Model theory

:This article discusses model theory as a mathematical discipline and not the informally used term mathematical model as used in other parts of mathematics and science.

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In mathematics, model theory is the study of the representation of mathematical concepts in terms of set theory, or the study of the models which underlie mathematical systems. It assumes that there are some pre-existing mathematical objects out there, and asks questions regarding how or what can be proven given the objects, some operations or relations amongst the objects, and a set of axioms.

Related Topics:
Mathematics - Set theory - Mathematical system

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The independence of the axiom of choice and the continuum hypothesis from the other axioms of set theory (proved by Paul Cohen and Kurt Gödel) are the two most famous results arising from model theory. It was proven that both the axiom of choice and its negation are consistent with the Zermelo-Fraenkel axioms of set theory; the same result holds for the continuum hypothesis. These results are a part of axiomatic set theory, a particular application of model theory.

Related Topics:
Axiom of choice - Continuum hypothesis - Set theory - Paul Cohen - Kurt Gödel - Zermelo-Fraenkel axioms - Axiomatic set theory

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An example of the concepts of model theory is provided by the theory of the real numbers. We start with a set of individuals, where each individual is a real number, and a set of relations and/or functions, such as { ×, +, −, ., 0, 1 }. If we ask a question such as "∃ y (y × y = 1 + 1)" in this language, then it is clear that the sentence is true for the reals - there is such a real number y, namely the square root of 2; for the rational numbers, however, the sentence is false. A similar proposition, "∃ y (y × y = 0 − 1)", is false in the reals, but is true in the complex numbers, where i × i = 0 − 1.

Related Topics:
Real number - Square root - Rational number - Complex number

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Model theory is then concerned with what is provable within given mathematical systems, and how these systems relate to each other. It is particularly concerned with what happens when we try to extend some system by the addition of new axioms or new language constructs.

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