Saturated model

In model theory, a structure is saturated if many types are realized in it.

Notations

For a lot of call, as usual, their thickness, for a language is the cardinality of the union of the symbols of the language. For a structure call their support amount.

Definition

Be an arbitrary ( possibly finite) cardinal number and structure.

Ie - saturated if for any amount with each complete (and thus any ) 1- type is realized via in.

Is called saturated if - saturated is.

Sets

Existence of kappa - sated extensions

That saturated extensions exist, shows the following sentence:

• For every cardinal and every infinite L- structure with a - saturated elementary extension with. 125

Ultra products

Countable products are ultra - saturated. The following applies:

• Let be a countable language and is a structure. Then the Utraprodukt is - saturated ultrafilter for a free. 148

In particular, therefore, follows from the continuum hypothesis ( and the next block, see below) that countable Utraprodukte of structures of cardinality not exceeding about countable languages ​​are isomorphic. These include, for example, the hyper- real numbers.

Uniqueness of saturated structures

We have the following isomorphism:

• Let and be two elementary equivalent L- structures of the same cardinality. If both structures saturated, then they are isomorphic. 132

Countable saturated models

A complete theory without finite models has a countable saturated model if and only if the theory is small.

Examples

• An infinite structure is apparently never - saturated if
• Is saturated. Full 1 type via a finite set states just where the position of X relative to the finite amount. ( There are so over an n- element set exactly 2n 1 complete 1- types.)
• Is - saturated, but not saturated. The type is not realized.