The synthetic approach axiomatically defines the real number system as a complete ordered field. Precisely, this means the following. A model for the real number system consists of a set R, two distinct elements 0 and 1 of R, two binary operations + and * on R (called addition and multiplication, resp.), a total order ≤ on R, satisfying the following properties. 1. (R, +, *) forms a field. In other words, For all x, y, and z in RSee also:

Construction of real numbers, Construction of real numbers - Synthetic approach, Construction of real numbers - Explicit constructions of models, Construction of real numbers - Construction from Cauchy sequences, Construction of real numbers - Construction by Dedekind cuts, Construction of real numbers - Construction by decimal expansions, Construction of real numbers - Construction from ultrafilters, Construction of real numbers - Construction from surreal numbers, Construction of real numbers - Construction from the group of integers

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In mathematics, specifically in algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a cochain complex. That is, cohomology is defined as the abstract study of cochains, cocycles and coboundaries. Cohomology can be viewed as a method of assigning algebraic invariants to a topological space that has a more refined algebraic structure than does homology. Cohomology arises from the algebraic dualization of the construction of homology. In less abstract language, cochains in the fundamental sense should assign 'qua ...

Including:

• Cohomology - History
• Cohomology - Cohomology theories
• Cohomology - Eilenberg-Steenrod theories
• Cohomology - Extraordinary cohomology theories
• Cohomology - Other cohomology theories

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In mathematics, the term well-defined is used to specify that a certain concept (a function, a property, a relation, etc.) is defined in a mathematical or logical way using a set of base axioms in an entirely unambiguous way. One of the most common places in mathematics in which the term well-defined is used is in dealing with cosets in group theory. It is as important that we check that we get the same result regardless of which representative of the coset we choose as it is that we always get the same result when we perform arithmet ...

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Many arguments about the existence of God have been proposed by philosophers, theologians, and other thinkers. This article lists some of the more common arguments, especially those covered in the area of philosophy of religion. In philosophical terminology, this article introduces schools of thought on the epistemology of the ontology of God. Existence of God - What is God? Definition of God's existence. See main articles: Definition, God, Deity, Ontology What does it mean to ass ...

Including:

• Existence of God - What is God? Definition of God's existence
• Existence of God - The problem of the supernatural
• Existence of God - How do we know?
• Existence of God - Positions on the Existence of God
• Existence of God - Theism
• Existence of God - Atheism
• Existence of God - Agnosticism
• Existence of God - Arguments for the existence of God
• Existence of God - Inductive arguments for
• Existence of God - Subjective arguments for
• Existence of God - Arguments against the existence of God
• Existence of God - Notes

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One meaning of the term common sense (or as an adjective, commonsense) on a strict construction of the term, is what people in common would agree; that which they "sense" in common as their common natural understanding. Some use the phrase to refer to beliefs or propositions that in their opinion they consider would in most people's experience be prudent and of sound judgment, without dependence upon esoteric knowledge or study or research, but based upon what is believed to be knowledge held by people "in common". The knowledg ...

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• Common sense - Philosophy and common sense
• Common sense - Other uses
• Common sense - Projects to collect common sense

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Albert Einstein (March 14, 1879–April 18, 1955) was a Jewish theoretical physicist, born in Ulm, Germany, who is widely regarded as the greatest scientist of the 20th century. He proposed the theory of relativity and also made major contributions to the development of quantum mechanics, statistical mechanics, and cosmology. He was awarded the 1921 Nobel Prize for Physics for his explanation of the photoelectric effect in 1905 (his "miracle ...

Including:

• 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

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A vector space (or linear space) is the basic object of study in the branch of mathematics called linear algebra. If one considers geometrical vectors, and the operations one can perform upon these vectors such as addition of vectors, scalar multiplication, with some natural constraints such as closure of these operations, associativity of these and combinations of these operations, and so on, we arrive at a description of a m ...

Including:

• 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

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Empiricism comes from the Greek word εμπειρισμός, a noun meaning a "test" or "trial". The -pir- is ultimately related to the -per- of the Latin words experientia and experimentum, both of which mean "experiment," and from which our words "experiment" and "experience" come. (Interestingly, it is also related to the Latin word periculum, "essay, trial, danger," which gives the English word "peril".) Empiricism is therefore the philosophical doctrine (-ism) of "testing" or "experimentation," and has taken ...

