Axiomatic system: Difference between revisions

Revision as of 20:58, 1 December 2005

Latest revision as of 20:09, 3 May 2025

(292 intermediate revisions by more than 100 users not shown)

Line 1:

{{short description|Mathematical term; concerning axioms used to derive theorems}}

In [[mathematics]], an '''axiomatic system''' is any [[set]] of [[axiom]]s from which some or all axioms can be used in conjunction to logically derive [[theorem]]s. 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''' typically means an axiomatic system, for example formulated within [[model theory]]. A '''formal proof''' is a complete rendition of a [[proof]] within a formal system.

{{more footnotes|date=March 2013}}

==Properties==

{{Expert needed|1=Mathematics|reason=Few citations despite the degree of detail|date=January 2025}}

In [[mathematics]] and [[logic]], an '''axiomatic system''' is a [[Set (mathematics)|set]] of [[Formal language|formal statements]] (i.e. [[axiom]]s) used to logically derive other statements such as [[Lemma (mathematics)|lemma]] or [[theorem]]s. A [[formal proof|proof]] within an axiom system is a sequence of [[Deductive reasoning|deductive steps]] that establishes a news statement as a consequence of the axioms. An axiom system is called [[Completeness (logic)|complete]] with respect to a property if every formula with the property can be derived using the axioms. The more general term [[Theory (mathematical logic)|theory]] is at times used to refer to an axiomc system and all its derived theorems.

An axiomatic system is said to be ''consistent'' if it lacks ''contradiction'', i.e. the ability to derive both a statement and its negation from the system's axioms.

In its pure form, an axiom system is effectively a syntactic construct and does not by itself refer to (or depend on) a [[Structure (mathematical logic)|formal structure]], although axioms are often defined for that purpose. The more modern field of [[model 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.

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.

== Properties ==

Although independence is not a necessary requirement for a system, consistency is. An axiomatic system will be called ''complete'' if every statement is either derivable or its negation is derivable.

Four typical properties of an axiom system are consistency, relative consistency, completeness and independence. An axiomatic system is said to be ''[[Consistency|consistent]]'' if it lacks [[contradiction]]. That is, it is impossible to derive both a statement and its negation from the system's axioms.<ref name=howson>A. G. Howson A Handbook of Terms Used in Algebra and Analysis, Cambridge UP, ISBN 0521084342 1972 pp 6</ref>

==Models==

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 axion system. However, in some cases we can show that an axion system A is consistent if another

axiom set B is consistent.<ref name=howson />

In an axiomatic system, an axiom is called ''[[Independence (mathematical logic)|independent]]'' 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.<ref name=howson /> 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 called ''[[Completeness (logic)|complete]]'' 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.<ref name=howson /><ref>{{Cite web|url=http://mathworld.wolfram.com/CompleteAxiomaticTheory.html|title=Complete Axiomatic Theory|last=Weisstein|first=Eric W.|website=mathworld.wolfram.com|language=en|access-date=2019-10-31}}</ref> However, note that in some cases it may be [[Undecidable problem|undecidible]] if a statement can be proven or not.

A ''[[Model theory|mathematical 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. The existence of a ''concrete model''* proves the ''consistency'' of a system.

== Axioms and models ==

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.

A [[Model theory|model]] for an axiomatic system is a well-defined [[Structure (mathematical logic)|formal 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. The existence of a {{Em|concrete model}} proves the [[consistency proof|consistency]] of a system{{Disputed inline||date=April 2020}}. A model is called concrete if the meanings assigned are objects and relations from the real world{{clarify|date=March 2018|reason=What does 'real world' mean in this context?}}, as opposed to an {{Em|abstract model}} which is based on other axiomatic systems.

Two models are said to be [[isomorphism|isomorphic]] if a one-to-one correspondence can be found between their elements, in a manner that preserves their relationship. An axiomatic system for which every model is isomorphic to another is called ''categorial'', and the property of ''categoriality'' ensures the ''completeness'' 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.

