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.

The independence of the axiom of choice and the continuum hypothesis from the other axioms of set theory (proven 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.

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 "∃ x (x × x = 1 + 1)" in this language, then it's clear that the sentence is true for the reals - there is such a real number x; for the rationals, however, the sentence is false. Conversely, "∃ x (x × x = 0 - 1)" is false in the reals - to make it true we can add a constant symbol i and a new axiom "i × i = 0 - 1", which gives us the complex numbers.

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.

A model is formally defined in context of some language L, following Tarski's concept of truth. The model consists of two things:

  1. A universe set U which contains all the objects of interest (the "domain of discourse"), and
  2. a mapping from L to U (called the evaluation mapping or interpretation function) which has as its domain all constant, predicate and function symbols in the language.

A theory is defined as a set of sentences which is consistent; often it is also defined to be closed under logical consequence. Under this definition a theory is thus a maximally consistent set of sentences. For example, the set of all sentences true in some particular model (e.g. the reals) or class of models is a theory.

Completeness in model theory is defined as the property that every statement in a language or its opposite is provable from some theory. Complete theories are desirable since they describe fully some model.

The compactness theorem states that a set of sentences S is satisfiable, i.e., has a model, if every finite subset of S is satisfiable. In the context of proof theory the analogous statement is trivial, since every proof can have only a finite number of antecedents used in the proof; in the context of model theory however, this proof is somewhat more difficult. There are two well known proofs, one by Gödel (which goes via proofs) and one by Malcev (which is more direct).

Model theory is concerned with first order logic, and to first order logic all cardinals look the same. This is expressed in the Lowenheim-Skolem theorems - which state that any theory with an infinite model A has models of all infinite cardinalities (at least that of the language) which agree with A on all sentences - they are "elementarily equivalent".

So in particular, set theory (whose language is countable) has a countable model - this is known as Skolem's Paradox, even though it's true! To see why it was thought paradoxical, consider that there are sentences in set theory which postulate the existence of uncountable sets - and these sentences are true in our countable model.

TODO - Vaught's test. Extensions, Embeddings and Diagrams. To give a flavor, mentioning the hyperreals would be good. (All of these need substantial filling out)

Note: The unrelated term 'mathematical model' is also used informally in other parts of mathematics and science.

See also: