisfy the following properties (Extensivity) (Idempotence) (Preservation of binary u ...

See also:

Kuratowski closure axioms, Kuratowski closure axioms - Definition, Kuratowski closure axioms - Notes, Kuratowski closure axioms - Recovering topological definitions

Read more here: » Kuratowski closure axioms: Encyclopedia II - Kuratowski closure axioms - Definition

axioms: Encyclopedia II - David Hilbert - Major contributions

Hilbert solved several important problems in the theory of invariants. Hilbert's basis theorem solved the principal problem in nineteenth century invariant theory by showing that any form of a given number of variables and of a given degree has a finite, yet complete system of independent rational integral invariants and covariants. He also unified the field of algebraic number theory with his 1897 treatis ...

See also:

David Hilbert, David Hilbert - Major contributions, David Hilbert - Miscellaneous talks essays and contributions, David Hilbert - Hilbert's program, David Hilbert - Later years, David Hilbert - Notes

Read more here: » David Hilbert: Encyclopedia II - David Hilbert - Major contributions

axioms: Encyclopedia II - Magma algebra - Free magma

A free magma on a set X is the "most general possible" magma generated by the set X (that is there are no relations or axioms imposed on the generators; see free object). It can be described, in terms familiar in computer science, as the magma of binary trees with leaves labelled by elements of X. The operation is that of joining trees at the root. It therefore has a foundational role in syntax. ...

See also:

Magma algebra, Magma algebra - Types of magmas, Magma algebra - Free magma, Magma algebra - More definitions

Read more here: » Magma algebra: Encyclopedia II - Magma algebra - Free magma

axioms: Encyclopedia II - Category theory - Background

The study of categories is an attempt to capture what is commonly found in various classes of related mathematical structures. Consider the following example. The class Grp of groups consists of all objects having a "group structure". More precisely, Grp consists of all sets G endowed with a binary operation satisfying a certain set of axioms. One can proceed to prove theorems about groups by making logical deductions from the set of axioms. For example, it is immediately proved from the axioms t ...

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Category theory, Category theory - Background, Category theory - Historical notes, Category theory - Categories objects and morphisms, Category theory - Some properties of morphisms, Category theory - Functors, Category theory - Natural transformations and isomorphisms, Category theory - Universal constructions limits and colimits, Category theory - Equivalent categories, Category theory - Further concepts and results, Category theory - Higher-dimensional categories

Read more here: » Category theory: Encyclopedia II - Category theory - Background

axioms: Encyclopedia II - Boolean algebra - Formal definition

A Boolean algebra is a set A, supplied with two binary operations (logical AND), (logical OR), a unary operation (logical NOT) and two elements 0 (logical FALSE) and 1 (logical TRUE), such that, for all elements a, b and c of set A, the following axioms hold: associativity commutativity absorption distributivity ...

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Boolean algebra, Boolean algebra - Formal definition, Boolean algebra - Examples, Boolean algebra - Order theoretic properties, Boolean algebra - Principle of duality, Boolean algebra - Other notation, Boolean algebra - Homomorphisms and isomorphisms, Boolean algebra - Boolean rings ideals and filters, Boolean algebra - Representing Boolean algebras, Boolean algebra - Axiomatics for Boolean algebras

Read more here: » Boolean algebra: Encyclopedia II - Boolean algebra - Formal definition

axioms: Encyclopedia II - Incidence geometry - Intersection of a pair of lines

Let L1 and L2 be a pair of lines, both in a projective plane and expressed in homogeneous coordinates: L1:[m1:b1:1]L L2:[m2:b2:1]L where m1 and m2 are slopes and b1 and b2 are y-interce ...

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Incidence geometry, Incidence geometry - Intersection of a pair of lines, Incidence geometry - Determining the line passing through a pair of points, Incidence geometry - Checking for incidence of a line on a point, Incidence geometry - Concurrence, Incidence geometry - Collinearity

Read more here: » Incidence geometry: Encyclopedia II - Incidence geometry - Intersection of a pair of lines

axioms: Encyclopedia II - Morphism - Definition

A category C is given by two pieces of data: a class of objects and a class of morphisms. There are two operations defined on every morphism, the domain (or source) and the codomain (or target). Morphisms are often depicted as arrows from their domain to their codomain, e.g. if a morphism f has domain X and codomain Y, it is denoted f : X → Y. The set of all morphisms from X to Y is denoted homC(X ...

