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Inner model
From Wikipedia, the free encyclopedia
In mathematical logic, suppose T is a theory in the language
If M is a model of L describing a set theory and N is a class of M such that
is a model of T containing all ordinals of M then we say that N is an inner model of T (in M).
This term inner model is sometimes applied to models which are proper classes; the term set model is used for models which are sets.
A model of set theory is called "standard" or "transitive" when the base class is a transitive class of sets and the element relation of the model is the actual element relation restricted to the base class. A model of set theory is assumed to be standard unless it is explicitly stated that it is non-standard. Inner models are usually standard because their ordinals are actual ordinals. Standard models are well-founded.
Usually when one talks about inner models of a theory, the theory one is discussing is ZFC or some extension of ZFC (like ZFC+
a measurable cardinal). When no theory is mentioned, it is usually assumed that the model under discussion is an inner model of ZFC. However, it is not uncommon to talk about inner models of subtheories of ZFC (like ZF or KP) as well.
It was proved by Kurt Gödel that any model of ZF has a least inner model of ZF (which is also an inner model of ZFC + GCH), called the constructible universe, or L.
There is a branch of set theory called inner model theory which studies ways of constructing least inner models of theories extending ZF. Inner model theory has led to the discovery of the exact consistency strength of many important set theoretical properties.