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.
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.
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 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 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.
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.
Not every consistent body of propositions can be captured by a describable collection of axioms. Call a collection of axioms recursive if a computer program 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.
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, traceable back to these axioms.