ONT Re: Model Theory
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Note 31
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| 1. Introduction
|
| 1.3. Languages, Models, and Satisfaction (cont.)
|
| We conclude this section by stating a number of important results
| without proofs, but whose proofs will be given in the next chapter.
|
| 1.3.20. Theorem. (Gödel's Completeness Theorem).
|
| Given any sentence !s!,
| !s! is a theorem
| if and only if
| !s! is valid.
|
| 1.3.21. Theorem. (Extended Completeness Theorem).
|
| Let !S! be any set of sentences.
| Then !S! is consistent
| iff !S! has a model.
|
| 1.3.22. Theoreom. (Compactness Theorem).
|
| A set of sentences !S! has a model iff
| every finite subset of !S! has a model.
|
| As in Section 1.2, we conclude with a table of equivalent notions.
|
| Table 1.3.1
| o-----------------------------o-----------------------------o
| | Syntax | Semantics |
| o-----------------------------o-----------------------------o
| | | |
| | p is a theorem | p is valid |
| | | |
| | |- p | |= p |
| | | |
| o-----------------------------o-----------------------------o
| | | |
| | !S! is consistent | !S! has a model |
| | | |
| o-----------------------------o-----------------------------o
| | | |
| | p is deducible from !S! | p is a consequence of !S! |
| | | |
| | !S! |- p | !S! |= p |
| | | |
| o-----------------------------o-----------------------------o
|
| Chang & Keisler, 'Model Theory', pages 32-33.
|
| C.C. Chang and H.J. Keisler, 'Model Theory',
| North-Holland, Amsterdam, Netherlands, 1973.
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