axioms - Article Index

Encyclopedia - Construction Of Real Numbers: Encyclopedia Ii - Construction Of Real Numbers - Synthetic Approach
The synthetic approach axiomatically defines the real number system as a complete ordered field. Precisely, this means the following. A m...   » Read the article

Encyclopedia - Cohomology: Encyclopedia - Cohomology
In mathematics, specifically in algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a cochain ...   » Read the article

Encyclopedia - Well-defined: Encyclopedia - Well-defined
In mathematics, the term well-defined is used to specify that a certain concept (a function, a property, a relation, etc.) is defined in ...   » Read the article

Encyclopedia - Existence Of God: Encyclopedia - Existence Of God
Many arguments about the existence of God have been proposed by philosophers, theologians, and other thinkers. This article lists some of...   » Read the article

Encyclopedia - Common Sense: Encyclopedia - Common Sense
One meaning of the term common sense (or as an adjective, commonsense) on a strict construction of the term, is what people in common wou...   » Read the article

Encyclopedia - Albert Einstein: Encyclopedia - Albert Einstein
Albert Einstein (March 14, 1879–April 18, 1955) was a Jewish theoretical physicist, born in Ulm, Germany, who is widely regarded as the...   » Read the article

Encyclopedia - Vector Space: Encyclopedia - Vector Space
A vector space (or linear space) is the basic object of study in the branch of mathematics called linear algebra. If one considers geomet...   » Read the article

Encyclopedia - Empiricism: Encyclopedia - Empiricism
Empiricism comes from the Greek word εμπειρισμός, a noun meaning a "test" or "trial". The -pir- is ultimately related to the -...   » Read the article

Encyclopedia - Dogma: Encyclopedia - Dogma
Dogma (the plural is either dogmata or dogmas) is belief or doctrine held by a religion or any kind of organization to be authoritative a...   » Read the article

Encyclopedia - Value Judgment: Encyclopedia - Value Judgment
A value judgment is a judgment of the rightness or wrongness of something based on a particular set of values or on a particular value sy...   » Read the article

Encyclopedia - David Hilbert: Encyclopedia - David Hilbert
David Hilbert (January 23, 1862 – February 14, 1943) was a German mathematician born in Wehlau, near Königsberg, Prussia (now Znamensk...   » Read the article

Encyclopedia - Charles Peirce: Encyclopedia - Charles Peirce
Charles Sanders Santiago Peirce (pronounced purse), (September 10, 1839, Cambridge, Massachusetts – April 19, 1914, Milford, Pennsylvan...   » Read the article

Encyclopedia - Abelian Category: Encyclopedia - Abelian Category
In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist an...   » Read the article

Encyclopedia - Universal Algebra: Encyclopedia - Universal Algebra
Universal algebra is the field of mathematics that studies the ideas common to all algebraic structures. Universal algebra - Basic idea. ...   » Read the article

Encyclopedia - Mathematics: Encyclopedia - Mathematics
Mathematics is often defined as the study of topics such as quantity, structure, space, and change. Another view, held by many mathematic...   » Read the article

Encyclopedia - Convex Set: Encyclopedia - Convex Set
In Euclidean space, an object is convex if for any pair of points within the object, any point on the straight line segment that joins th...   » Read the article

Encyclopedia - Theorem: Encyclopedia - Theorem
A theorem is a proposition that has been or is to be proved on the basis of explicit assumptions. Proving theorems is a central activity ...   » Read the article

Encyclopedia - Functor: Encyclopedia - Functor
In category theory, a functor is a special type of mapping between categories. Functors can be thought of as morphisms in the category of...   » Read the article

Encyclopedia - Construction Of Real Numbers: Encyclopedia - Construction Of Real Numbers
In mathematics, there are a number of ways of defining the real number system as an ordered field. The synthetic approach gives a list of...   » Read the article

Encyclopedia - Bertrand Russell: Encyclopedia - Bertrand Russell
The Right Honourable Bertrand Arthur William Russell, 3rd Earl Russell, OM, FRS (18 May 1872 – 2 February 1970), was an influential Bri...   » Read the article

Encyclopedia - Axiomatization: Encyclopedia - Axiomatization
In mathematics, axiomatization is the process of defining the basic axiomatic systems from which mathematical theories can be derived. Ev...   » Read the article

Encyclopedia - Axiomatic Set Theory: Encyclopedia - Axiomatic Set Theory
Set theory is a branch of mathematics created principally by the German mathematician Georg Cantor at the end of the 19th century. Initia...   » Read the article

Encyclopedia - Boolean Algebra: Encyclopedia - Boolean Algebra
In abstract algebra, a Boolean algebra is an algebraic structure (a collection of elements and operations on them obeying defining axioms...   » Read the article

Encyclopedia - Category Theory: Encyclopedia - Category Theory
Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. It is...   » Read the article

