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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. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In traditional Logic, an axiom or postulate is a proposition that is not proved or demonstrated but considered to be either self-evident, or subject In Mathematics, a theorem is a statement proven on the basis of previously accepted or established statements A mathematical theory consists of an axiomatic system and all its derived theorems. The word theory has many distinct meanings in different fields of Knowledge, depending on their methodologies and the context of discussion. 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. In formal logic, a formal system (also called a logical system, a logistic system, or simply a logic Formal systems in mathematics consist Therefore discussion of axiomatic systems is normally only semi-formal. A formal theory typically means an axiomatic system, for example formulated within model theory. In Mathematics, model theory is the study of (classes of mathematical structures such as groups, Fields graphs or even models A formal proof is a complete rendition of a mathematical proof within a formal system. In Mathematics, a proof is a convincing demonstration (within the accepted standards of the field that some Mathematical statement is necessarily true

Contents

Properties

An axiomatic system is said to be consistent if it lacks contradiction, i. In Classical logic, a contradiction consists of a logical incompatibility between two or more Propositions It occurs when the propositions taken together yield e. the ability to derive both a statement and its negation from the system's axioms.

In an axiomatic system, an axiom is called independent if it is not a theorem that can be derived from other axioms in the system. A system will be called independent if each of its underlying axioms is independent.

Although independence is not a necessary requirement for a system, consistency is. An axiomatic system will be called complete if for every statement, either itself or its negation is derivable. This is very difficult to achieve, however, and as shown by the combined works of Kurt Gödel and Paul Cohen, impossible for axiomatic systems involving infinite sets. Kurt Gödel (kʊɐ̯t ˈgøːdl̩ (April 28 1906 – January 14 1978 was an Austrian American Logician, Mathematician and Philosopher Paul Joseph Cohen ( April 2, 1934 &ndash March 23, 2007) was an American Mathematician best known for his proof of So, along with consistency, relative consistency is also the mark of a worthwhile axiom system. This is when the undefined terms of a first axiom system are provided definitions from a second such that the axioms of the first are theorems of the second.

A good example is the relative consistency of neutral geometry or absolute geometry with respect to the theory of the real number system. Lines and points are undefined terms in absolute geometry, but assigned meanings in the theory of real numbers in a way that is consistent with both axiom systems.

Models

A model for an axiomatic system is a well-defined set, which assigns meaning for the undefined terms presented in the system, in a manner that is correct with the relations defined in the system. In Mathematics, model theory is the study of (classes of mathematical structures such as groups, Fields graphs or even models The existence of a concrete model* proves the consistency of a system.

Models can also be used to show the independence of an axiom in the system. By constructing a valid model for a subsystem without a specific axiom, we show that the omitted axiom is independent if its correctness does not necessarily follow from the subsystem.

Two models are said to be isomorphic if a one-to-one correspondence can be found between their elements, in a manner that preserves their relationship. In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective An axiomatic system for which every model is isomorphic to another is called categorial (sometimes categorical), and the property of categoriality (categoricity) ensures the completeness of a system.

* A model is called concrete if the meanings assigned are objects and relations from the real world, as opposed to an abstract model which is based on other axiomatic systems.

The first axiomatic system was Euclidean geometry. Euclidean geometry is a mathematical system attributed to the Greek Mathematician Euclid of Alexandria.

Axiomatic method

The axiomatic method involves replacing a coherent body of propositions (i. e. a mathematical theory) by a simpler collection of propositions (i. e. axioms). The axioms are designed so that the original body of propositions can be deduced from the axioms.

