ONT Re: Model Theory
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Note 35
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| 1. Introduction
|
| 1.4. Theories and Examples of Theories (cont.)
|
| 1.4.6. Example. [The Theory of Groups].
|
| Let $L$ = {+, 0},
|
| where + is a 2-placed function symbol
| and 0 is a constant symbol.
|
| The theory of 'groups' has the following axioms:
|
| 1. Associativity
|
| x + (y + z) = (x + y) + z
|
| 2. Identity
|
| x + 0 = x
|
| 0 + x = x
|
| 3. Existence of Inverse
|
| (`E`y) (x + y = 0 & y + x = 0)
|
| A model <G, +, 0> of this theory is a 'group'.
| We obtain the theory of 'Abelian groups' when
| we add the axiom:
|
| 4. Commutativity
|
| x + y = y + x
|
| The 'order' of an element x of a group is the least n such that
| x + x + ... + x (n times) = 0. If no such n exists, the order
| of x is infinity. For a fixed n >= 1, we can write down the
| abbreviation nx for the expression:
|
| x + (x + ( ... (x + x) ... )), n times.
|
| Suppose p is a prime. The theory of 'Abelian groups
| with all elements of order p' has the extra axiom:
|
| 5_p. Prime Order
|
| px = 0
|
| 1.4.7. Proposition.
|
| Any two models of the theory of Abelian groups
| with all elements of order p of the same power
| are isomorphic.
|
| To obtain the theory of 'Abelian groups with all elements
| of order infinity (torsion-free)' we need an infinite list
| of axioms. For each n >= 1, we add the axiom:
|
| 6_n. Torsion Free
|
| x =/= 0 => nx =/= 0
|
| This theory is our first example of a non-finitely-axiomatizable theory.
| If we add a further infinite list of axioms, one for each n >= 1, thus:
|
| 7_n. Divisibility
|
| (`E`y)(ny = x)
|
| we have the theory of 'divisible torsion-free Abelian groups'.
|
| 1.4.8. Proposition.
|
| Any two uncountable divisible torsion-free Abelian groups of the
| same power are isomorphic. There are countably many such groups
| which are countable and not isomorphic.
|
| Chang & Keisler, 'Model Theory', pages 39-40.
|
| C.C. Chang and H.J. Keisler, 'Model Theory',
| North-Holland, Amsterdam, Netherlands, 1973.
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