<small>* 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.</small>

Two models are said to be [[isomorphism|isomorphic]] if a one-to-one correspondence can be found between their elements, in a manner that preserves their relationship.<ref>{{Citation|last1=Hodges|first1=Wilfrid|title=First-order Model Theory|date=2018|url=https://plato.stanford.edu/archives/win2018/entries/modeltheory-fo/|encyclopedia=The Stanford Encyclopedia of Philosophy|editor-last=Zalta|editor-first=Edward N.|edition=Winter 2018|publisher=Metaphysics Research Lab, Stanford University|access-date=2019-10-31|last2=Scanlon|first2=Thomas}}</ref> An axiomatic system for which every model is isomorphic to another is called {{Em|categorial}} (sometimes {{Em|categorical}}). 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 the [[semantics]] of the system.

The first axiomatic system was [[Euclidean geometry]].

==Axiomatic= methodExample ===

As an example, observe the following axiomatic system, based on [[first-order logic]] with additional semantics of the following [[countably infinite]]ly many axioms added (these can be easily formalized as an [[axiom schema]]):

The '''axiomatic method''' is often discussed as if it were a unitary approach, or uniform procedure. With the example of [[Euclid]] to appeal to, it was indeed treated that way for many centuries: 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).

:<math>\exist x_1: \exist x_2: \lnot (x_1=x_2)</math> (informally, there exist two different items).

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. For example, [[group theory]] was first put on an axiomatic basis towards the end of that century. Once the axioms were clarified (that [[inverse element]]s should be required, for example), the subject could proceed autonomously, without reference to the [[transformation group]] origins of those studies.

:<math>\exist x_1: \exist x_2: \exist x_3: \lnot (x_1=x_2) \land \lnot (x_1=x_3) \land \lnot (x_2=x_3)</math> (informally, there exist three different items).

Therefore, there are at least three 'modes' of axiomatic method current in mathematics, and in the fields it influences. In caricature, possible attitudes are:

:<math>...</math>

#Accept my axioms and you must accept their consequences;

#I reject one of your axioms and accept extra models;

#My set of axioms defines a research programme.

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

The first case is the classic [[deductive method]]. The second goes by the slogan ''be wise, generalise''; it may go along with the assumption that concepts can or should be expressed at some intrinsic 'natural level of generality'. The third was very prominent in the mathematics of the [[twentieth century]], in particular in subjects based around [[homological algebra]].

The system has at least two different models – one is the [[natural number]]s (isomorphic to any other countably infinite set), and another is the real numbers (isomorphic to any other set with the [[cardinality 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 the [[Łoś–Vaught test]].

It is easy to see that the axiomatic method has limitations outside mathematics. For example, in [[political philosophy]] axioms that lead to unacceptable conclusions are likely to be rejected wholesale; so that no one really assents to version 1 above.

== Axiomatic method ==

==See also==

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 avoid [[infinite regress]]. This way of doing mathematics is called the '''axiomatic method'''.<ref>"''Set Theory and its Philosophy, a Critical Introduction'' S.6; Michael Potter, Oxford, 2004</ref>

*[[Axiomatization]]

*[[Model theory]]

*[[Gödel's incompleteness theorem]]

A common attitude towards the axiomatic method is [[logicism]]. 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. 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 around [[homological algebra]].

[[Category:Mathematical logic]]

[[Category:Systems]]

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 [[Ring (mathematics)|ring]]s need not be [[Commutative ring|commutative]], which differed from [[Emmy Noether]]'s original formulation. Mathematicians decided to consider [[topological space]]s more generally without the [[separation axiom]] which [[Felix Hausdorff]] originally formulated.

[[de:Axiomensystem]]

[[gl:Sistema axiomático]]

The [[Zermelo–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 of [[Naive set theory|naïve set theory]]. One such problem was the [[continuum hypothesis]]. Zermelo–Fraenkel set theory, with the historically controversial [[axiom of choice]] included, is commonly abbreviated [[ZFC]], where "C" stands for "choice". Many authors use [[Zermelo–Fraenkel set theory|ZF]] to refer to the axioms of Zermelo–Fraenkel set theory with the axiom of choice excluded.<ref>{{Cite web|url=http://mathworld.wolfram.com/Zermelo-FraenkelAxioms.html|title=Zermelo-Fraenkel Axioms|last=Weisstein|first=Eric W.|website=mathworld.wolfram.com|language=en|access-date=2019-10-31}}</ref> Today ZFC is the standard form of [[axiomatic set theory]] and as such is the most common [[foundations of mathematics|foundation of mathematics]].

