ONT Re: Model Theory
¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤
Note 2
¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤
| 1. Introduction
|
| 1.1. What Is Model Theory? (cont.)
|
| What kinds of theorems are proved in model theory?
| We can already give a few examples. Perhaps the earliest
| theorem in model theory is Löwenheim's theorem (Löwenheim, 1915):
| If a sentence has an infinite model, then it has a countable model.
| Another classical result is the compactness theorem, due to Gödel (1930)
| and Malcev (1936): If each finite subset of a set !S! of sentences has a
| model, then the whole set !S! has a model. As a third example, we may state
| a more recent result, due to Morley (1965). Let us say that a set !S! of
| sentences is 'categorical' in power !a! iff there is, up to isomorphism,
| only one model of !S! of power !a!. Morley's theorem states that, if
| !S! is categorical in one uncountable power, then !S! is categorical
| in every uncountable power.
|
| These theorems are typical results of model theory. They say something
| negative about the "power of expression" of first-order predicate logic.
| Thus Löwenheim's theorem shows that no consistent sentence can imply
| that a model is uncountable. Morley's theorem shows that first-order
| predicate logic cannot, as far as categoricity is concerned, tell
| the difference between one uncountable power and another. And the
| compactness theorem has been used to show that many interesting
| properties of models cannot be expressed by a set of first-order
| sentences -- for instance, there is no set of sentences whose
| models are precisely all the finite models.
|
| The three theorems we have stated also say something positive about the
| existence of models having certain properties. Indeed, in almost all
| of the deeper theorems in model theory the key to the proof is to
| construct the right kind of a model. For instance, look again
| at Löwenheim's theorem. To prove that theorem, we must begin
| with an uncountable model of a given sentence and construct
| from it a countable model of the sentence. Likewise,
| to prove the compactness theorem we must construct
| a single model in which each sentence of !S! is
| true. Even Morley's theorem depends vitally
| on the construction of a model. To prove
| it we begin with the assumption that
| !S! has two different models of one
| uncountable power and construct
| two different models of every
| other uncountable power.
|
| Chang & Keisler, 'Model Theory', page 2.
|
| C.C. Chang and H.J. Keisler, 'Model Theory',
| North-Holland, Amsterdam, Netherlands, 1973.
¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