# axiomatic method

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. 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 mathematical proof within a formal system.

## Properties

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 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. 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. 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. 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.

## 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. In their book Principia Mathematica, Alfred North Whitehead and Betrand 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 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 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. Mathematics decided to consider topological spaces more generally without the separation axiom which Felix Hausdorff originally formulated.

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. One such problem was the Continuum hypothesis.

### 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. 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). 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. 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.

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.

### 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 programme can recognize whether a given proposition in the language is an axiom. Gödel's First Incompleteness Theorem then tells us that there are certain consistent bodies of propositions with no recursive axiomatization. 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. 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. 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, ... is based on an axiomatic system that was first written down by the mathematician Peano in 1901. 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:
• 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 ab, then SaSb.
• 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.

### Outside of mathematics

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 [Confusing — Please clarify].

In mathematics, axiomatization 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, tracable back to these axioms.

## References

Mathematics (colloquially, maths or math) is the body of knowledge centered on such concepts as quantity, structure, space, and change, and also the academic discipline that studies them. Benjamin Peirce called it "the science that draws necessary conclusions".
SET may stand for:
• Sanlih Entertainment Television, a television channel in Taiwan
• Secure electronic transaction, a protocol used for credit card processing,

axiom is a sentence or proposition that is not proved or demonstrated and is considered as self-evident or as an initial necessary consensus for a theory building or acceptation.
theorem is a statement, often stated in natural language, that can be proved on the basis of explicitly stated or previously agreed assumptions. In logic, a theorem is a statement in a formal language that can be derived by applying rules and axioms from a deductive system.
The word theory has a number of distinct meanings in different fields of knowledge, depending on their methodologies and the context of discussion.

In common usage, people often use the word theory to signify a conjecture, an opinion, or a speculation.
This article or section is in need of attention from an expert on the subject.
This article discusses model theory as a mathematical discipline and not the informally used term mathematical model as used in other parts of mathematics and science.

In mathematics, a proof is a demonstration that, assuming certain axioms, some statement is necessarily true. A proof is a logical argument, not an empirical one. That is, one must demonstrate that a proposition is true in all cases before it is considered a theorem of mathematics.
In logic, a contradiction consists of a logical incompatibility between two or more propositions. It occurs when the propositions, taken together, yield two conclusions which form the logical inversions of each other.
Kurt Gödel

Kurt Gödel
Born March 28 1906
Brünn (Brno) Austria-Hungary
Died January 14 1978 (aged 73)
Paul J. Cohen
Born March 2 1934
Long Branch, New Jersey
Died March 23 2007 (aged 74)
Stanford, California
This article discusses model theory as a mathematical discipline and not the informally used term mathematical model as used in other parts of mathematics and science.

SET may stand for:
• Sanlih Entertainment Television, a television channel in Taiwan
• Secure electronic transaction, a protocol used for credit card processing,

In mathematical logic, a formal system is consistent if it does not contain a contradiction, or, more precisely, for no proposition φ are both φ and ¬φ provable.

A consistency proof is a formal proof that a formal system is consistent.
In mathematics, an isomorphism (Greek: isos "equal", and morphe "shape") is a bijective map f such that both f and its inverse f −1 are homomorphisms, i.e., structure-preserving mappings.
Euclidean geometry is a mathematical system attributed to the Greek mathematician Euclid of Alexandria. Euclid's text Elements is the earliest known systematic discussion of geometry.
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 some or all mathematics is reducible to logic.
Principia Mathematica is a 3-volume work on the foundations of mathematics, written by Alfred North Whitehead and Bertrand Russell and published in 1910–1913. It is an attempt to derive all mathematical truths from a well-defined set of axioms and inference rules in
Alfred North Whitehead, OM (February 15 1861, Ramsgate, Kent, England – December 30 1947, Cambridge, Massachusetts, U.S.) was an English-born mathematician who became a philosopher.
Bertrand Arthur William Russell, 3rd Earl Russell, OM, FRS, (18 May 1872 – 2 February 1970), was a British philosopher, historian, logician, mathematician, advocate for social reform, pacifist, and prominent rationalist.
twentieth century of the Common Era began on January 1, 1901 and ended on December 31, 2000, according to the Gregorian calendar. Some historians consider the era from about 1914 to 1991 to be the Short Twentieth Century.
Homological algebra is the branch of mathematics which studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precursor to algebraic topology) and abstract algebra (theory of
In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. The branch of abstract algebra which studies rings is called ring theory.
In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation obeys the commutative law. This means that if a and b are any elements of the ring, then a×b=b×a.
Amalie Emmy Noether[1] (March 23 1882 – April 14 1935) was a German-born Jewish mathematician, said by Einstein in eulogy to be "[i]n the judgment of the most competent living mathematicians, [...

Topological spaces are mathematical structures that allow the formalization of concepts such as convergence, connectedness and continuity.
separation axioms. These are sometimes called Tychonoff separation axioms, after Andrey Tychonoff.

The separation axioms are axioms only in the sense that, when defining the notion of topological space, you could add these conditions as extra axioms to get a more
Felix Hausdorff

Born November 8 1868
Breslau, Germany
Died January 26 1942 (aged 75)
Bonn, Germany