ONT Re: Model Theory
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Note 20
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| 1. Introduction
|
| 1.3. Languages, Models, and Satisfaction (cont.)
|
| Turning now to the models for a given language $L$, we first point out that
| the situation here is more complicated than for the sentential logic $S$
| in Section 1.2. There, each S in $S$ could take on at most two values,
| true or false. Thus the set of intended interpretations for $S$ has
| rather simple properties, as the reader discovered. This time, each
| n-placed relation symbol has as its intended interpretations all
| n-placed relations among the objects, each m-placed function
| symbol has as its intended interpretations all m-placed
| functions from objects to objects, and, finally, each
| constant symbol has as intended interpretations
| fixed or constant objects.
|
| Therefore, a "possible world", or model for $L$ consists, first of all,
| of a 'universe' A, a nonempty set. In this universe, each n-placed P
| corresponds to an n-placed 'relation' R c A^n on A, each m-placed F
| corresponds to an m-placed 'function' G : A^m -> A on A, and each
| constant symbol 'c' corresponds to a 'constant' x in A.
|
| This correspondence is given by an 'interpretation' function $I$ mapping
| the symbols of $L$ to appropriate relations, functions, and constants in A.
|
| A 'model' for $L$ is a pair <A, $I$>.
|
| We use Gothic [$Script$] letters to range over models. Thus we write
| $A$ = <A, $I$>, $B$ = <B, $J$>, $C$ = <C, $K$>, etc., with appropriate
| subscripts and superscripts. We shall try to be quite consistent in this
| respect, so that the universes of the models $B$’, $B$”, $B$_i, $B$_j, etc.,
| are precisely the sets B’, B”, B_i, B_j, etc. The relations, functions,
| and constants of $A$ are, respectively, the images under $I$ of the
| relation symbols, function symbols, and constant symbols of $L$.
|
| Chang & Keisler, 'Model Theory', pages 19-20.
|
| C.C. Chang and H.J. Keisler, 'Model Theory',
| North-Holland, Amsterdam, Netherlands, 1973.
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