Inner model

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In mathematical logic, suppose T is a theory in the language

L = \langle \in \rangle

of set theory.

If M is a model of L describing a set theory and N is a class of M such that

 \langle N, \in_M, \ldots \rangle

is a model of T containing all ordinals of M then we say that N is an inner model of T (in M).[1] Ordinarily these models are transitive subsets or subclasses of the von Neumann universe V, or sometimes of a generic extension of V.

This term inner model is sometimes applied to models that are proper classes; the term set model is used for models that are sets.

A model of set theory is called standard if the element relation of the model is the actual element relation restricted to the model. A model is called transitive when it is standard and the base class is a transitive class of sets. A model of set theory is often assumed to be transitive unless it is explicitly stated that it is non-standard. Inner models are transitive, transitive models are standard, and standard models are well-founded.

The assumption that there exists a standard model of ZFC (in a given universe) is stronger than the assumption that there exists a model. In fact, if there is a standard model, then there is a smallest standard model called the minimal model contained in all standard models. The minimal model contains no standard model (as it is minimal) but (assuming the consistency of ZFC) it contains some model of ZFC by the Gödel completeness theorem. This model is necessarily not well founded otherwise its Mostowski collapse would be a standard model. (It is not well founded as a relation in the universe, though it satisfies the axiom of foundation so is "internally" well founded. Being well founded is not an absolute property.[2]) In particular in the minimal model there is a model of ZFC but there is no standard model of ZFC.

Use[edit]

Usually when one talks about inner models of a theory, the theory one is discussing is ZFC or some extension of ZFC (like ZFC + \exists 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.

Related ideas[edit]

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 that 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.

References[edit]

  1. ^ Jech, Thomas (2002). Set Theory. Berlin: Springer-Verlag. ISBN 3-540-44085-2. 
  2. ^ Kunen, Kenneth (1980). Set Theory. Amsterdam: North-Holland Pub. Co. ISBN 0-444-86839-9. , Page 117

See also[edit]