ONT Model Theory
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Note 1
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| Model Theory
|
| 1. Introduction
|
| 1.1. What Is Model Theory?
|
| Model theory is the branch of mathematical logic which deals with the
| relation between a formal language and its interpretations, or models.
| We shall concentrate on the model theory of first-order predicate logic,
| which may be called "classical model theory".
|
| Let us now take a short introductory tour of model theory.
| We begin with the models which are structures of the kind which
| arise in mathematics. For example, the cyclic group of order 5,
| the field of rational numbers, and the partially-ordered structure
| consisting of all sets of integers ordered by inclusion, are models
| of the kind we consider. At this point we could, if we wish, study
| our models at once without bringing the formal language into the
| picture. We would then be in the area known as universal algebra,
| which deals with homomorphisms, substructures, free structures,
| direct products, and the like. The line between universal
| algebra and model theory is sometimes fuzzy; our own
| usage is explained by the equation:
|
| universal algebra + logic = model theory.
|
| To arrive at model theory, we set up our formal language, the
| first-order logic with identity. We specify a list of symbols
| and then give precise rules by which sentences can be built up
| from the symbols. The reason for setting up a formal language is
| that we wish to use the sentences to say things about the models.
| This is accomplished by giving a basic 'truth definition', which
| specifies for each pair consisting of a sentence and a model one
| of the truth values 'true' or 'false'.
|
| The truth definition is the bridge connecting the formal language with
| its interpretation by means of models. If the truth value "true" goes
| with the sentence !p! and model !A!, we say that !p! is 'true' in !A!
| and also that !A! is a 'model' of !p!. Otherwise we say that !p! is
| 'false' in !A! and that !A! is not a model of !p!. Moreover, we say
| that !A! is a 'model' of a set !S! of sentences iff !A! is a model
| of each sentence in the set !S!.
|
| Chang & Keisler, 'Model Theory', pages 1-2.
|
| C.C. Chang and H.J. Keisler, 'Model Theory',
| North-Holland, Amsterdam, Netherlands, 1973.
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