Including:

• Empiricism - Empiricism and Science
• Empiricism - Empiricism in history
• Empiricism - Classical Empiricism
• Empiricism - Modern Empiricism
• Empiricism - Radical Empiricism
• Empiricism - Moderate Empiricism
• Empiricism - Other forms
• Empiricism - Criticisms
• Empiricism - Kuhn's The Structure of Scientific Revolutions
• Empiricism - Constructivism
• Empiricism - Quantum mechanics

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Dogma (the plural is either dogmata or dogmas) is belief or doctrine held by a religion or any kind of organization to be authoritative and not to be disputed or doubted. 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 thought of as concepts or "givens" so fundamental that disputing them would be unimaginable; dogmata are also fundamental (e.g. "God exists") y ...

Including:

• Dogma - Dogma faith and logic
• Dogma - Dogma in religion
• Dogma - Dogma outside of religion

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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 system. The term is usually employed, often in a disparaging sense, to imply that a statement is not objectively true, but is 'merely' a value judgment, distinguishing between an objective statement of fact or a conclusion reached logically through rational analysis, and a statement or conclusion reached owing to the specific values or value system held by the one asserting it. Since it can be a ...

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David Hilbert (January 23, 1862 – February 14, 1943) was a German mathematician born in Wehlau, near Königsberg, Prussia (now Znamensk, near Kaliningrad, Russia) who is recognized as one of the most influential mathematicians of the 19th and early 20th centuries. He established his reputation in a broad range of fields including invariant theory, the axiomization of geometry and the foundations of functional analysis. Later in life, he became a world leader in mathematics, exemplified by his presentation, in 1900, of a set of probl ...

Including:

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

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Charles Sanders Santiago Peirce (pronounced purse), (September 10, 1839, Cambridge, Massachusetts – April 19, 1914, Milford, Pennsylvania) was an American polymath. Although educated as a chemist and employed as a scientist for 30 years, he is now mostly seen as a philosopher. He is the greatest American builder of architectonic systems, and his admirers deem him the most important systemat ...

Including:

• Charles Peirce - Life
• Charles Peirce - Reception
• Charles Peirce - Works
• 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

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In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have nice properties. The motivating prototype example of an abelian category is the category of abelian groups, Ab. Abelian category - Definitions. A category is abelian if it has a zero object, it has all pullbacks and pushouts, and all monomorphisms and epimorphisms are normal. By a theorem of Pe ...

Including:

• Abelian category - Definitions
• Abelian category - Examples
• Abelian category - Elementary properties
• Abelian category - Related concepts
• Abelian category - History

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Universal algebra is the field of mathematics that studies the ideas common to all algebraic structures. Universal algebra - Basic idea. From the point of view of universal algebra, an algebra (or abstract algebra) is a set A together with a collection of operations on A. An n-ary operation on A is a function that takes n elements of A and returns a single element of A. Thus, a 0-ary operation (or nullary operation) is simply an el ...

Including:

• Universal algebra - Basic idea
• Universal algebra - Examples
• Universal algebra - Groups
• Universal algebra - Modules
• Universal algebra - Further issues

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Mathematics is often defined as the study of topics such as quantity, structure, space, and change. Another view, held by many mathematicians, is that mathematics is the body of knowledge justified by deductive reasoning, starting from axioms and definitions. Practical mathematics, in nearly every society, is used for such purposes as accounting, measuring land, or predicting astronomical events. Mathematical discovery or research often involves discovering and cataloging patterns, without regard for application. Today, the natural sciences, engineering, economics, and medici ...

