Saturation (model theory)
In model theory , a structure is saturated when a large number of types are realized in it.
Notations
For a set denotes, as usual, its power , for a language it is the power of the union of the symbols of the language. For a structure denote its support amount.
definition
Let be any (possibly finite) cardinal number and a structure.
is called -saturated , if for every set with every complete (and thus every) 1-type via in is realized.
is called saturated when -saturated.
sentences
Existence of kappa-saturated extensions
The following sentence shows that saturated extensions exist:
- For every cardinal number and every infinite L-structure with there is a -saturated elementary extension with .
Universality and homogeneity
According to a theorem of Michael D. Morley and Robert Vaught , a structure is saturated if and only if it is universal and homogeneous .
Ultra products
Countable ultra products are -saturated. The following applies:
- Let be a countable language and for be a -structure. Then the ultra-product is saturated after a free ultrafilter .
In particular, it follows from the continuum hypothesis (and the next sentence, see below) that countable ultra-products of structures of the power of at most countable languages are isomorphic. These include B. the hyper real numbers .
Uniqueness of saturated structures
The following isomorphism applies:
- Be and two elementary equivalent L-patterns of the same thickness. If both structures are saturated, then they are isomorphic.
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. A complete 1-type over a finite set says exactly where the position of x is with respect to the finite set. (So there are exactly 2n + 1 complete 1-types over an n-element set.) See also: Dense order
- is -saturated, but not saturated. The type is not realized.
literature
- Gerald E. Sacks : Saturated Model Theory . WA Benjamin, 1972, ISBN 0-8053-8380-8 .
- Chang, CC; Keisler, HJ Model theory. Third edition. Studies in Logic and the Foundations of Mathematics, 73. North-Holland Publishing Co., Amsterdam, 1990. ISBN 0-444-88054-2