Standard model (set theory)

In set theory, a branch of mathematical logic, an inner model for a theory T is a substructure of a model M of a set theory that is both a model for T and contains all the ordinals of M.

Let ${\displaystyle L=\langle \in \rangle }$ be the language of set theory. Let S be a particular set theory, for example the ZFC axioms and let T (possibly the same as S) also be a theory in ${\displaystyle L}$.

If M is a model for S, and N is a substructure of M such that

then we say that N is an inner model of T (in M). Usually T will equal (or subsume) S, so that N is a model for S 'inside' the model M.

If only conditions 1 and 2 hold, N is called a standard model. A model N is called transitive when it is standard and condition 3 holds. If the axiom of foundation is not assumed (that is, is not in S) all three of these concepts are given the additional condition that N be well-founded. Hence inner models are transitive, transitive models are standard, and standard models are well-founded.

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