Inmathematics andlogic, anaxiomatic system is aset offormal statements (i.e.axioms) used to logically derive other statements such aslemmas ortheorems. Aproof within an axiom system is a sequence ofdeductive steps that establishes a new statement as a consequence of the axioms. An axiom system is calledcomplete with respect to a property if every formula with the property can be derived using the axioms. The more general termtheory is at times used to refer to an axiomatic system and all its derived theorems.

In its pure form, an axiom system is effectively a syntactic construct and does not by itself refer to (or depend on) aformal structure, although axioms are often defined for that purpose. The more modern field ofmodel theory refers to mathematical structures. The relationship between an axiom systems and the models that correspond to it is often a major issue of interest.

Four typical properties of an axiom system are consistency, relative consistency, completeness and independence. An axiomatic system is said to beconsistent if it lackscontradiction. That is, it is impossible to derive both a statement and its negation from the system's axioms.^[1]Consistency is a key requirement for most axiomatic systems, as the presence of contradiction would allow any statement to be proven (principle of explosion).Relative consistency comes into play when we can not prove the consistency of an axiom system. However, in some cases we can show that an axiom system A is consistent if anotheraxiom set B is consistent.^[1]

In an axiomatic system, an axiom is calledindependent if it cannot be proven or disproven from other axioms in the system. A system is called independent if each of its underlying axioms is independent.^[1] Unlike consistency, in many cases independence is not a necessary requirement for a functioning axiomatic system — though it is usually sought after to minimize the number of axioms in the system.

An axiomatic system is calledcomplete if for every statement, either itself or its negation is derivable from the system's axioms, i.e. every statement can be proven true or false by using the axioms.^[1][2] However, note that in some cases it may beundecidable if a statement can be proven or not.

Amodel for an axiomatic system is aformal structure, which assigns meaning for the undefined terms presented in the system, in a manner that is correct with the relations defined in the system. If an axiom system has a model, the axioms are said to have beensatisfied.^[3] The existence of a model which satisfies an axiom system, proves theconsistency of the system.^[4]

Models can also be used to show the independence of an axiom in the system. By constructing a model for a subsystem (without a specific axiom) shows that the omitted axiom is independent if its correctness does not necessarily follow from the subsystem.^[3]

Two models are said to beisomorphic if a one-to-one correspondence can be found between their elements, in a manner that preserves their relationship.^[5] An axiomatic system for which every model is isomorphic to another is calledcategorical or categorial. However, this term should not be confused with the topic ofcategory theory. The property of categoriality (categoricity) ensures the completeness of a system, however the converse is not true: Completeness does not ensure the categoriality (categoricity) of a system, since two models can differ in properties that cannot be expressed by thesemantics of the system.

As an example, observe the following axiomatic system, based onfirst-order logic with additional the followingcountably infinitely many axioms added (these can be easily formalized as anaxiom schema):

${\displaystyle \mathrm{\exists }{x}_1:\mathrm{\exists }{x}_2:\mathrm{\neg }(x_1=x_2)}$  (informally, there exist two different items).

${\displaystyle \mathrm{\exists }{x}_1:\mathrm{\exists }{x}_2:\mathrm{\exists }{x}_3:\mathrm{\neg }(x_1=x_2)\wedge \mathrm{\neg }(x_1=x_3)\wedge \mathrm{\neg }(x_2=x_3)}$  (informally, there exist three different items).

${\displaystyle ...}$ 

Informally, this infinite set of axioms states that there are infinitely many different items. However, the concept of aninfinite set cannot be defined within the system — let alone thecardinality of such a set.

The system has at least two different models – one is thenatural numbers (isomorphic to any other countably infinite set), and another is the real numbers (isomorphic to any other set with thecardinality of the continuum). In fact, it has an infinite number of models, one for each cardinality of an infinite set. However, the property distinguishing these models is their cardinality — a property which cannot be defined within the system. Thus the system is not categorial. However it can be shown to be complete, for example by using theŁoś–Vaught test.