The axiomatic method, brought to the extreme, results in logicism. Logicism is one of the schools of thought in the Philosophy of mathematics, putting forth the theory that Mathematics is an extension of Logic and therefore In their book Principia Mathematica, Alfred North Whitehead and Bertrand Russell attempted to show that all mathematical theory could be reduced to some collection of axioms. The Principia Mathematica is a 3-volume work on the Foundations of mathematics, written by Alfred North Whitehead and Bertrand Russell Alfred North Whitehead, OM ( February 15 1861, Ramsgate, Kent, England &ndash December 30 1947, Bertrand Arthur William Russell 3rd Earl Russell, OM, FRS (18 May 1872 – 2 February 1970 was a British Philosopher, Historian More generally, the reduction of a body of propositions to a particular collection of axioms belies the mathematician's research program. This was very prominent in the mathematics of the twentieth century, in particular in subjects based around homological algebra. The twentieth century of the Common Era began on Homological algebra is the branch of Mathematics which studies homology in a general algebraic setting

The explication of the particular axioms used in a theory can help to clarify a suitable level of abstraction that the mathematician would like to work with. For example, mathematicians opted that rings need not be commutative, which differed from Emmy Noether's original formulation. In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real In Ring theory, a branch of Abstract algebra, a commutative ring is a ring in which the multiplication operation has the commutative property Amalie Emmy Noether, ˈnøːtɐ (23 March 1882 – 14 April 1935 was a German Mathematician known for her groundbreaking contributions to Abstract algebra and Mathematics decided to consider topological spaces more generally without the separation axiom which Felix Hausdorff originally formulated. Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. In Topology and related fields of Mathematics, there are several restrictions that one often makes on the kinds of Topological spaces that one wishes to consider Felix Hausdorff ( November 8, 1868 &ndash January 26, 1942) was a German Mathematician who is considered to be one of the founders

The Zermelo-Franekel axioms, the result of the axiomatic method applied to set theory, allowed the proper formulation of set theory problems and helped avoided the paradoxes of naïve set theory. Zermelo–Fraenkel set theory with the axiom of choice, commonly abbreviated ZFC, is the standard form of Axiomatic set theory and as such is the most common Naive set theory is one of several theories of sets used in the discussion of the Foundations of mathematics. One such problem was the Continuum hypothesis. In Mathematics, the continuum hypothesis (abbreviated CH) is a Hypothesis, advanced by Georg Cantor, about the possible sizes of Infinite

History

The earliest record we have of such a practice dates back to Euclid (circa 300 BC) in his attempt to axiomatize Euclidean geometry and elementary number theory. Euclid ( Greek:.) fl 300 BC also known as Euclid of Alexandria, is often referred to as the Father of Geometry Events By place Egypt Pyrrhus, the King of Epirus, is taken as a hostage to Egypt after the Battle of Ipsus Euclidean geometry is a mathematical system attributed to the Greek Mathematician Euclid of Alexandria. Number theory is the branch of Pure mathematics concerned with the properties of Numbers in general and Integers in particular as well as the wider classes Up until the beginning of the nineteenth century it was generally assumed, in European mathematics and philosophy (for example in Spinoza's work) that the heritage of Greek mathematics represented the highest standard of intellectual finish (development more geometrico, in the style of the geometers). The 19th century of the Common Era began on January 1, 1801 and ended on December 31, 1900, according to the Gregorian calendar Baruch or Benedict de Spinoza (ברוך שפינוזה Bento de Espinosa Benedictus de Spinoza ( November 24, 1632 – February 21, Greek mathematics, as that term is used in this article is the Mathematics written in Greek, developed from the 6th century BC to the 5th century As such, modern mathematician often discuss the axiomatic method as if it were a unitary approach.

This traditional approach, in which axioms were supposed to be self-evident and so indisputable, was swept away during the course of the nineteenth century, by the development of non-Euclidean geometry, the foundations of real analysis, Cantor's set theory and Frege's work on foundations, and Hilbert's 'new' use of axiomatic method as a research tool. In mathematics non-Euclidean geometry describes how this all works--> hyperbolic and Elliptic geometry, which are contrasted with Euclidean geometry Real analysis is a branch of Mathematical analysis dealing with the set of Real numbers In particular it deals with the analytic properties of real Georg Ferdinand Ludwig Philipp Cantor ( – January 6 1918) was a German Mathematician, born in Russia. Friedrich Ludwig Gottlob Frege ( 8 November 1848, Wismar, Grand Duchy of Mecklenburg-Schwerin  &ndash 26 July 1925 David Hilbert ( January 23, 1862 &ndash February 14, 1943) was a German Mathematician, recognized as one of the most For example, group theory was first put on an axiomatic basis towards the end of that century. Group theory is a mathematical discipline the part of Abstract algebra that studies the Algebraic structures known as groups. Once the axioms were clarified (that inverse elements should be required, for example), the subject could proceed autonomously, without reference to the transformation group origins of those studies. In Mathematics, the idea of inverse element generalises the concepts of negation, in relation to Addition, and reciprocal, in relation to The Symmetry group of an object ( Image, signal, etc eg in 1D 2D or 3D is the group of all Isometries under which it is

