ONT Re: Model Theory
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Note 22
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| 1. Introduction
|
| 1.3. Languages, Models, and Satisfaction (cont.)
|
| We next introduce some simple but basic notions and operations on models.
| The reader should go through the exercises at the end of this section in
| order to be familiar with them.
|
| Two models $A$ and $A$’ for $L$ are 'isomorphic' iff
| there is a 1-1 function f mapping A onto A’ satisfying:
|
| 1. For each n-placed relation R of $A$ and
| the corresponding relation R’ of $A$’
|
| R(x_1 ... x_n) if and only if R’(f(x_1) ... f(x_n))
|
| for all x_1, ..., x_n in A.
|
| 2. For each m-placed function G of $A$ and
| the corresponding function G’ of $A$’
|
| f(G(x_1 ... x_m)) = G’(f(x_1) ... f(x_m))
|
| for all x_1, ..., x_m in A.
|
| 3. For each constant x of $A$ and the
| corresponding constant x’ of $A$’
|
| f(x) = x’.
|
| A function f that satisfies the above is called an 'isomorphism' of $A$ onto $A$’,
| or an 'isomorphism' between $A$ and $A$’. We use the notation f : $A$ ~=~ $A$’
| to denote that f is an isomorphism of $A$ onto $A$’, and we use $A$ ~=~ $A$’
| for $A$ is isomorphic to $A$’. For convenience we use ~=~ to denote the
| 'isomorphism relation' between models for $L$. It is quite clear that
| ~=~ is an equivalence relation. Furthermore, it preserves powers,
| that is, if $A$ ~=~ $B$, then |A| = |B|. Indeed, unless we wish
| to consider the particular structure of each element of A or B,
| for all practical purposes $A$ and $B$ are the same if they
| are isomorphic.
|
| Chang & Keisler, 'Model Theory', page 21.
|
| C.C. Chang and H.J. Keisler, 'Model Theory',
| North-Holland, Amsterdam, Netherlands, 1973.
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