In
mathematics and
logic, an axiomatic system is any
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
of
axiom
An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or f ...
s from which some or all axioms can be used in conjunction to logically derive
theorems. A
theory is a consistent, relatively-self-contained body of knowledge which usually contains an axiomatic system and all its derived theorems. An axiomatic system that is completely described is a special kind of
formal system. A formal theory is an axiomatic system (usually formulated within
model theory) that describes a set of sentences that is closed under logical implication. A
formal proof is a complete rendition of a
mathematical proof
A mathematical proof is an inferential argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use other previously established statements, such as theorems; but every proo ...
within a formal system.
Properties
An axiomatic system is said to be ''
consistent
In classical deductive logic, a consistent theory is one that does not lead to a logical contradiction. The lack of contradiction can be defined in either semantic or syntactic terms. The semantic definition states that a theory is consistent i ...
'' if it lacks
contradiction
In traditional logic, a contradiction occurs when a proposition conflicts either with itself or established fact. It is often used as a tool to detect disingenuous beliefs and bias. Illustrating a general tendency in applied logic, Aristotle's ...
. That is, it is impossible to derive both a statement and its negation from the system's axioms. Consistency is a key requirement for most axiomatic systems, as the presence of contradiction would allow any statement to be proven (
principle of explosion).
In an axiomatic system, an axiom is called ''
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. Unlike consistency, 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 ''
complete
Complete may refer to:
Logic
* Completeness (logic)
* Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable
Mathematics
* The completeness of the real numbers, which implies t ...
'' if for every statement, either itself or its negation is derivable from the system's axioms (equivalently, every statement is capable of being proven true or false).
Relative consistency
Beyond consistency, relative consistency is also the mark of a worthwhile axiom system. This describes the scenario where 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
absolute geometry
Absolute geometry is a geometry based on an axiom system for Euclidean geometry without the parallel postulate or any of its alternatives. Traditionally, this has meant using only the first four of Euclid's postulates, but since these are not suf ...
with respect to the theory of the
real number system
In mathematics, a real number is a number that can be used to measure a ''continuous'' one- dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Eve ...
.
Lines and
points
Point or points may refer to:
Places
* Point, Lewis, a peninsula in the Outer Hebrides, Scotland
* Point, Texas, a city in Rains County, Texas, United States
* Point, the NE tip and a ferry terminal of Lismore, Inner Hebrides, Scotland
* Points ...
are undefined terms (also called
primitive notions) 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
A model is an informative representation of an object, person or system. The term originally denoted the plans of a building in late 16th-century English, and derived via French and Italian ultimately from Latin ''modulus'', a measure.
Models c ...
for an axiomatic system is a well-defined
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
, 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 proves the
consistency of a system. A model is called concrete if the meanings assigned are objects and relations from the real world, as opposed to an which is based on other axiomatic systems.
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. An axiomatic system for which every model is isomorphic to another is called (sometimes ). 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.
Example
As an example, observe the following axiomatic system, based on
first-order logic
First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantifi ...
with additional semantics of the following
countably infinite
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural number ...
ly many axioms added (these can be easily formalized as an
axiom schema In mathematical logic, an axiom schema (plural: axiom schemata or axiom schemas) generalizes the notion of axiom.
Formal definition
An axiom schema is a formula in the metalanguage of an axiomatic system, in which one or more schematic variables ap ...
):
:
(informally, there exist two different items).
:
(informally, there exist three different items).
:
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
In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ...
of such as set.
The system has at least two different models – one is the
natural numbers (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 set theory, the cardinality of the continuum is the cardinality or "size" of the set of real numbers \mathbb R, sometimes called the continuum. It is an infinite cardinal number and is denoted by \mathfrak c (lowercase fraktur "c") or , \mathb ...
). 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.
Axiomatic method
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
An infinite regress is an infinite series of entities governed by a recursive principle that determines how each entity in the series depends on or is produced by its predecessor. In the epistemic regress, for example, a belief is justified bec ...
. This way of doing mathematics is called the axiomatic method.
A common attitude towards the axiomatic method is
logicism. In their book ''
Principia Mathematica
The ''Principia Mathematica'' (often abbreviated ''PM'') is a three-volume work on the foundations of mathematics written by mathematician–philosophers Alfred North Whitehead and Bertrand Russell and published in 1910, 1912, and 1913. ...
'',
Alfred North Whitehead
Alfred North Whitehead (15 February 1861 – 30 December 1947) was an English mathematician and philosopher. He is best known as the defining figure of the philosophical school known as process philosophy, which today has found applicat ...
and
Bertrand Russell
Bertrand Arthur William Russell, 3rd Earl Russell, (18 May 1872 – 2 February 1970) was a British mathematician, philosopher, logician, and public intellectual. He had a considerable influence on mathematics, logic, set theory, linguistics, ar ...
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.
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
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...
s need not be
commutative, which differed from
Emmy Noether's original formulation. Mathematicians decided to consider
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
s more generally without the
separation axiom
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. Some of these restrictions are given by the separation axioms. These are sometim ...
which
Felix Hausdorff
Felix Hausdorff ( , ; November 8, 1868 – January 26, 1942) was a German mathematician who is considered to be one of the founders of modern topology and who contributed significantly to set theory, descriptive set theory, measure theory, and f ...
originally formulated.