Encyclopedia - Completeness: Encyclopedia - Completeness
In mathematics and related technical fields, a mathematical object is complete if nothing needs to be added to it. This is made precise i...   » Read the article

Encyclopedia - Zermelo-fraenkel Set Theory: Encyclopedia - Zermelo-fraenkel Set Theory
The Zermelo-Fraenkel axioms of set theory together with the axiom of choice are the standard axioms of axiomatic set theory. All of ordin...   » Read the article

Encyclopedia - Theory: Encyclopedia - Theory
Theory has a number of distinct meanings in different fields of knowledge, depending on the context and their methodologies. Theory - Ety...   » Read the article

Encyclopedia - Axiomatic System: Encyclopedia - Axiomatic System
In mathematics, an axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derive the...   » Read the article

Encyclopedia - Axiom Of Infinity: Encyclopedia - Axiom Of Infinity
In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of infinity is one of the axi...   » Read the article

Encyclopedia - Answers In Genesis: Encyclopedia - Answers In Genesis
Answers in Genesis (AiG) is a not-for-profit Christian apologetics ministry with a particular focus on Young Earth Creationism, and a lit...   » Read the article

Encyclopedia - Areas Of Mathematics: Encyclopedia - Areas Of Mathematics
The aim of this page is to list all areas of modern mathematics, with a brief explanation about their scope and links to other parts of t...   » Read the article

Encyclopedia - Axiom Schema: Encyclopedia - Axiom Schema
In symbolic logic, it is sometimes inconvenient or impossible to express an axiomatic system in a finite number of axioms. For this reaso...   » Read the article

Encyclopedia - Sublime Philosophy: Encyclopedia - Sublime Philosophy
In aesthetics, the sublime (from the Latin sublimis (exalted)) is the quality of transcendent greatness, whether physical, moral, intelle...   » Read the article

Encyclopedia - Annus Mirabilis Papers: Encyclopedia - Annus Mirabilis Papers
The Annus Mirabilis Papers (from Annus mirabilis, Latin for 'year of wonders') are the papers of Albert Einstein submitted to the "Annale...   » Read the article

Encyclopedia - Axiomatic: Encyclopedia - Axiomatic
Axiomatic. In mathematics, an axiomatic theory is one based on axioms. Axiomatic is a collection of short stories by Greg Egan. Axiomat...   » Read the article

Encyclopedia - Group Mathematics: Encyclopedia Ii - Group Mathematics - Some Elementary Examples And Nonexamples
Group mathematics - An abelian group: the integers under addition. A group that we are introduced to in elementary school is the intege...   » Read the article

Encyclopedia - Controversy Over Cantor's Theory: Encyclopedia Ii - Controversy Over Cantor's Theory - Objections To Hume's Principle
As argued above, many naïve objections depend on implicitly denying Hume's principle, and are therefore question-begging. Wittgenstein e...   » Read the article

Encyclopedia - Euclid's Elements: Encyclopedia Ii - Euclid's Elements - First Principles
Euclid based his work in Book I on 23 definitions, such as point, line and surface, five postulates and five "common notions" (both of wh...   » Read the article

Encyclopedia - Euclid: Encyclopedia Ii - Euclid - The Elements
Although many of the results in Elements originated with earlier mathematicians, one of Euclid's accomplishments was to present them in a...   » Read the article

Encyclopedia - Hoare Logic: Encyclopedia Ii - Hoare Logic - Partial Correctness
Hoare logic - Assignment axiom schema. The assignment axiom states that after the assignment any predicate holds for the variable that ...   » Read the article

Encyclopedia - Euclidean Geometry: Encyclopedia Ii - Euclidean Geometry - Axiomatic Approach
The traditional presentation of Euclidean geometry is as an axiomatic system, setting out to prove all the "true statements" as theorems ...   » Read the article

Encyclopedia - List Of 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...   » Read the article

Encyclopedia - Cissoid Of Diocles: 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 l...   » Read the article

Encyclopedia - Physical Law: 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 cha...   » Read the article

Encyclopedia - Charles Peirce: 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 ...   » Read the article

Encyclopedia - Inner Product Space: 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 ...   » Read the article

Encyclopedia - Vector Space: 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...   » Read the article

Encyclopedia - Imre Lakatos: 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...   » Read the article

Encyclopedia - Kuratowski Closure 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 h...   » Read the article

Encyclopedia - David Hilbert: 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...   » Read the article

Encyclopedia - Magma Algebra: 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 ...   » Read the article

Encyclopedia - Category Theory: 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 t...   » Read the article

Encyclopedia - Boolean Algebra: 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...   » Read the article

Encyclopedia - Incidence Geometry: 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 wh...   » Read the article

Encyclopedia - Morphism: 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 morph...   » Read the article

Encyclopedia - David Hilbert: 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...   » Read the article

