In mathematics, axiomatization is the process of defining the basic axiomatic systems from which mathematical theories can be derived. Every mathematical theory is based on a set of axioms. Usually these axioms are not mentioned when a mathematical equation is presented. Mathematicians know from their education on which axioms mathematical theories are based. Indeed, mathematical theories usually are based on very few axioms. Some of them are mentioned in the example below. Axiomatization - E ...

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• Axiomatization - Example: The axiomatization of natural numbers

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Set theory is a branch of mathematics created principally by the German mathematician Georg Cantor at the end of the 19th century. Initially controversial, set theory has come to play the role of a foundational theory in modern mathematics, in the sense of a theory invoked to justify assumptions made in mathematics concerning the existence of mathematical objects (such as numbers or functions) and their properties. Formal versions of set theory also have a foundational role to play as specifying a theoretical ideal of mathematical rig ...

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• Axiomatic set theory - The origins of rigorous set theory
• Axiomatic set theory - Axioms for set theory
• Axiomatic set theory - Independence in ZFC
• Axiomatic set theory - Set theory ZFC foundations for mathematics
• Axiomatic set theory - Well-foundedness and hypersets
• Axiomatic set theory - Objections to set theory

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In abstract algebra, a Boolean algebra is an algebraic structure (a collection of elements and operations on them obeying defining axioms) that captures essential properties of both set operations and logic operations. Specifically, it deals with the set operations of intersection, union, complement; and the logic operations of AND, OR, NOT. For example, the logical assertion that a statement a and its negation ¬a cannot both be true, parallels the set-theory assertion that a subset Including:

• 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

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Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. It is half-jokingly known as "generalized abstract nonsense". Categories appear in most branches of mathematics, in some areas of theoretical computer science and mathematical physics, and have been a unifying notion. Categories were first introduced by Samuel Eilenberg and Saunders Ma ...

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• 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

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In mathematics and related technical fields, a mathematical object is complete if nothing needs to be added to it. This is made precise in various ways, several of which have a related notion of completion. It should be noted that "complete" here is just a term that takes on specific meanings in specific situations, and not every situation in which a type of "completion" occurs is called a "completion". See, for example, algebraically closed field, compactification, or Gödel's incompleteness theorem. A metric spac

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The Zermelo-Fraenkel axioms of set theory together with the axiom of choice are the standard axioms of axiomatic set theory. All of ordinary mathematics can be based on this axiom system. The Zermelo-Fraenkel axioms without the axiom of choice are usually denoted by ZF. The ZF axioms together with the axiom of choice (AC) are denoted ZFC. The axioms are the result of work by Thoralf Skolem in 1922, based on earlier work by Abraham Fraenkel in the same year, which was based on the axi ...

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• Zermelo-Fraenkel set theory - The Axioms

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Theory has a number of distinct meanings in different fields of knowledge, depending on the context and their methodologies. Theory - Etymology. The word ‘theory’ derives from the Greek ‘theorein’, which means ‘to look at’. According to some sources, it was used frequently in terms of ‘looking at’ a theatre stage, which may explain why sometimes the word ‘theory’ is used as something provisional or not completely resembling real. The term ‘theoria’ (a noun) was already used by ...

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• Theory - Etymology
• Theory - Science
• Theory - Models
• Theory - Types of theories
• Theory - Further explanation of a scientific theory
• Theory - Characteristics
• Theory - Mathematics
• Theory - Other fields
• Theory - List of famous theories
• Theory - Reference

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In mathematics, an axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems. A mathematical theory consists of an axiomatic system and all its derived theorems. An axiomatic system that is completely described is a special kind of formal system; usually though the effort towards complete formalisation brings diminishing returns in certainty, and a lack of readability for humans. Therefore discussion of axiomatic systems is normally only semi-formal. A formal theory< ...

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• Axiomatic system - Properties
• Axiomatic system - Models
• Axiomatic system - Axiomatic method

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In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of infinity is one of the axioms of Zermelo-Fraenkel set theory. In the formal language of the Zermelo-Fraenkel axioms, the axiom reads: or in words: There is a set N, such that the empty set is in N and such that whenever x is a member of N, the set formed by taking the union of x with its singleton {x} is also a member of

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Answers in Genesis (AiG) is a not-for-profit Christian apologetics ministry with a particular focus on Young Earth Creationism, and a literal interpretation of the first chapters of the Book of Genesis. They state that this is 'secondary in importance to the proclamation of the Gospel of Jesus Christ' [1]. AiG employs a staff exclusively of Christian evangelicals, some who have earned Ph.D. degrees from secular universities in various sciences including biology, geology, and astrophysics. Their literal interpretation of ...

