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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 "∃ 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. Conversely, "∃ y (y × y = 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. . The model consists of two things:
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 theoremModel theory Godel's completeness theorem is a fundamental theorem in mathematical logic proved by Kurt Godel in 1929. It states, in its most familiar form, that in first-order predicate calculus every universally valid formula can be proved. The word "pr says that a theory has a model if and only if it is consistentConsistency has three technical meanings: In mathematics and logic, as well as in theoretical physics, it refers to the proposition that a formal theory or a physical theory contains no contradictions. See consistency proof. In statistics, consistency (st, 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 theoremModel theory The compactness theorem is a basic fact in symbolic logic and model theory and asserts that a set of first-order sentences is satisfiable, i. has a model, if and only if every finite subset of it is satisfiable. The compactness theorem for th 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 completenessModel theory Godel's completeness theorem is a fundamental theorem in mathematical logic proved by Kurt Godel in 1929. It states, in its most familiar form, that in first-order predicate calculus every universally valid formula can be proved. The word "pr and compactnessModel theory The compactness theorem is a basic fact in symbolic logic and model theory and asserts that a set of first-order sentences is satisfiable, i. has a model, if and only if every finite subset of it is satisfiable. The compactness theorem for th 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 - 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 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.
TODO - Vaught's test. Extensions, Embeddings and Diagrams. To give a flavor, mentioning the hyperreals and/or the extension of the concepts of basis and dimension to strongly minimal theories 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.
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