Stating definitions and propositions in a way such that each new term can be formally eliminated by the priorly introduced terms requires primitive notions (axioms) to avoidinfinite regress. This way of doing mathematics is called theaxiomatic method.^[6]

A common attitude towards the axiomatic method islogicism. In their bookPrincipia Mathematica,Alfred North Whitehead andBertrand Russell attempted to show that all mathematical theory could be reduced to some collection of axioms. More generally, the reduction of a body of propositions to a particular collection of axioms underlies the mathematician's research program. This was very prominent in the mathematics of the twentieth century, in particular in subjects based aroundhomological algebra.

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 thatrings need not becommutative, which differed fromEmmy Noether's original formulation. Mathematicians decided to considertopological spaces more generally without theseparation axiom whichFelix Hausdorff originally formulated.

TheZermelo–Fraenkel set theory, a result of the axiomatic method applied to set theory, allowed the "proper" formulation of set-theory problems and helped avoid the paradoxes ofnaïve set theory. One such problem was thecontinuum hypothesis. Zermelo–Fraenkel set theory, with the historically controversialaxiom of choice included, is commonly abbreviatedZFC, where "C" stands for "choice". Many authors useZF to refer to the axioms of Zermelo–Fraenkel set theory with the axiom of choice excluded.^[7] Today ZFC is the standard form ofaxiomatic set theory and as such is the most commonfoundation of mathematics.

Mathematical methods developed to some degree of sophistication in ancient Egypt, Babylon, India, and China, apparently without employing the axiomatic method.

Euclid ofAlexandria authored the earliest extant axiomatic presentation ofEuclidean geometry andnumber theory. His idea begins with five undeniable geometric assumptions calledaxioms. Then, using these axioms, he established the truth of other propositions byproofs, hence the axiomatic method.^[8]

Many axiomatic systems were developed in the nineteenth century, includingnon-Euclidean geometry, the foundations ofreal analysis,Cantor'sset theory,Frege's work on foundations, andHilbert's 'new' use of axiomatic method as a research tool. For example,group theory was first put on an axiomatic basis towards the end of that century. Once the axioms were clarified (thatinverse elements should be required, for example), the subject could proceed autonomously, without reference to thetransformation group origins of those studies.

The mathematical system ofnatural numbers 0, 1, 2, 3, 4, ... is based on an axiomatic system first devised by the mathematicianGiuseppe Peano in 1889. He chose the axioms, in the language of a single unary function symbolS (short for "successor"), for the set of natural numbers to be:

• There is a natural number 0.
• Every natural numbera has a successor, denoted bySa.
• There is no natural number whose successor is 0.
• Distinct natural numbers have distinct successors: ifa ≠b, thenSa ≠Sb.
• If a property is possessed by 0 and also by the successor of every natural number it is possessed by, then it is possessed by all natural numbers ("Induction axiom").

Inmathematics,axiomatization is the process of taking a body of knowledge and working backwards towards its axioms. It is the formulation of a system of statements (i.e.axioms) that relate a number of primitive terms — in order that aconsistent body ofpropositions may be deriveddeductively from these statements. Thereafter, theproof of any proposition should be, in principle, traceable back to these axioms.

If the formal system is notcomplete not every proof can be traced back to the axioms of the system it belongs. 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 totopology orcomplex analysis. It might not be immediately clear whether another proof can be found that derives itself solely from the Peano axioms.

• Axiom schema – Short notation for a set of statements that are taken to be true
• Formal system – Mathematical model for deduction or proof systems
• Gödel's incompleteness theorems – Limitative results in mathematical logic
• Hilbert-style deduction system – System of formal deduction in logicPages displaying short descriptions of redirect targets
• History of logic
• List of logic systems
• Logicism – Programme in the philosophy of mathematics
• Zermelo–Fraenkel set theory – Standard system of axiomatic set theory, an axiomatic system for set theory and today's most common foundation for mathematics.

• "Axiomatic method",Encyclopedia of Mathematics,EMS Press, 2001 [1994]
• Eric W. Weisstein,Axiomatic System, From MathWorld—A Wolfram Web Resource.Mathworld.wolfram.com &Answers.com