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
naïve set theory
Naive set theory is any of several theories of sets used in the discussion of the foundations of mathematics.
Unlike axiomatic set theories, which are defined using formal logic, naive set theory is defined informally, in natural language. It de ...
. One such problem was the
continuum hypothesis
In mathematics, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets. It states that
or equivalently, that
In Zermelo–Fraenkel set theory with the axiom of choice (ZFC), this is equivalent to ...
. Zermelo–Fraenkel set theory, with the historically controversial
axiom of choice
In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection ...
included, is commonly abbreviated
ZFC, where "C" stands for "choice". Many authors use
ZF to refer to the axioms of Zermelo–Fraenkel set theory with the axiom of choice excluded.
Today ZFC is the standard form of
axiomatic set theory
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concer ...
and as such is the most common
foundation of mathematics.
History
Mathematical methods developed to some degree of sophistication in ancient Egypt, Babylon, India, and China, apparently without employing the axiomatic method.
Euclid
Euclid (; grc-gre, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the ''Elements'' treatise, which established the foundations of ge ...
of
Alexandria
Alexandria ( or ; ar, ٱلْإِسْكَنْدَرِيَّةُ ; grc-gre, Αλεξάνδρεια, Alexándria) is the second largest city in Egypt, and the largest city on the Mediterranean coast. Founded in by Alexander the Great, Alexandri ...
authored the earliest extant axiomatic presentation of
Euclidean geometry and
number theory. Many axiomatic systems were developed in the nineteenth century, including
non-Euclidean geometry, the foundations of
real analysis,
Cantor
A cantor or chanter is a person who leads people in singing or sometimes in prayer. In formal Jewish worship, a cantor is a person who sings solo verses or passages to which the choir or congregation responds.
In Judaism, a cantor sings and lead ...
's
set theory,
Frege
Friedrich Ludwig Gottlob Frege (; ; 8 November 1848 – 26 July 1925) was a German philosopher, logician, and mathematician. He was a mathematics professor at the University of Jena, and is understood by many to be the father of analytic p ...
's work on foundations, and
Hilbert's 'new' use of axiomatic method as a research tool. For example,
group theory
In abstract algebra, group theory studies the algebraic structures known as groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as ...
was first put on an axiomatic basis towards the end of that century. Once the axioms were clarified (that
inverse element
In mathematics, the concept of an inverse element generalises the concepts of opposite () and reciprocal () of numbers.
Given an operation denoted here , and an identity element denoted , if , one says that is a left inverse of , and that ...
s should be required, for example), the subject could proceed autonomously, without reference to the
transformation group origins of those studies.
Issues
Not every consistent body of propositions can be captured by a describable collection of axioms. In recursion theory, a collection of axioms is called
recursive if a computer program can recognize whether a given proposition in the language is a theorem.
Gödel's first incompleteness theorem then tells us that there are certain consistent bodies of propositions with no recursive axiomatization. Typically, the computer can recognize the axioms and logical rules for deriving theorems, and the computer can recognize whether a proof is valid, but to determine whether a proof exists for a statement is only soluble by "waiting" for the proof or disproof to be generated. 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, which is only partially axiomatized by the
Peano axioms (described below).
In practice, not every proof is traced back to the axioms. At times, it is not even clear which collection of axioms a proof appeals 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. 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 it is possible that although they may appear arbitrary when viewed only from the point of view of the canons of deductive logic, that appearance is due to a limitation on the purposes that deductive logic serves.
Example: The Peano axiomatization of natural numbers
The mathematical system of
natural numbers 0, 1, 2, 3, 4, ... is based on an axiomatic system first devised by the mathematician
Giuseppe Peano
Giuseppe Peano (; ; 27 August 1858 – 20 April 1932) was an Italian mathematician and glottologist. The author of over 200 books and papers, he was a founder of mathematical logic and set theory, to which he contributed much notation. The stan ...
in 1889. He chose the axioms, in the language of a single unary function symbol ''S'' (short for "
successor
Successor may refer to:
* An entity that comes after another (see Succession (disambiguation))
Film and TV
* ''The Successor'' (film), a 1996 film including Laura Girling
* ''The Successor'' (TV program), a 2007 Israeli television program Musi ...
"), 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 ("''
Induction axiom''").
Axiomatization
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
An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or f ...
s) that relate a number of primitive terms — in order that a
consistent
In classical deductive logic, a consistent theory is one that does not lead to a logical contradiction. The lack of contradiction can be defined in either semantic or syntactic terms. The semantic definition states that a theory is consistent i ...
body of
propositions
In logic and linguistics, a proposition is the meaning of a declarative sentence. In philosophy, " meaning" is understood to be a non-linguistic entity which is shared by all sentences with the same meaning. Equivalently, a proposition is the ...
may be derived
deductively from these statements. Thereafter, the
proof of any proposition should be, in principle, traceable back to these axioms.
See also
*
*
*
*
*
*
*
* , an axiomatic system for set theory and today's most common foundation for mathematics.
References
Further reading
*
* Eric W. Weisstein, ''Axiomatic System'', From MathWorld—A Wolfram Web Resource
Mathworld.wolfram.comAnswers.com
{{Mathematical logic
*
Conceptual systems
Formal systems
Methods of proof