In mathematics, and in particular model theory, a prime model is a model is prime if it admits an elementary embedding to which it is elementarily equivalent satisfying the same complete theory).

Cardinality

In contrast with the notion of saturated model, prime models are restricted to very specific cardinalities by the Löwenheim–Skolem theorem is a first-order language can have larger cardinality since at the very least it must be elementarily embedded in such a model. This still leaves much ambiguity in the actual cardinality. In the case of countable languages, all prime models are at most countably infinite.

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 typess as possible, a prime model realizes as few as possible: it is an atomic model, realizing only the types that 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 that is optional is ignored in it.

is a prime model of the theory of the natural numbers N with a successor operation S meaning that there is a copy of the full integers that 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):

1. There is a unique element that is not the successor of any element;

2. No two distinct elements have the same successor;

3. No element satisfies Sn(x) = x with n>0.

These are, in fact, two of Peano's axioms is a prime model.

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