Standard model (set theory)

In set theory, a standard model for a theory [math]\displaystyle{ T }[/math] is a model [math]\displaystyle{ M }[/math] for [math]\displaystyle{ T }[/math] where the membership relation [math]\displaystyle{ \in_M }[/math] is the same as the membership relation [math]\displaystyle{ \in }[/math] of the set theoretical universe [math]\displaystyle{ V }[/math] (restricted to the domain of [math]\displaystyle{ M }[/math]). In other words, [math]\displaystyle{ M }[/math] is a substructure of [math]\displaystyle{ V }[/math]. A standard model [math]\displaystyle{ M }[/math] that satisfies the additional transitivity condition that [math]\displaystyle{ x \in y \in M }[/math] implies [math]\displaystyle{ x \in M }[/math] is a standard transitive model (or simply a transitive model).

Usually, when one talks about a model [math]\displaystyle{ M }[/math] of set theory, it is assumed that [math]\displaystyle{ M }[/math] is a set model, i.e. the domain of [math]\displaystyle{ M }[/math] is a set in [math]\displaystyle{ V }[/math]. If the domain of [math]\displaystyle{ M }[/math] is a proper class, then [math]\displaystyle{ M }[/math] is a class model. An inner model is necessarily a class model.

References

• Cohen, P. J. (1966). Set theory and the continuum hypothesis. Addison–Wesley. ISBN 978-0-8053-2327-6. 
• Chow, Timothy Y. (2007). "A beginner's guide to forcing". arXiv:0712.1320 [math.LO].