Encyclopedia - First-order Logic: 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...   » Read the article

Encyclopedia - List Of Publications In Mathematics: 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 ...   » Read the article

Encyclopedia - Albert Einstein: 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 St...   » Read the article

Encyclopedia - Edgar Allan Poe: 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 o...   » Read the article

Encyclopedia - Erlangen Program: 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 ...   » Read the article

Encyclopedia - Dedekind-infinite Set: 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...   » Read the article

Encyclopedia - Coalgebra: 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) = ...   » Read the article

Encyclopedia - Socratic Method: Encyclopedia Ii - Socratic Method - Application
Socrates generally applied his method of examination to concepts that seem to lack any concrete definition; e.g., the key moral concepts ...   » Read the article

Encyclopedia - Parallel Postulate: Encyclopedia Ii - Parallel Postulate - Logically Equivalent Properties
Several properties of Euclidean geometry are logically equivalent to Euclid's parallel postulate, meaning that they can be proven in a sy...   » Read the article

Encyclopedia - Topological Space: Encyclopedia Ii - Topological Space - Definition
A topological space is a set X together with a collection T of subsets of X satisfying the following axioms: The empty set and X are in ...   » Read the article

Encyclopedia - Bertrand Russell: Encyclopedia Ii - Bertrand Russell - Russell's Philosophical Work
Bertrand Russell - Analytic philosophy. Russell is generally recognised as one of the founders of analytic philosophy, indeed, even of ...   » Read the article

Encyclopedia - Numerical Analysis: Encyclopedia Ii - Numerical Analysis - General Introduction
Many problems in continuous mathematics do not possess a closed-form solution. Examples are finding the integral of exp(−x2) (see error...   » Read the article

Encyclopedia - Separation Axiom: Encyclopedia Ii - Separation Axiom - Definitions Of The Axioms
Many of these names have alternative meanings in some of mathematical literature, as explained on History of the separation axioms; for e...   » Read the article

Encyclopedia - Non-euclidean Geometry: Encyclopedia Ii - Non-euclidean Geometry - History
While Euclidean geometry (named for the Greek mathematician Euclid) includes some of the oldest known mathematics, non-Euclidean geometri...   » Read the article

Encyclopedia - Peano Axioms: Encyclopedia Ii - Peano Axioms - Categorical Interpretation
The Peano axioms may be interpreted in the general context of category theory. Let US1 be the category of pointed unary systems; i.e. US1...   » Read the article

Encyclopedia - Theoretical Physics: Encyclopedia Ii - Theoretical Physics - Overview
Theoretical physics is the study of matter and energy through the development of theory and was begun 2300 years ago by the ancient Greek...   » Read the article

Encyclopedia - Vector Space: Encyclopedia Ii - Vector Space - Formal Definition
A vector space over a field F (such as the field of real or of complex numbers) is a set V together with two operations: vector addition...   » Read the article

Encyclopedia - Zermelo-fraenkel Set Theory: Encyclopedia Ii - Zermelo-fraenkel Set Theory - The Axioms
The axioms of ZFC are: Axiom of extensionality: Two sets are the same if and only if they have the same elements. Axiom of empty s...   » Read the article

Encyclopedia - Proof Theory: Encyclopedia Ii - Proof Theory - History
Although the formalisation of logic was much advanced by the work of such figures as Gottlob Frege, Peano, Russell and Dedekind, conventi...   » Read the article

Encyclopedia - Principles Of Nlp: Encyclopedia Ii - Principles Of Nlp - Specific Principles Within Nlp
NLP is a divergent subject, and so different individuals will have different formulations for what they consider "principles of NLP". How...   » Read the article

Encyclopedia - Universal Algebra: Encyclopedia Ii - Universal Algebra - Examples
Universal algebra - Groups. To see how this is supposed to work, let's consider the definition of a group. Normally a group is defined ...   » Read the article

Encyclopedia - Presuppositional Apologetics: Encyclopedia Ii - Presuppositional Apologetics - Varieties Of Presuppositionalism
Presuppositional apologetics - Van Tillian presuppositionalism. Apologists who follow Van Til earned the label "presuppositional" becau...   » Read the article

Encyclopedia - Naive Set Theory: Encyclopedia Ii - Naive Set Theory - Unions Intersections And Relative Complements
Given two sets A and B, we may construct their union. This is the set consisting of all objects which are elements of A or of B or of bot...   » Read the article

Encyclopedia - Quantum Field Theory: Encyclopedia Ii - Quantum Field Theory - Quantizing A Classical Field Theory
Quantum field theory - Canonical quantization. Quantum field theory solves these problems by consistently quantizing a field. By interp...   » Read the article

Encyclopedia - Albert Einstein: 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 St...   » Read the article

