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.

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 theorems can be proven given the objects, some operations or relations amongst the objects, and a set of axioms.

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 proved 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 applications of model-theoretic methods to axiomatic set theory.

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