See also:

Morphism, Morphism - Definition, Morphism - Types of morphisms, Morphism - Examples

Read more here: » Morphism: Encyclopedia II - Morphism - Definition

axioms: Encyclopedia II - David Hilbert - Major contributions

Hilbert solved several important problems in the theory of invariants. Hilbert's basis theorem solved the principal problem in nineteenth century invariant theory by showing that any form of a given number of variables and of a given degree has a finite, yet complete system of independent rational integral invariants and covariants. He also unified the field of algebraic number theory with his 1897 treatis ...

See also:

David Hilbert, David Hilbert - Major contributions, David Hilbert - Miscellaneous talks essays and contributions, David Hilbert - Hilbert's program, David Hilbert - Later years

Read more here: » David Hilbert: Encyclopedia II - David Hilbert - Major contributions

axioms: Encyclopedia II - First-order logic - Formation rules

The formation rules define the terms, formulas, and the free variables in them as follows. The set of terms is recursively defined by the following rules: Any constant is a term (with no free variables). Any variable is a term (whose only free variable is itself). Any expression f(t1,...,tn) of n≥1 arguments (where each argument ti is a term and f is a function symbol of valence n) is a term. I ...

See also:

First-order logic, First-order logic - Defining first-order logic, First-order logic - Vocabulary, First-order logic - Formation rules, First-order logic - Equality, First-order logic - Inference rules, First-order logic - Quantifier axioms, First-order logic - The predicate calculus, First-order logic - Metalogical theorems of first-order logic, First-order logic - Comparison with other logics

Read more here: » First-order logic: Encyclopedia II - First-order logic - Formation rules

axioms: Encyclopedia II - List of publications in mathematics - Geometry

List of publications in mathematics - Euclid's Elements. Euclid Publication data: c. 300 BC Online version: Interactive Java version Description: This is probably not only the most important work in geometry but the most important work in mathematics. It contains many important results in geometry, number theory and the first algorithm as well. The Elements is still a valuable resource and a good introduction to algorithm. More tha ...

