Relationship With Saturated Models
There is a duality between the definitions of prime and saturated models. Half of this duality is discussed in the article on saturated models, while the other half is as follows. While a saturated model realizes as many types as possible, a prime model realizes as few as possible: it is an atomic model, realizing only the types which cannot be omitted and omitting the remainder. This may be interpreted in the sense that a prime model admits "no frills": any characteristic of a model which is optional is ignored in it.
For example, the model is a prime model of the theory of the natural numbers N with a successor operation S; a non-prime model might be meaning that there is a copy of the full integers which lies disjoint from the original copy of the natural numbers within this model; in this add-on, arithmetic works as usual. These models are elementarily equivalent; their theory admits the following axiomatization (verbally):
- There is a unique element which is not the successor of any element;
- No two distinct elements have the same successor;
- No element satisfies Sn(x) = x with n>0.
These are, in fact, two of Peano's axioms, while the third follows from the first by induction (another of Peano's axioms). Any model of this theory consists of disjoint copies of the full integers in addition to the natural numbers, since once one generates a submodel from 0 all remaining points admit both predecessors and successors indefinitely. This is the outline of a proof that is a prime model.
Read more about this topic: Prime Model
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