ONT Re: Model Theory
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Note 26
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| 1. Introduction
|
| 1.3. Languages, Models, and Satisfaction (cont.)
|
| Following standard usage, |- p means that p is a theorem of $L$.
|
| If !S! is a set of sentences of $L$, then !S! |- p means
| that there is a proof of p from the logical axioms and !S!.
| If !S! = {!s!_1, ..., !s!_n} is finite, we write:
|
| !s!_1 ... !s!_n |- p.
|
| As the logical axioms are always assumed,
| we say that 'there is a proof' of p from !S!,
| or p is 'deducible' from !S!, whenever !S! |- p.
|
| !S! is 'inconsistent' iff every formula of $L$ can
| be deduced from !S!. Otherwise !S! is 'consistent'.
| A sentence !s! is consistent iff {!s!} is.
|
| !S! is 'maximal consistent' (in $L$) iff
| !S! is consistent and no set of sentences
| (of $L$) properly containing !S! is consistent.
|
| We list in the proposition below some useful, though
| simple, properties of consistent and maximal consistent
| sets of sentences. (Many of these properties are found
| also in Proposition 1.2.8.)
|
| 1.3.10. Proposition.
|
| 1. !S! is consistent if and only if every
| finite subset of !S! is consistent.
|
| 2. Let !s! be a sentence.
|
| !S! |_| {!s!} is inconsistent
|
| if and only if
|
| !S! |- ~!s!.
|
| Whence !S! |_| {!s!} is consistent
|
| if and only if
|
| ~!s! is not deducible from !S!.
|
| 3. If !S! is maximal consistent, then, for any sentences !s! and !t!:
|
| !S! |- !s! iff !s! belongs to !S!.
|
| !s! is not in !S! iff ~!s! belongs to !S!.
|
| !s! & !t! is in !S! iff !s! and !t! belong to !S!.
|
| 4. Deduction Theorem.
|
| !S! |_| {!s!} |- !t! if and only if !S! |- !s! => !t!.
|
| (Here, !s! is a sentence, although !t! need not be one.)
|
| Chang & Keisler, 'Model Theory', page 25.
|
| C.C. Chang and H.J. Keisler, 'Model Theory',
| North-Holland, Amsterdam, Netherlands, 1973.
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