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In mathematical logic, and in particular model theory, a model <math>M<math> is <math>\kappa<math>-saturated if and only if it realizes all elements
- <math>p\in S(A)<math>, for <math>A\subseteq M<math>
with <math>|A|<\kappa<math>.
A model <math>M<math> is saturated if and only if it is <math>|M|<math>-saturated, that is, it realizes all complete types over sets of parameters of size less than <math>|M|<math>.
Saturated models exist: for instance, <math>\langle {\mathbb Q},<\rangle<math> is saturated, the countable random graph is saturated. The natural numbers are not saturated: the type <math>p(x)<math> containing
- <math>\{ x>0,x>S(0),x>S(S(0)),\cdots,x>S(\cdots (S(0)) \cdots ),\cdots \}<math>
is not realized in <math>\langle {\mathbb N},<,+,\cdot,S,0,1\rangle <math>.
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