ONT Re: Model Theory
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Note 16
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| 1. Introduction
|
| 1.2. Model Theory for Sentential Logic (cont.)
|
| We now turn to another kind of sentence.
| By a 'conditional sentence' we mean a
| sentence p_1 & ... & p_n, where each
| p_i is one of the following kinds:
|
| 1. S,
|
| 2. ~S_1 v ~S_2 v ... v ~S_k,
|
| 3. ~S_1 v ~S_2 v ... v ~S_k v S.
|
| A set !S! of sentences is said to be
| 'preserved under finite intersections' iff
| A |= !S! and B |= !S! implies A |^| B |= !S!.
|
| !S! is said to be 'preserved under arbitrary intersections'
| iff for every nonempty set {A_i : i in I} of models of !S!,
| the intersection |^|_(i in I) A_i is also a model of !S!.
|
| 1.2.17. Lemma.
|
| A theory !C! is preserved under finite intersections
|
| if and only if
|
| !C! is preserved under arbitrary intersections.
|
| Proof. Let !C! be preserved under finite intersections, let {A_i : i in I}
| be a nonempty set of models of !C!, and let B = |^|_(i in I) A_i.
| Let !S! be the set of all sentences of the form S or ~S which hold
| in B. We show that !C! |_| !S! is satisfiable. Let !S!_0 be an
| arbitrary finite subset of !S!, and let the negative sentences in
| !S!_0 be ~S_1, ..., ~S_k. If k = 0, all the sentences in !S!_0 are
| positive, and each of the models A_i is a model of !S!_0, because
| B c A_i. Let k > 0 and choose models A_i_1, ..., A_i_k from among
| the A_i such that S_1 is not in A_i_1, ..., S_k is not in A_i_k.
| Then A = A_i_1 |^| ... |^| A_i_k is a model of !S!_0. Since !C!
| is preserved under finite intersections, A is also a model of !C!.
| We have shown that !C! |_| !S! is finitely satisfiable. By the
| compactness theorem, !C! |_| !S! has a model. But the only model
| of !S! is B, so B is a model of !C!. -|
|
| In view of the above lemma, we may as well simply say from now on
| that !C! is 'preserved under intersections', since it makes no
| difference whether we say finite or arbitrary intersections.
|
| Chang & Keisler, 'Model Theory', pages 14-15.
|
| C.C. Chang and H.J. Keisler, 'Model Theory',
| North-Holland, Amsterdam, Netherlands, 1973.
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