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. ${\displaystyle X}$ ${\displaystyle |X|}$ ${\displaystyle L}$ ${\displaystyle |L|}$ ${\displaystyle {\mathfrak {M}}}$ ${\displaystyle M}$

definition

Let be any (possibly finite) cardinal number and a structure. ${\displaystyle \kappa }$ ${\displaystyle {\mathfrak {M}}}$

${\displaystyle {\mathfrak {M}}}$is called -saturated , if for every set with every complete (and thus every) 1-type via in is realized. ${\displaystyle \kappa }$ ${\displaystyle A\subseteq M}$ ${\displaystyle |A|<\kappa }$ ${\displaystyle A}$ ${\displaystyle {\mathfrak {M}}}$

${\displaystyle {\mathfrak {M}}}$is called saturated when -saturated. ${\displaystyle {\mathfrak {M}}\ |M|}$

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 . ${\displaystyle \kappa \geqslant |L|}$ ${\displaystyle {\mathfrak {M}}}$ ${\displaystyle |M|<2^{\kappa }}$ ${\displaystyle \kappa ^{+}}$ ${\displaystyle {\mathfrak {M}}^{*}}$ ${\displaystyle |M^{*}|\leqslant 2^{\kappa }}$

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: ${\displaystyle \aleph _{1}}$

• Let be a countable language and for be a -structure. Then the ultra-product is saturated after a free ultrafilter . ${\displaystyle L}$ ${\displaystyle i\in \mathbb {N} }$ ${\displaystyle {\mathfrak {M}}^{i}}$ ${\displaystyle L}$ ${\displaystyle \aleph _{1}}$

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 . ${\displaystyle 2^{\aleph _{0}}}$

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. ${\displaystyle {\mathfrak {M}}}$ ${\displaystyle {\mathfrak {M}}'}$

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 ${\displaystyle {\mathfrak {M}}}$ ${\displaystyle \kappa }$ ${\displaystyle \kappa >|M|}$
• ${\displaystyle (\mathbb {Q} ,<)}$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
• ${\displaystyle (\mathbb {R} ,<)}$is -saturated, but not saturated. The type is not realized. ${\displaystyle \aleph _{0}}$ ${\displaystyle \{x>1,x>2,x>3,\ldots \}}$

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

Individual evidence

1. ↑ ^{a b c} A. Prestel: Introduction to Mathematical Logic and Model Theory . Braunschweig 1986
2. ↑ Sacks, p. 112.
3. ↑ Philipp Rothmaler: Introduction to the model theory , Spektrum Akademischer Verlag 1995, ISBN 978-3-86025-461-5 , sentence 12.3