ONT Re: Model Theory
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Note 32
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| 1. Introduction
|
| 1.4. Theories and Examples of Theories
|
| A (first-order) theory T of $L$ is a collection of sentences of $L$.
| T is said to be 'closed' iff it is closed under the |= relation.
| In view of Table 1.3.1, this is the same as requiring that T
| be closed under |- . Since theories are sets of sentences
| of $L$, we may apply the expressions:
|
| a model of a theory,
|
| consistent theory,
|
| satisfiable theory,
|
| as introduced in Section 1.3.
|
| A theory T is called 'complete' (in $L$) if and only if its set of
| consequences is maximal consistent. If T is a theory of $L$, with
| $L$ c $L$’ and $L$ =/= $L$’, then T is not a closed theory of $L$’.
| On the other hand, it is easy to see that if $L$’ c $L$, then the
| 'restriction' of a closed theory T to $L$’, in symbols T | $L$’,
| is always a closed theory of $L$’. T is a 'subtheory' of T’ iff
| T c T’. If T is a subtheory of T’, then T’ is an 'extension' of T.
|
| A 'set of axioms' of a theory T is a set of sentences with the
| same consequences as T. Clearly, T is a set of axioms of T, and
| the empty set is a set of axioms of T if and only if T is a set
| of valid sentences of $L$. Every set of sentences !S! is a set
| of axioms for the closed theory T = {p : !S! |= p}. A theory T
| is 'finitely axiomatizable' iff it has a finite set of axioms.
|
| The most convenient and standard way of giving a theory T is by
| listing a finite or infinite set of axioms for it. Another way
| to give a theory is as follows: Let $A$ be a model for $L$;
| then the 'theory of' $A$ is the set of all sentences which
| hold in $A$. The theory of any model $A$ is obviously
| a complete theory.
|
| Historically, the importance of theories stems from the following
| two facts. Once the axioms of a theory are given, then by using
| the relation |- we can find out, in a syntactical manner, all the
| consequences of T. On the other hand, by using the satisfaction
| relation, we can also study all the models of T.
|
| By the extended completeness theorem, these two approaches
| give basically the same results about consequences of T.
| However, owing to the fact that models of T also have
| non-first-order properties, such as isomorphism,
| submodels, extensions, plus many others, the
| second approach leads to the field now
| known as model theory.
|
| We shall give in the rest of this section some examples of theories
| and their models to show the intimate connections that model theory
| has with other branches of mathematics. In each example we describe
| a closed theory by a set of axioms. Some classical results will be
| stated without proof.
|
| Chang & Keisler, 'Model Theory', pages 36-37.
|
| C.C. Chang and H.J. Keisler, 'Model Theory',
| North-Holland, Amsterdam, Netherlands, 1973.
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