Axiomatization

In mathematics and logic, an axiomatic system is a set of formal statements (i.e. axioms) used to logically derive other statements such as lemmas or theorems. A proof within an axiom system is a sequence of deductive steps that establishes a new statement as a consequence of the axioms. An axiom system is called complete with respect to a property if every formula with the property can be derived using the axioms. The more general term theory 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) a 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.

Properties

Four typical properties of an axiom system are consistency, relative consistency, completeness and independence. An axiomatic system is said to be ''consistent'' if it lacks contradiction. That is, it is impossible to derive both a statement and its negation from the system's axioms.A. G. Howson A Handbook of Terms Used in Algebra and Analysis, Cambridge UP, ISBN 0521084342 1972 pp 6
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 another
axiom set B is consistent.

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, 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 '' 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. However, note that in some cases it may be undecidable if a statement can be proven or not.

Axioms and models

A model for an axiomatic system is a well-defined 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. If an axiom system has a model, the axioms are said to have been satisfied. C. C. Chang and H. J. Keisler "Model Theory" Elsevier 1990, pp 1-7 The existence of a model which satisfies an axiom system, proves the consistency of the system.

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.

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 categorical or categorial. However, this term should not be confused with the topic of category 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 the semantics of the system.

Example

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

:$\exist x_1: \exist x_2: \lnot (x_1=x_2)$ (informally, there exist two different items).

:$\exist x_1: \exist x_2: \exist x_3: \lnot (x_1=x_2) \land \lnot (x_1=x_3) \land \lnot (x_2=x_3)$ (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 of such a 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 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.

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. This way of doing mathematics is called the axiomatic method.

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.

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. Mathematicians decided to consider topological spaces more generally without the separation axiom which Felix Hausdorff 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. 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 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 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 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 proofs, hence the axiomatic method.

Many axiomatic systems were developed in the nineteenth century, including non-Euclidean geometry, the foundations of real analysis, Cantor's set theory, 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 elements 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

The mathematical system of natural numbers 0, 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"), 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 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. axioms) that relate a number of primitive terms — in order that a consistent body of propositions may be derived deductively from these statements. Thereafter, the proof of any proposition should be, in principle, traceable back to these axioms.

If the formal system is not 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

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* , an axiomatic system for set theory and today's most common foundation for mathematics.

References

Further reading

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* Eric W. Weisstein, ''Axiomatic System'', From MathWorld—A Wolfram Web Resource
Mathworld.wolfram.com

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