If one were to look at non-Western mathematics, one would observe that mathematics developed to some sophistication in the ancient civilizations in the Near East, India and China without employing the axiomatic method. Although many disciplines in modern mathematics, notably abstract algebra and topology, are conceived within the framework of the axiomatic method, the flourishing of ancient mathematics provides a viable alternate epistemology towards the practice of mathematics. Abstract algebra is the subject area of Mathematics that studies Algebraic structures such as groups, rings, fields, modules Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of Epistemology (from Greek επιστήμη - episteme, "knowledge" + λόγος, " Logos " or theory of knowledge

Issues

Not every consistent body of propositions can be captured by a describable collection of axioms. Call a collection of axioms recursive if a computer program can recognize whether a given proposition in the language is an axiom. In Computability theory, a set of Natural numbers is called recursive, computable or decidable if there is an Algorithm Gödel's First Incompleteness Theorem then tells us that there are certain consistent bodies of propositions with no recursive axiomatization. In Mathematical logic, Gödel's incompleteness theorems, proved by Kurt Gödel in 1931 are two Theorems stating inherent limitations of all but the most If the computer cannot recognize the axioms, the computer also will not be able to recognize whether a proof is valid. The result is that one will not know which propositions are theorems and the axiomatic method breaks down. An example of such a body of propositions is the theory of the natural numbers. In Mathematics, a natural number (also called counting number) can mean either an element of the set (the positive Integers or an The Peano Axioms (described below) thus only partially axiomatize this theory.

In practice, not every proof is traced back to the axioms. At times, it is not clear which collection of axioms does a proof appeal to. For example, a number-theoretic statement might be expressible in the language of arithmetic (i. e. the language of the Peano Axioms) and a proof might be given that appeals to topology or complex analysis. Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of Mathematics investigating functions of Complex It might not be immediately clear whether another proof can be found that derives itself solely from the Peano Axioms.

Any more-or-less arbitrarily chosen system of axioms is the basis of some mathematical theory, but such an arbitrary axiomatic system will not necessarily be free of contradictions, and even if it is, it is not likely to shed light on anything. Philosophers of mathematics sometimes assert that mathematicians choose axioms "arbitrarily", but the truth is that although they may appear arbitrary when viewed only from the point of view of the canons of deductive logic, that is merely a limitation on the purposes that deductive logic serves.

Example: The axiomatization of natural numbers

The mathematical system of natural numbers 0, 1, 2, 3, 4, . In Mathematics, a natural number (also called counting number) can mean either an element of the set (the positive Integers or an . . is based on an axiomatic system that was first written down by the mathematician Peano in 1901. Giuseppe Peano ( August 27, 1858 &ndash April 20, 1932) was an Italian Mathematician, whose work was of exceptional Year 1901 ( MCMI) was a Common year starting on Tuesday (link will display calendar of the Gregorian calendar (or a Common year starting He chose the axioms (see Peano axioms), in the language of a single unary function symbol S (short for "successor"), for the set of natural numbers to be:

Outside of mathematics

In mathematics, axiomatization is the formulation of a system of statements (i. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and e. axioms) that relate a number of primitive terms in order that a consistent body of propositions may be derived deductively from these statements. In traditional Logic, an axiom or postulate is a proposition that is not proved or demonstrated but considered to be either self-evident, or subject A boolean-valued function, in some usages a predicate or a proposition, is a function of the type f: X → B, where X is an arbitrary set Deductive reasoning is Reasoning which uses deductive Arguments to move from given statements ( Premises to Conclusions which must be true if the Thereafter, the proof of any proposition should be, in principle, traceable back to these axioms. In Mathematics, a proof is a convincing demonstration (within the accepted standards of the field that some Mathematical statement is necessarily true

See also

References

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