ONT Re: Model Theory
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Note 3
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| 1. Introduction
|
| 1.1. What Is Model Theory? (cont.)
|
| There are a small number of extremely important ways in which models
| have been constructed. For example, for various purposes they can
| be constructed from individual constants, from functions, from
| Skolem terms, or from unions of chains. These constructions
| give the subject of model theory unity. To a large extent,
| we have organized this book according to these ways of
| constructing models.
|
| Another point which gives model theory unity is
| the distinction between 'syntax' and 'semantics'.
| Syntax refers to the purely formal structure of the
| language -- for instance, the length of a sentence
| and the collection of symbols occurring in a sentence,
| are syntactical properties. Semantics refers to the
| interpretation, or meaning, of the formal language --
| the truth or falsity of a sentence in a model is
| a semantical property. As we shall soon see,
| much of model theory deals with the interplay
| of syntactical and semantical ideas.
|
| We now turn to a brief historical sketch.
| The mathematical world was forced to observe
| that a theory may have more than one model in the
| 19th century, when Bolyai and Lobachevsky developed
| non-Euclidean geometry, and Riemann constructed a model
| in which the parallel postulate was false but all the
| other axioms were true. Later in the 19th century,
| Frege formally developed the predicate logic, and
| Cantor developed the intuitive set theory in which
| our models live.
|
| Model theory is a young subject. It was not clearly
| visible as a separate area of research in mathematics
| until the early 1950's. However, its historical roots
| go back to the older subjects of logic, universal algebra,
| and set theory -- and some of the early work, such as
| Löwenheim's theorem, is now classified as model theory.
| Other important early developments which contributed to
| the theory are: the extension of Löwenheim's theorem by
| Skolem (1920) and Tarski; the completeness theorem of
| Gödel (1930) and its generalization by Malcev (1936);
| the characterization of definable sets of real numbers,
| the rigorous definition of the truth of a sentence
| in a model, and the study of relational systems by
| Tarski (1931, 1933, 1935a); the construction of a
| nonstandard model of number theory by Skolem (1934);
| and the study of equational classes initiated by
| Birkhoff (1935). Model theory owes a great deal to
| general methods which were originally developed for
| special purposes in older branches of mathematics.
| We shall come across many instances of this in our
| book; to mention just one, the important notion
| of a saturated model (Chapter 5) goes back to the
| !h!_!a! [eta sub alpha]-structures in the theory
| of simple order, due to Hausdorff (1914). The
| subject grew rapidly after 1950, stimulated by
| the papers of Henkin (1949), Tarski (1950), and
| Robinson (1950). The phrase "theory of models"
| is due to Tarski (1954). Today the literature in
| the subject is quite extensive. There is a rather
| complete bibliography in Addison, Henkin, and Tarski
| (1965). In recent years, the theory of models has been
| applied to obtain significant results in other fields,
| notably set theory, algebra, and analysis. However,
| until now only a tiny part of the potential strength
| of model theory has been used in such applications.
| It will be interesting to see what happens when
| (and if) the full strength is used.
|
| Chang & Keisler, 'Model Theory', pages 2-4.
|
| C.C. Chang and H.J. Keisler, 'Model Theory',
| North-Holland, Amsterdam, Netherlands, 1973.
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