ONT Re: Model Theory
¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤
Note 21
¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤
| 1. Introduction
|
| 1.3. Languages, Models, and Satisfaction (cont.)
|
| Notice that in a given universe A there are many different
| permissible interpretations of the symbols of $L$. Suppose
| that $A$ = <A, $I$> and $A$’ = <A’, $I$’> are models for $L$
| and that R and R’ are relations of $A$ and $A$’, respectively.
| We say that R’ is the 'corresponding relation' to R if they are
| the interpretations of the same relation symbol in $L$, that is:
|
| $I$(P) = R and $I$’(P) = R’ for some P in $L$.
|
| We introduce similar conventions as regards the functions and constants.
|
| When
|
| $L$ = {P_0, ..., P_n, F_0, ..., F_m, c_0, ..., c_k},
|
| we write the models for $L$ in displayed form as:
|
| $A$ = <A, R_0, ..., R_m, G_0, ..., G_m, x_0, ..., x_k>.
|
| When the symbols of $L$ are familiar, we shall agree to use, for instance,
|
| $A$ = <A, =<, +, ·>
|
| for models of the language
|
| $L$ = {=<, +, ·}.
|
| We may resort to
|
| $A$ = <A, =<_A, +_A, ·_A>,
|
| $B$ = <B, =<_B, +_B, ·_B>,
|
| etc.,
|
| if the context of the discussion requires it.
|
| If we start with a model $A$ for the language $L$
| we can always expand it to a model for the language
| $L$’ = $L$ |_| X by giving appropriate interpretations
| for the symbols in X. If $I$’ is any interpretation for
| the symbols of X in $A$, and X is disjoint from $L$, then
| $A$’ = <A, $I$ |_| $I$’> is a model for $L$’. In this case
| we say that $A$’ is an 'expansion' of $A$ to $L$’, and $A$ is
| the 'reduct' of $A$’ to $L$. Sometimes we use the shorter
| notation ($A$, $I$’) for $A$’. Clearly, there are many
| ways a model $A$ for $L$ can be expanded to a model
| $A$’ for $L$’. On the other hand, given a model
| $A$’ for $L$’, it has only one reduction $A$ to
| $L$. Namely, we form $A$ by restricting the
| interpretation function $I$’ on $L$ |_| X
| to $L$. The processes of expansion and
| reduction do not change the universe
| of the model.
|
| The 'cardinal', or 'power', of the model $A$ is the cardinal |A|.
| $A$ is said to be finite, countable, or uncountable if |A| is
| finite, countable, or uncountable. Notice that on a finite
| universe A, while there can be only finitely many different
| relations, functions, and constants, the number of different
| interpretation functions $I$ can be very large and depends
| on |$L$|.
|
| Chang & Keisler, 'Model Theory', pages 20-21.
|
| C.C. Chang and H.J. Keisler, 'Model Theory',
| North-Holland, Amsterdam, Netherlands, 1973.
¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