[[pt:Sistema axiomático]]

[[zh:公理系统]]

=== History ===

{{Further|History of mathematics}}

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

[[Euclid]] of [[Alexandria]] authored the earliest extant axiomatic presentation of [[Euclidean geometry]] and [[number theory]]. His idea begins with five undeniable geometric assumptions called [[axioms]]. Then, using these axioms, he established the truth of other propositions by [[Mathematical proof|proofs]], hence the axiomatic method.<ref>{{cite book |last1=Lehman |first1=Eric |last2=Meyer |first2=Albert R |last3=Leighton |first3=F Tom |title=Mathematics for Computer Science |url=https://courses.csail.mit.edu/6.042/spring17/mcs.pdf |access-date=2 May 2023}}</ref>

Many axiomatic systems were developed in the nineteenth century, including [[non-Euclidean geometry]], the foundations of [[real analysis]], [[Georg Cantor|Cantor]]'s [[set theory]], [[Gottlob Frege|Frege]]'s work on foundations, and [[David Hilbert|Hilbert]]'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 (that [[inverse element]]s should be required, for example), the subject could proceed autonomously, without reference to the [[transformation group]] origins of those studies.

=== Example: The Peano axiomatization of natural numbers ===

{{Main|Peano axioms}}

The mathematical system of [[natural number]]s 0,&thinsp;1, 2, 3, 4, ... is based on an axiomatic system first devised by the mathematician [[Giuseppe Peano]] in 1889. He chose the axioms, in the language of a single unary function symbol ''S'' (short for "[[Successor function|successor]]"), for the set of natural numbers to be:

* There is a natural number 0.

* Every natural number ''a'' has a successor, denoted by ''Sa''.

* There is no natural number whose successor is 0.

* Distinct natural numbers have distinct successors: if ''a'' ≠ ''b'', then ''Sa'' ≠ ''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 ("''[[Mathematical induction#Axiom of induction|Induction axiom]]''").

=== Axiomatization and proof ===

In [[mathematics]], '''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. [[axiom]]s) that relate a number of primitive terms — in order that a [[consistency proof|consistent]] body of [[Boolean-valued function|propositions]] may be derived [[deductive reasoning|deductively]] from these statements. Thereafter, the [[mathematical proof|proof]] of any proposition should be, in principle, traceable back to these axioms.

If the formal system is not [[Completeness (logic)|complete]] 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 to [[topology]] or [[complex analysis]]. It might not be immediately clear whether another proof can be found that derives itself solely from the Peano axioms.

== See also ==

{{Portal|Philosophy|Mathematics}}

{{wikiquote}}

* {{annotated link|Axiom schema}}

* {{annotated link|Formalism (philosophy of mathematics)|Formalism}}

* {{annotated link|Gödel's incompleteness theorems}}

* {{annotated link|Hilbert-style deduction system}}

* {{annotated link|History of logic}}

* {{annotated link|List of logic systems}}

* {{annotated link|Logicism}}

* {{annotated link|Zermelo–Fraenkel set theory}}, an axiomatic system for set theory and today's most common foundation for mathematics.

== References ==

{{reflist}}

== Further reading ==

* {{springer|title=Axiomatic method|id=p/a014300}}

* Eric W. Weisstein, ''Axiomatic System'', From MathWorld—A Wolfram Web Resource. [http://mathworld.wolfram.com/AxiomaticSystem.html Mathworld.wolfram.com] & [http://www.answers.com/topic/axiomatic-system Answers.com]

{{Mathematical logic}}

[[Category:Mathematical axioms|*]]

[[Category:Formal systems]]

[[Category:Methods of proof]]