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 preexisting 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 in 1963) 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 ZermeloFraenkel axioms of set theory; the similar result holds for the continuum hypothesis.
In the case of the real numbers, one would start with a set of individuals, where each individual is a real number, and a set of relations, such as {×,+,,.,0,1}. If we ask a question such as "∃ x (x × x = 1 + 1)" in this language, then it's clear that there's an answer in the reals; there is however no answer in the rational numbers. This model is not large enough to support a query such as "∃ x (x × x = 0  1  1)"; to do that an additional symbol "i" defined as the constant obtained from "∃ x (x × x = 0  1)" must be added to the language.
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. Restricting the cardinality of models and noting that facts are true may allow the compactness theorem to be invoked, showing some theorem is true in a model of larger cardinality.
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. Under this definition a theory is thus a maximally consistent set of sentences.
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 and Malcev[?].
Elementary equivalence LST theorem and Vaught's test.
Extensions, Embeddings and Diagrams. Upward and downward LowenheimSkolem theorems. To give a flavor, mentioning the hyperreals[?] would be good. (All of these need substantial filling out)
Note: The term 'mathematical model' is also used informally in other parts of mathematics and science.
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