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Axiomatic_system.
An introduction to axiomatic systems by Burnett Meyer
Axiomatic Utility Theory Under Risk: Non-Archimedean Representations and Application to Insurance Economics (Lecture Notes in Economics and Mathematical Systems by Ulrich Schmidt
Final report on the study of the relationship between probabilistic design and axiomatic design methodology NASA grant no. NAG3-1479 (SuDoc NAS 1.26:2064 by Chinyere Onwubiko
Economic Theory of Fuzzy Equilibria: An Axiomatic Analysis (Lecture Notes in Economics and Mathematical Systems, Vol 373) by Antoine Billot
Axiomatic system
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.
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 no additional axiom can be added to the system without making the new system either dependent or inconsistent.
Models
A mathematical 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, and the property of categoriality 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.
See also
The above article is adapted from from Wikipedia All Wikipedia article text is available under the terms of the GNU Free Documentation License
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