Prime model
Encyclopedia
In mathematics
, and in particular model theory
, a prime model is a model which is as simple as possible. Specifically, a model
is prime if it admits an elementary embedding into any model 
to which it is elementarily equivalent (that is, into any model 
satisfying the same complete theory
as
).
, prime models are restricted to very specific cardinalities by the Löwenheim-Skolem theorem. If
is a first-order language with cardinality 
and 
a complete theory over 
then this theorem guarantees a model for 
of cardinality 
therefore no prime model of 
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 unless 
which admits no smaller cardinalities; therefore one often talks about countable languages, in which all prime models are also countable.
s, 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):
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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, and in particular model theory
Model theory
In mathematics, model theory is the study of mathematical structures using tools from mathematical logic....
, a prime model is a model which is as simple as possible. Specifically, a model
is prime if it admits an elementary embedding into any model to which it is elementarily equivalent (that is, into any model satisfying the same complete theoryComplete theory
In mathematical logic, a theory is complete if it is a maximal consistent set of sentences, i.e., if it is consistent, and none of its proper extensions is consistent...
as
).Cardinality
In contrast with the notion of saturated modelSaturated model
In mathematical logic, and particularly in its subfield model theory, a saturated model M is one which realizes as many complete types as may be "reasonably expected" given its size...
, prime models are restricted to very specific cardinalities by the Löwenheim-Skolem theorem. If
is a first-order language with cardinality and a complete theory over then this theorem guarantees a model for of cardinality therefore no prime model of 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 unless which admits no smaller cardinalities; therefore one often talks about countable languages, in which all prime models are also countable.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 modelSaturated model
In mathematical logic, and particularly in its subfield model theory, a saturated model M is one which realizes as many complete types as may be "reasonably expected" given its size...
s, 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.