Editing Saturated model (section)

==Definition==

Let ''κ'' be a [[finite set|finite]] or [[Infinity|infinite]] [[cardinal number]] and ''M'' a model in some [[first-order language]].  Then ''M'' is called '''''κ''-saturated''' if  for all subsets ''A'' ⊆ ''M'' of [[cardinality]] less than ''κ'', the model ''M'' realizes all [[Type (model theory)|complete types]]  over ''A''. The model ''M'' is called '''saturated''' if  it is |''M''|-saturated where |''M''| denotes the cardinality of ''M''. That is, it realizes all complete types over sets of parameters of size less than |''M''|. According to some authors, a model ''M'' is called '''countably saturated''' if  it is [[aleph-1 | <math>\aleph_1</math>]]-saturated; that is, it realizes all complete types over countable sets of parameters.<ref>{{Cite journal|last=Morley|first=Michael|authorlink = Michael D. Morley|date=1963|title=On theories categorical in uncountable powers|journal=[[Proceedings of the National Academy of Sciences of the United States of America]]|volume=49|issue=2 |pages=213–216|doi=10.1073/pnas.49.2.213 |pmid=16591050 |pmc=299780 |bibcode=1963PNAS...49..213M |doi-access=free }}</ref> According to others, it is countably saturated if it is countable and saturated.<ref>Chang and Keisler 1990</ref>