In set theory

Set theory

Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...

, inner model theory is the study of certain model

Model theory

In mathematics, model theory is the study of mathematical structures using tools from mathematical logic....

s of ZFC or some fragment or strengthening thereof. Ordinarily these models are transitive

Transitive set

In set theory, a set A is transitive, if* whenever x ∈ A, and y ∈ x, then y ∈ A, or, equivalently,* whenever x ∈ A, and x is not an urelement, then x is a subset of A....

 subset

Subset

In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...

s or subclasses of the von Neumann universe

Von Neumann universe

In set theory and related branches of mathematics, the von Neumann universe, or von Neumann hierarchy of sets, denoted V, is the class of hereditary well-founded sets...

 V, or sometimes of a generic extension of V. Inner model theory studies the relationships of these models to determinacy

Determinacy

In set theory, a branch of mathematics, determinacy is the study of under what circumstances one or the other player of a game must have a winning strategy, and the consequences of the existence of such strategies.-Games:...

, large cardinals, and descriptive set theory

Descriptive set theory

In mathematical logic, descriptive set theory is the study of certain classes of "well-behaved" subsets of the real line and other Polish spaces...

. Despite the name, it is considered more a branch of set theory than of model theory

Model theory

In mathematics, model theory is the study of mathematical structures using tools from mathematical logic....

.

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
  Constructible universe
  In mathematics, the constructible universe , denoted L, is a particular class of sets which can be described entirely in terms of simpler sets. It was introduced by Kurt Gödel in his 1938 paper "The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis"...
  
   L developed by Kurt Gödel
  Kurt Gödel
  Kurt Friedrich Gödel was an Austrian logician, mathematician and philosopher. Later in his life he emigrated to the United States to escape the effects of World War II. One of the most significant logicians of all time, Gödel made an immense impact upon scientific and philosophical thinking in the...
  
  . Every model M of ZFC has an inner model L^M satisfying the axiom of constructibility
  Axiom of constructibility
  The axiom of constructibility is a possible axiom for set theory in mathematics that asserts that every set is constructible. The axiom is usually written as V = L, where V and L denote the von Neumann universe and the constructible universe, respectively.- Implications :The axiom of...
  
  , and this will be the smallest inner model of M containing all the ordinals of M. Regardless of the properties of the original model, L^M will satisfy the generalized continuum hypothesis and combinatorial axioms such as ◊
  Diamondsuit
  In mathematics, and particularly in axiomatic set theory, the diamond principle ◊ is a combinatorial principle introduced by that holds in the constructible universe and that implies the continuum hypothesis...
  
  .
• 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.

Consistency results

One important use of inner models is the proof of consistency results. If it can be shown that every model of an axiom A has an inner model satisfying axiom B, then if A is consistent, B must also be consistent. This analysis is most useful when A is an axiom independent of ZFC, for example a large cardinal axiom; it is one of the tools used to rank axioms by consistency strength.