Axiomatic systems

From New World Encyclopedia

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.

Properties

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 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 it 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, and 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.

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.

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.

Axiomatic method

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).

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. One important episodes in this was the development of Non-Euclidean geometry, based on denial of Euclid's parallel postulate. It was found that this postulate is not self-evident, and that consistent geometries can be constructed by denying it, 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.


, 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.

Therefore, there are at least three 'modes' of axiomatic method current in mathematics, and in the fields it influences. In caricature, possible attitudes are:

  1. Accept my axioms and you must accept their consequences;
  2. I reject one of your axioms and accept extra models;
  3. My set of axioms defines a research programme.

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. 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.

See also

  • Axiomatization
  • Model theory
  • Gödel's incompleteness theorem

References
ISBN links support NWE through referral fees

  • Bernays, Paul, Axiomatic Set Theory, With a hist. introd. by Abraham A. Fraenkel, Amsterdam: North-Holland Pub. Co., 1958.
  • Henkin, Leon, The Axiomatic Method With Special Reference to Geometry and Physics, Amsterdam: North-Holland Pub. Co., 1959.
  • Lightstone, A. H., The Axiomatic Method; An Introduction to Mathematical Logic, Englewood Cliffs, N.J.: Prentice-Hall, 1964.
  • Meyer, Burnett, An Introduction to Axiomatic Systems, Boston: Prindle, Weber & Schmidt, 1974. ISBN 0871501740
  • Quine, W. V., Set Theory and Its Logic, Rev. Ed., Cambridge: Belknap Press of Harvard University Press, 1969.
  • Suppes, Patrick, Axiomatic Set Theory, New York: Dover Publications, 1972. ISBN 0486616304
  • Takeuti, Gaisi and W.M. Zaring, Axiomatic Set Theory, New York: Springer-Verlag, 1973. ISBN 0387900519 0387900500

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