In mathematics and set theory, an axiomatic system is any set of specified axioms from which some or all of those axioms can be used, in conjunction along with derivation rules or procedures, to logically derive theorems. A mathematical theory or set 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, however, the effort towards complete formalization 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 or set-theoritical proof within a formal system.
An axiomatic system is said to be consistent if it lacks contradiction (i.e. it is not possible 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.
The most important criterion for assessment of an axiomatic system is that particular system's consistency. Inconsistency in an axiomatic system is universally regarded as being a fatal flaw for that system.
Independence is also a desireable property, but its lack is not a fatal flaw. Lack of independence means that the system has redundancy in its axioms, meaning that one or more of its axioms is not needed. This is usually considered to be a flaw because reducing the number of axioms of a system to the minimum necessary for deriving all the needed or desired theorems of that system is considered to be a virtue, because axioms are unproved and unprovable; having as little of that as possible means that as few unproved assumptions as possible are being made in that system.
An axiomatic system will be called complete if for every statement, either itself or its negation, is derivable in that system. This is very difficult to achieve, however, and as shown by the combined works of Gödel and Coen, impossible for axiomatic systems involving infinite sets. So, along with consistency, relative consistency is also the mark of a worthwhile axiom system. This occurs when the undefined terms of a first axiom system are given 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.
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.
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.
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).
That approach, in which axioms were supposed to be self-evident and thus indisputable, was swept away during the course of the nineteenth century. One important episode in this was the development of Non-Euclidean geometry, based on denial of Euclid's parallel postulate (or axiom). It was found that consistent geometries can be constructed by denying that postulate, taking as an axiom that more than one parallel to a given line can be drawn through a point outside that line, or a different axiom that no parallel can be drawn—both of those result in different and consistent geometric systems that may or may not be applicable to an experienced world.
Other challenges to the supposed self-evidence of axioms came from the foundations of real analysis, from Georg Cantor's set theory, and from the failure of Frege's work on foundations. Russell was able to derive a paradox—a kind of contradiction—from Frege's axioms for set theory, thus showing that Frege's axiomatic system was not consistent, and this showed that the supposed self-evidence of Frege's axioms was mistaken.
Another challenge came from David 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's origins of those studies.
Therefore, there are at least three 'modes' of axiomatic method current in mathematics, and in the fields it influences. In caricature, possible attitudes are:
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.
It is easy to see that the axiomatic method has limitations outside mathematics and set theory. 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.
Historically, the most important purpose of an axiom system was to reach an overview of some science or part of science. Euclid succeeded in doing that for geometry. Beyond its successful use in set theory and mathematics, there have been attempts to use the axiomatic method in physics (by Ludwig Boltzmann, Heinrich Hertz, and some members of the Vienna Circle), biology (by J. H. Woodger), quantum mechanics (by Günther Ludwig), and possibly other sciences.
Those attempts have been at best only partly successful. If such efforts at axiomatization for sciences were to be successful, this would make it possible to study these sciences simply by drawing logical inferences from the axioms, without needing any new empirical input. When conclusions are drawn from general scientific laws or principles that method is in fact employed, empirical testing of such theoretical derivations is still always needed. Thus, the axiomatization and formalization of the system is incomplete and does not solve the problem of whether the system yields actual scientific knowledge.
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