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• Answers in Genesis - History
• Answers in Genesis - Tax-exempt status
• Answers in Genesis - The Creation Museum
• Answers in Genesis - Facts and figures
• Answers in Genesis - Teachings and beliefs
• Answers in Genesis - Methodology
• Answers in Genesis - Mission
• Answers in Genesis - Apologetic method
• Answers in Genesis - AiG's views on cosmology and astronomy
• Answers in Genesis - AiG's views on moral and social issues
• Answers in Genesis - Life issues
• Answers in Genesis - Homosexuality
• Answers in Genesis - Evolution and race
• Answers in Genesis - Culture and media
• Answers in Genesis - Criticism
• Answers in Genesis - Is Answers in Genesis a scientific organization?
• Answers in Genesis - Criticisms of specific claims made by Answers in Genesis
• Answers in Genesis - Controversy over Interview with Richard Dawkins
• Answers in Genesis - Answers in Genesis and September 11 the Indian Ocean tsunami and hurricane Katrina disasters
• Answers in Genesis - Do Answers in Genesis and mainstream scientists agree about defining evolution?

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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 this encyclopedia, set out in a systematic way. The way research-level mathematics is internally organised is mostly determined by practitioners, and does change over time; this is in contrast with the apparently timeless syllabus divisions used in mathematics education, where calculus can seem to be much the same over a time scale of a century. Calculus itself does not appear as a major heading — m ...

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• Areas of mathematics - Foundations / general
• Areas of mathematics - Algebra
• Areas of mathematics - Analysis
• Areas of mathematics - Geometry
• Areas of mathematics - Applied mathematics
• Areas of mathematics - Probability and statistics
• Areas of mathematics - Computational sciences
• Areas of mathematics - Physical sciences
• Areas of mathematics - Non-physical sciences

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In symbolic logic, it is sometimes inconvenient or impossible to express an axiomatic system in a finite number of axioms. For this reason, an axiom schema is used. Formally, an axiom schema is a set (usually infinite) of well formed formulae, each of which is taken to be an axiom. Often, this set is constructed recursively. A well known axiom schema is the axiom schema of replacement. There is debate among metamathematicians as to whether an axiomatic system containing an axiom schema should be considered elegant. Some logicians thus pre ...

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In aesthetics, the sublime (from the Latin sublimis (exalted)) is the quality of transcendent greatness, whether physical, moral, intellectual or artistic. The term especially references a greatness with which nothing else can be compared and which is beyond all possibility of calculation, measurement or imitation. The first study of the value of the sublime is the treatise ascribed to Longinus: On the Sublime. For Longinus, artistic genius was the skill of metaphor. Prior to the eighteenth century sublime was a r ...

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• Sublime philosophy - Eighteenth Century
• Sublime philosophy - Romantic Period
• Sublime philosophy - Post Romantic and Twentieth Century

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The Annus Mirabilis Papers (from Annus mirabilis, Latin for 'year of wonders') are the papers of Albert Einstein submitted to the "Annalen der Physik" journal in 1905. The four articles provided a foundation for modern physics. Annus Mirabilis Papers - Papers. Three of those papers (on Brownian motion, the photoelectric effect, and special relativity) deserved Nobel Prizes according to some physicists. Only the paper on the photoelectric effect would win one. What makes these papers remarkable ...