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List of publications in mathematics, List of publications in mathematics - Geometry, List of publications in mathematics - Euclid's Elements, List of publications in mathematics - La Géométrie, List of publications in mathematics - Logic, List of publications in mathematics - Begriffsschrift, List of publications in mathematics - Formulario mathematico, List of publications in mathematics - Principia Mathematica, List of publications in mathematics - Gödel's incompleteness theorem, List of publications in mathematics - Information theory, List of publications in mathematics - Number theory, List of publications in mathematics - Disquisitiones Arithmeticae, List of publications in mathematics - On the Number of Primes Less Than a Given Magnitude, List of publications in mathematics - Vorlesungen über Zahlentheorie, List of publications in mathematics - Number Theory An approach through history from Hammurapi to Legendre, List of publications in mathematics - Calculus, List of publications in mathematics - Philosophiae Naturalis Principia Mathematica, List of publications in mathematics - Newton's Principia for the Common Reader, List of publications in mathematics - Calculus and Calculus on Manifolds, List of publications in mathematics - Numerical analysis, List of publications in mathematics - Method of Fluxions, List of publications in mathematics - Game theory, List of publications in mathematics - Evolution and the Theory of Games, List of publications in mathematics - Theory of Games and Economic Behavior, List of publications in mathematics - On Numbers and Games, List of publications in mathematics - Winning Ways for your Mathematical Plays, List of publications in mathematics - Fractals, List of publications in mathematics - How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension, List of publications in mathematics - Early manuscripts, List of publications in mathematics - Rhind Mathematical Papyrus, List of publications in mathematics - The Nine Chapters on the Mathematical Art, List of publications in mathematics - Archimedes Palimpsest, List of publications in mathematics - The Sand Reckoner, List of publications in mathematics - Textbooks, List of publications in mathematics - Course of Pure Mathematics, List of publications in mathematics - Art of Problem Solving, List of publications in mathematics - Metalogic: an Introduction to the Metatheory of Standard First Order Logic, List of publications in mathematics - Popular writing, List of publications in mathematics - Gödel Escher Bach, List of publications in mathematics - The World of Mathematics, List of publications in mathematics - Arithmetic, List of publications in mathematics - Arithmetick: or The Grounde of Arts, List of publications in mathematics - The Schoolmaster's Assistant Being a Compendium of Arithmetic both Practical and Theoretical, List of publications in mathematics - Abstract algebra, List of publications in mathematics - Moderne Algebra, List of publications in mathematics - Linear algebra, List of publications in mathematics - Algebraic geometry, List of publications in mathematics - Faisceaux Algébriques Cohérents, List of publications in mathematics - Géométrie Algébrique et Géométrie Analytique, List of publications in mathematics - Éléments de géométrie algébrique, List of publications in mathematics - Séminaire de géométrie algébrique, List of publications in mathematics - Algebraic Geometry, List of publications in mathematics - Universal algebra, List of publications in mathematics - Universal algebra, List of publications in mathematics - Group theory, List of publications in mathematics - Monoid, List of publications in mathematics - Topology, List of publications in mathematics - Topologie, List of publications in mathematics - Topology, List of publications in mathematics - General Topology, List of publications in mathematics - Graph theory, List of publications in mathematics - Category theory, List of publications in mathematics - Categories for the Working Mathematician, List of publications in mathematics - Category Theory for Computing Science, List of publications in mathematics - Order theory, List of publications in mathematics - Trigonometry, List of publications in mathematics - Differential geometry, List of publications in mathematics - Differential topology, List of publications in mathematics - Topology from the Differentiable Viewpoint, List of publications in mathematics - Algebraic topology, List of publications in mathematics - Algebraic Topology, List of publications in mathematics - Fractal geometry, List of publications in mathematics - Discrete mathematics, List of publications in mathematics - Combinatorics, List of publications in mathematics - Set theory, List of publications in mathematics - Grundzüge der Mengenlehre, List of publications in mathematics - Naive Set Theory, List of publications in mathematics - Cardinal and Ordinal Numbers, List of publications in mathematics - The Consistency of the Continuum Hypothesis, List of publications in mathematics - Set Theory and the Continuum Hypothesis, List of publications in mathematics - Set Theory: An Introduction to Independence Proofs, List of publications in mathematics - Optimization, List of publications in mathematics - The New Variational Method, List of publications in mathematics - Decomposition Principle for Linear Programs., List of publications in mathematics - Network Flows and General Matchings, List of publications in mathematics - Paths trees and Flowers, List of publications in mathematics - The complexity of theorem proving procedures, List of publications in mathematics - Reducibility among combinatorial problems, List of publications in mathematics - How good is the simplex algorithm?, List of publications in mathematics - Linear Programming and Polynomial time algorithms, List of publications in mathematics - New polynomial-time algorithm for linear programming, List of publications in mathematics - Interior Point Polynomial Algorithms in Convex Programming

Read more here: » List of publications in mathematics: Encyclopedia II - List of publications in mathematics - Geometry

axioms: Encyclopedia II - Albert Einstein - Biography

Albert Einstein - Youth and college. Einstein was born on March 14, 1879 at Ulm in Baden-Württemberg, Germany, about 100 km east of Stuttgart. His parents were Hermann Einstein, a featherbed salesman who later ran an electrochemical works, and Pauline, whose maiden name was Koch. They were married in Stuttgart-Bad Cannstatt. The family was Jewish (non-observant); Albert attended a Catholic elementary school and, at the insist ...

