Inner Model Theory - Examples

Examples

  • The class of all sets is an inner model containing all other inner models.
  • The first non-trivial example of an inner model was the constructible universe L developed by Kurt Gödel. Every model M of ZFC has an inner model LM satisfying the axiom of constructibility, and this will be the smallest inner model of M containing all the ordinals of M. Regardless of the properties of the original model, LM will satisfy the generalized continuum hypothesis and combinatorial axioms such as ◊.
  • The sets that are hereditarily ordinal definable form an inner model
  • The sets that are hereditarily definable over a countable sequence of ordinals form an inner model, used in Solovay's theorem.

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