Encyclopedia - Nikolai Ivanovich Lobachevsky: Encyclopedia Ii - Nikolai Ivanovich Lobachevsky - Biography
Lobachevsky was born in Nizhny Novgorod, Russia. His parents were Ivan Maksimovich Lobachevsky, a clerk in a landsurveying office, and Pr...   » Read the article

Encyclopedia - Associative Algebra: Encyclopedia Ii - Associative Algebra - Definition
An associative algebra A over a field K is defined to be a vector space over K together with a K-bilinear multiplication A x A → A (whe...   » Read the article

Encyclopedia - Axiom Schema Of Specification: Encyclopedia Ii - Axiom Schema Of Specification - Relation To The Axiom Schema Of Replacement
The axiom schema of separation can almost be derived from the axiom schema of replacement. First, recall this axiom schema: for any fu...   » Read the article

Encyclopedia - Axiom Schema Of Replacement: Encyclopedia Ii - Axiom Schema Of Replacement - Statement
Suppose P is any predicate in two variables that doesn't use the symbol B. Then in the formal language of the Zermelo-Fraenkel axioms, th...   » Read the article

Encyclopedia - Axiom Of Extensionality: Encyclopedia Ii - Axiom Of Extensionality - In Predicate Logic Without Equality
The axiom given above assumes that equality is a primitive symbol in predicate logic. Some treatments of axiomatic set theory prefer to d...   » Read the article

Encyclopedia - Rule Of Inference: Encyclopedia Ii - Rule Of Inference - Admissibility And Derivability
In a set of rules, an inference rule could be redundant in the sense that it is admissible or derivable. A derivable rule is one whose co...   » Read the article

Encyclopedia - Romantic Music: Encyclopedia Ii - Romantic Music - Brief Chronology Of Musical Romanticism
Romantic music - Classical roots of Romanticism 1780-1815. In literature, the Romantic period is often said to begin in the 1770s or 17...   » Read the article

Encyclopedia - Monotonic Function: Encyclopedia Ii - Monotonic Function - Monotonicity In Calculus And Analysis
In calculus, there is often no need to call upon the abstract methods of order theory. As already noted, functions are usually mappings b...   » Read the article

Encyclopedia - Morphism: 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 morph...   » Read the article

Encyclopedia - Mereology: Encyclopedia Ii - Mereology - Mereology And Set Theory
It is possible to formulate a "naive mereology" analogous to naive set theory, and possible to generate paradoxes analogous to Russell's ...   » Read the article

Encyclopedia - Mathematics: Encyclopedia Ii - Mathematics - Major Themes In Mathematics
An alphabetical and subclassified list of mathematics articles is available. The following list of themes and links gives just one possib...   » Read the article

Encyclopedia - Reverse Mathematics: Encyclopedia Ii - Reverse Mathematics - Second-order Arithmetic
This section is somewhat technical and tries to describe precisely the usual framework for reverse mathematics (namely, subsystems of sec...   » Read the article

Encyclopedia - Blum Axioms: Encyclopedia Ii - Blum Axioms - Blum Axioms
A Blum complexity measure is a tuple with a Gödel numbering of the partial computable functions and a computable function so that ...   » Read the article

Encyclopedia - Elementary Group Theory: Encyclopedia Ii - Elementary Group Theory - Latin Square Property
Theorem 1.3: For all a,b in G, there exists a unique x in G such that a*x = b. Certainly, at least one such x exists, for if we let x = ...   » Read the article

Encyclopedia - Axiomatic System: Encyclopedia Ii - Axiomatic System - Axiomatic Method
The axiomatic method is often discussed as if it were a unitary approach, or uniform procedure. With the example of Euclid to appeal to, ...   » Read the article

Encyclopedia - Axiomatic Set Theory: Encyclopedia Ii - Axiomatic Set Theory - The Origins Of Rigorous Set Theory
The important idea of Cantor's, which got set theory going as a new field of study, was to define two sets A and B to have the same numbe...   » Read the article

Encyclopedia - Unexpected Hanging Paradox: Encyclopedia Ii - Unexpected Hanging Paradox - Discussion
This paradox is unsettling because the prisoner seems to show that the judge is being self-contradictory, yet in the end the judge ends u...   » Read the article

Encyclopedia - Existence Of God: Encyclopedia Ii - Existence Of God - Positions On The Existence Of God
Positions on the existence of God can be roughly divided into two camps: Theist and Atheist. Both of these camps can be further divided i...   » Read the article

Encyclopedia - Dogma: Encyclopedia Ii - Dogma - Dogma Faith And Logic
There are some conceptual similarities between dogma and the axioms used as the starting point for logical analysis. Axioms may be though...   » Read the article

Encyclopedia - Sublime Philosophy: Encyclopedia Ii - Sublime Philosophy - Eighteenth Century
The development of the concept of the sublime as an aesthetic quality distinct from beauty was first brought into prominence in the eight...   » Read the article