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• Annus Mirabilis Papers - Papers
• Annus Mirabilis Papers - Background
• Annus Mirabilis Papers - Photoelectric effect
• Annus Mirabilis Papers - Brownian motion
• Annus Mirabilis Papers - Special relativity
• Annus Mirabilis Papers - Matter and energy equivalence
• Annus Mirabilis Papers - Commemoration

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Axiomatic. In mathematics, an axiomatic theory is one based on axioms. Axiomatic is a collection of short stories by Greg Egan. Axiomatic is an album by Australian band Taxiride. Other related archivesAxiomatic, Greg Egan, Taxiride, axioms, mathematics

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Group mathematics - An abelian group: the integers under addition. A group that we are introduced to in elementary school is the integers under addition. For this example, let Z be the set of integers, {..., −4, −3, −2, −1, 0, 1, 2, 3, 4, ...}, and let the symbol "+" indicate the operation of addition. Then (Z,+) is a group (written additively). Proof: If a and b are integers then a + b is an integer. (Closure; + really is a binary operation)See also:

Group mathematics, Group mathematics - History, Group mathematics - Basic definitions, Group mathematics - Notation for groups, Group mathematics - Some elementary examples and nonexamples, Group mathematics - An abelian group: the integers under addition, Group mathematics - Not a group: the integers under multiplication, Group mathematics - An abelian group: the nonzero rational numbers under multiplication, Group mathematics - A finite nonabelian group: permutations of a set, Group mathematics - Further examples, Group mathematics - Simple theorems, Group mathematics - Constructing new groups from given ones

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As argued above, many naïve objections depend on implicitly denying Hume's principle, and are therefore question-begging. Wittgenstein explicitly denies the principle, arguing that our concept of number depends essentially on counting. "Where the nonsense starts is with our habit of thinking of a large number as closer to infinity than a small one" The expressions "divisible into two parts" and "divisible without limit" have completely different forms. This is, of course, the same case as the one in which someone operat ...

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Controversy over Cantor's theory, Controversy over Cantor's theory - Preface, Controversy over Cantor's theory - Introduction, Controversy over Cantor's theory - Cantor's argument, Controversy over Cantor's theory - Reception of the argument, Controversy over Cantor's theory - Naïve objections, Controversy over Cantor's theory - Objections to Cantor's theorem, Controversy over Cantor's theory - Objections to Hume's principle, Controversy over Cantor's theory - Objection to the axiom of infinity, Controversy over Cantor's theory - Objections to the power set axiom, Controversy over Cantor's theory - Footnote

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Euclid based his work in Book I on 23 definitions, such as point, line and surface, five postulates and five "common notions" (both of which are today called axioms). Postulates in Book I: A straight line segment can be drawn by joining any two points. A straight line segment can be extended indefinitely in a straight line. Given a straight line segment, a circle can be drawn using the segment as radius and one endpoint as center. All right angles are congruent. If two lines are drawn w ...

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Euclid's Elements, Euclid's Elements - First principles, Euclid's Elements - Parallel postulate, Euclid's Elements - History

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Although many of the results in Elements originated with earlier mathematicians, one of Euclid's accomplishments was to present them in a single, logically coherent framework. In addition to providing some missing proofs, Euclid's text also includes sections on number theory and three-dimensional geometry. In particular, Euclid's proof of the infinitude of prime numbers is in Book IX, Proposition 20. The geometrical system described in Elements was long known simply as "the" geometry. Today, however, it is often referred ...

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Euclid, Euclid - The Elements, Euclid - Other works, Euclid - Biographical sources

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Hoare logic - Assignment axiom schema. The assignment axiom states that after the assignment any predicate holds for the variable that was previously true for the right-hand side of the assignment: Here P[x / E] denotes the expression P in which all free occurrences of the variable x have been replaced with the expression E. An example of a valid triple is: Hoare l ...

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Hoare logic, Hoare logic - Partial correctness, Hoare logic - Assignment axiom schema, Hoare logic - Sequencing rule, Hoare logic - Conditional rule, Hoare logic - While rule, Hoare logic - Rule of consequence, Hoare logic - Total correctness

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The traditional presentation of Euclidean geometry is as an axiomatic system, setting out to prove all the "true statements" as theorems in geometry from a set of finite number of axioms. The five postulates of the Elements are as follows: Any two points can be joined by a straight line. Any straight line segment can be extended indefinitely in a straight line. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center. All right angles ...

See also:

Euclidean geometry, Euclidean geometry - Axiomatic approach, Euclidean geometry - Modern introduction to Euclidean geometry, Euclidean geometry - The construction, Euclidean geometry - Classical theorems

Read more here: » Euclidean geometry: Encyclopedia II - Euclidean geometry - Axiomatic approach