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Albert Einstein, Albert Einstein - Biography, Albert Einstein - Youth and college, Albert Einstein - Work and doctorate, Albert Einstein - Middle years, Albert Einstein - Final years, Albert Einstein - Personality, Albert Einstein - Religious views, Albert Einstein - Political views, Albert Einstein - Popularity and cultural impact, Albert Einstein - Entertainment, Albert Einstein - Licensing, Albert Einstein - Honors, Albert Einstein - Notes, Albert Einstein - Works by Albert Einstein

Read more here: » Albert Einstein: Encyclopedia II - Albert Einstein - Biography

axioms: Encyclopedia II - Edgar Allan Poe - Legacy and lore

Poe's works have had a broad influence on American and World literature (sometimes even despite those who tried to resist it), and even on the art world beyond literature. The scope of Poe's impact on art is evident when one sees the many and diverse artists who were directly and profoundly influenced by him. He is often credited as being an originator the genre of detective fiction with his three stories about Auguste Dupin, the most famous of which is "The Murders in the Rue Morgue." (Poe also wrote a satirical detective sto ...

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Edgar Allan Poe, Edgar Allan Poe - The life of Edgar Allan Poe, Edgar Allan Poe - Death, Edgar Allan Poe - Memoir - Griswold's biography of Edgar Allan Poe, Edgar Allan Poe - Literary and artistic theory, Edgar Allan Poe - Legacy and lore, Edgar Allan Poe - Notable works, Edgar Allan Poe - Poems, Edgar Allan Poe - Short Stories, Edgar Allan Poe - Longer Works, Edgar Allan Poe - Essays, Edgar Allan Poe - Play, Edgar Allan Poe - Adaptations, Edgar Allan Poe - Poe as a character, Edgar Allan Poe - Notes, Edgar Allan Poe - General references

Read more here: » Edgar Allan Poe: Encyclopedia II - Edgar Allan Poe - Legacy and lore

axioms: Encyclopedia II - Erlangen program - The problems of nineteenth century geometry

Was there one 'geometry' or many? Since Euclid, geometry had meant the geometry of Euclidean space of two dimensions (plane geometry) or of three dimensions (solid geometry). In the first half of the nineteenth century there had been several developments complicating the picture. Mathematical applications required geometry of four or more dimensions; the close scrutiny of the foundations of the traditional Euclidean geometry had revealed the independence of the Parallel Axiom from the others, and non-Euclidean geometry had been born; and in projective geometry new 'points' (p ...

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Erlangen program, Erlangen program - The problems of nineteenth century geometry, Erlangen program - Homogeneous spaces, Erlangen program - Examples: affine geometry, Erlangen program - Influence on later work, Erlangen program - Abstract returns from the Erlangen program

Read more here: » Erlangen program: Encyclopedia II - Erlangen program - The problems of nineteenth century geometry

axioms: Encyclopedia II - Dedekind-infinite set - Comparison with the usual definition of infinite set

This definition of "infinite set" should be compared and contrasted to the usual definition: a set A is finite if A is empty, or if there is a positive integer n such that A is equinumerous to the set {1, 2, 3, ..., n}. Explicitly, this means that there is a bijection between A and some member of ω, where ω is defined to be the intersection of all sets which contain the empty se ...

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Dedekind-infinite set, Dedekind-infinite set - Comparison with the usual definition of infinite set, Dedekind-infinite set - Dedekind-infinite sets in ZF, Dedekind-infinite set - Relation to AC and ACω, Dedekind-infinite set - History

Read more here: » Dedekind-infinite set: Encyclopedia II - Dedekind-infinite set - Comparison with the usual definition of infinite set

axioms: Encyclopedia II - Coalgebra - Further concepts and facts

A coalgebra (C,Δ,ε) is called co-commutative if σoΔ = Δ, where σ : C⊗C → C⊗C is the K-linear map defined by σ(c⊗d) = d⊗c for all c,d in C. In Sweedler's sumless notation, C is co-commutative if and only if for all c in C. (It's important to understand that the implied summation is significant here: we are not requiring that all the summands are pairwise equal, o ...

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Coalgebra, Coalgebra - Formal definition, Coalgebra - Examples, Coalgebra - Sweedler notation, Coalgebra - Further concepts and facts

Read more here: » Coalgebra: Encyclopedia II - Coalgebra - Further concepts and facts

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