Model theory

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 (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.

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.

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.

Definition

A model is formally defined in the context of some language L. 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. For example, the set of all sentences true in some particular model (e.g. the reals) is a theory.

Gödel's completeness theorem says that a theory has a model if and only if it is consistent, i.e. no contradiction is proved by the theory. This is the heart of model theory as it lets us answer questions about theories by looking at models and vice-versa. One should not confuse the completeness theorem with the notion of a complete theory. A complete theory is a theory which contains every sentence or its negation. Importantly, one can find a complete consistent theory extending any consistent theory.

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 and allows us to restrict the cardinality of the resulting model).

Model theory is usually concerned with first order logic, and many important results (such as the completeness and compactness theorems) fail in second order logic or other alternatives. In first order logic all infinite cardinals look the same to a language which is countable. This is expressed in the Löwenheim-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, i.e. 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 (providing you accept the axioms of set theory)! 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. Particularly the proof of the independence of the continuum hypothesis requires considering sets in models which appear to be uncountable when viewed from within the model, but are countable to someone outside the model.

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

See also

• Axiomatizable class
• Proof theory
• Hyperreals
• Compactness theorem
• Elementary embedding
• Saturated model
• Forcing (mathematics)
• Finite model theory
• Descriptive complexity
• Kripke semantics

References

• Wilfrid Hodges, A shorter model theory (1997) Cambridge University Press ISBN 0-521-58713-1

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Abstract algebra | Universal algebra | Graph theory | Category theory | Order theory | Model theory | Structural proof theory
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