ONT Re: Model Theory
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Note 33
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| 1. Introduction
|
| 1.4. Theories and Examples of Theories (cont.)
|
| 1.4.1. Let $L$ consist of the single 2-placed relation symbol =<.
| Using the usual notation for =<, we write x =< y for =<(x, y).
| The theory of 'partial order' has three axioms:
|
| 1. (`A`xyz)(x =< y & y =< z => x =< z),
|
| 2. (`A`xy) (x =< y & y =< x => x = y),
|
| 3. (`A`x) (x =< x).
|
| They are, respectively, the transitive, antisymmetric, and reflexive properties of
| partial orders. Any model <A, =< > of this theory consists of a nonempty set A
| and a partial order relation =< on A. If we add the comparability axiom:
|
| 4. (`A`xy) (x =< y or y =< x),
|
| we obtain the theory of 'simple order' (also called 'linear order').
| A model <A, =< > for this theory is a simply-ordered set. Adding
| two more axioms (writing x =/= y for ~(x = y)):
|
| 5. (`A`xy) (x =< y & x =/= y
|
| => (`E`z)(x =< z & z =/= x & z =< y & z =/= y)),
|
| 6. (`E`xy) (x =/= y),
|
| we then have the theory of 'dense (simple) order'.
| The rationals with the usual =< is an example of
| a model of this theory. The theory of dense order
| has no finite models. If we wish to consider only
| dense orders 'without endpoints', we add the axioms:
|
| 7. (`A`x)(`E`y)(x =< y & x =/= y),
|
| 8. (`A`x)(`E`y)(y =< x & x =/= y).
|
| 1.4.2. Proposition.
|
| Any two countable models of the theory
| of dense order without endpoints
| are isomorphic.
|
| Chang & Keisler, 'Model Theory', pages 37-38.
|
| C.C. Chang and H.J. Keisler, 'Model Theory',
| North-Holland, Amsterdam, Netherlands, 1973.
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