Including:

• Mathematics - History
• Mathematics - Inspiration pure and applied mathematics and aesthetics
• Mathematics - Notation language and rigor
• Mathematics - Is mathematics a science?
• Mathematics - Overview of fields of mathematics
• Mathematics - Major themes in mathematics
• Mathematics - Quantity
• Mathematics - Structure
• Mathematics - Space
• Mathematics - Change
• Mathematics - Foundations and methods
• Mathematics - Discrete mathematics
• Mathematics - Applied mathematics
• Mathematics - Important theorems
• Mathematics - Important conjectures
• Mathematics - History and the world of mathematicians
• Mathematics - Mathematics and other fields
• Mathematics - Common misconceptions

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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 them is also within the object. For example, a solid cube is convex, but anything that is hollow or has a dent in it is not convex. Convex set - Convex sets. Let C be a set in a real or complex vector space. C is said to be convex if, for all x and y in C and all t in the interval [0,1], the point (1 − t) ...

Including:

• Convex set - Convex sets
• Convex set - Properties of convex sets
• Convex set - Non-Euclidean geometry
• Convex set - Generalized convexity
• Convex set - Orthogonal convexity
• Convex set - Abstract axiomatic convexity

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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 of mathematicians. Note that "theorem" is distinct from "theory". A theorem generally has a set-up – a number of conditions, which may be listed in the theorem or described beforehand. Then it has a conclusion – a mathematical statement which is true under the given set up. The proof, though necessary to the statement's classification as a theor ...

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In category theory, a functor is a special type of mapping between categories. Functors can be thought of as morphisms in the category of small categories. Functors were first considered in algebraic topology, where algebraic objects (like the fundamental group) are associated to topological spaces, and algebraic homomorphisms are associated to continuous maps. Nowadays, functors are used throughout modern mathematics to relate various categories. Functor - Definition. Let C and D be ca ...

Including:

• Functor - Definition
• Functor - Covariance and contravariance
• Functor - Examples
• Functor - Properties
• Functor - Relation to other categorical concepts

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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 axioms for the real numbers as a complete ordered field. Under the usual axioms of set theory, one can show that these axioms are categorical, in the sense that there is a model for the axioms, and any two such models are isomorphic. Any one of these models must be explicitly constructed, and most of these models are built using the basic properties of the rational number system as an ordered ...

Including:

• Construction of real numbers - Synthetic approach
• Construction of real numbers - Explicit constructions of models
• Construction of real numbers - Construction from Cauchy sequences
• Construction of real numbers - Construction by Dedekind cuts
• Construction of real numbers - Construction by decimal expansions
• Construction of real numbers - Construction from ultrafilters
• Construction of real numbers - Construction from surreal numbers
• Construction of real numbers - Construction from the group of integers

Read more here: » Construction of real numbers: Encyclopedia - Construction of real numbers

The Right Honourable Bertrand Arthur William Russell, 3rd Earl Russell, OM, FRS (18 May 1872 – 2 February 1970), was an influential British logician, philosopher, and mathematician, working mostly in the 20th century. A prolific writer, Bertrand Russell was also a populariser of philosophy and a commentator on a large variety of topics, ranging from very serious issues to the mundane. Continuing a family tradition in political affairs, he was a prominent liberal as well as a socialist and anti-war activist for most of his long life. ...

Including:

• Bertrand Russell - Biography
• Bertrand Russell - Russell's philosophical work
• Bertrand Russell - Analytic philosophy
• Bertrand Russell - Epistemology
• Bertrand Russell - Ethics
• Bertrand Russell - Logical atomism
• Bertrand Russell - Logic and mathematics
• Bertrand Russell - Philosophy of language
• Bertrand Russell - Philosophy of science
• Bertrand Russell - Religion and theology
• Bertrand Russell - Influence on philosophy
• Bertrand Russell - Russell's activism
• Bertrand Russell - Pacifism war and nuclear weapons
• Bertrand Russell - Communism and socialism
• Bertrand Russell - Women's suffrage
• Bertrand Russell - Sexuality
• Bertrand Russell - Eugenics and race
• Bertrand Russell - Russell summing up his life
• Bertrand Russell - Comments about Russell
• Bertrand Russell - As a man
• Bertrand Russell - As a philosopher
• Bertrand Russell - As a writer and his place in history
• Bertrand Russell - As a mathematician and logician
• Bertrand Russell - As an activist
• Bertrand Russell - As a recipient of the Nobel Prize for Literature
• Bertrand Russell - From a daughter
• Bertrand Russell - Quotes
• Bertrand Russell - Asides
• Bertrand Russell - Succession

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