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.
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.
Read more about this topic: Axiomatic System
Famous quotes containing the word models:
“The greatest and truest models for all orators ... is Demosthenes. One who has not studied deeply and constantly all the great speeches of the great Athenian, is not prepared to speak in public. Only as the constant companion of Demosthenes, Burke, Fox, Canning and Webster, can we hope to become orators.”
—Woodrow Wilson (1856–1924)
“Today it is not the classroom nor the classics which are the repositories of models of eloquence, but the ad agencies.”
—Marshall McLuhan (1911–1980)
“The parents who wish to lead a quiet life I would say: Tell your children that they are very naughty—much naughtier than most children; point to the young people of some acquaintances as models of perfection, and impress your own children with a deep sense of their own inferiority. You carry so many more guns than they do that they cannot fight you. This is called moral influence and it will enable you to bounce them as much as you please.”
—Samuel Butler (1835–1902)