Girish Mahajan (Editor)

Cumulative hierarchy

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In mathematical set theory, a cumulative hierarchy is a family of sets Wα indexed by ordinals α such that

Contents

  • Wα⊆Wα+1
  • If α is a limit then Wα = ∪β<α Wβ

    • It is also sometimes assumed that Wα+1⊆P(Wα) or that W0 is empty.

    The union W of the sets of a cumulative hierarchy is often used as a model of set theory.

    The phrase "the cumulative hierarchy" usually refers to the standard cumulative hierarchy Vα of the Von Neumann universe with Vα+1=P(Vα) introduced by Zermelo (1930)

    Reflection principle

    A cumulative hierarchy satisfies a form of the reflection principle: any formula of the language of set theory that holds in the union W of the hierarchy also holds in some stages Wα.

    Examples

  • The Von Neumann universe is built from a cumulative hierarchy Vα.
  • The sets Lα of the constructible universe form a cumulative hierarchy.
  • The Boolean valued models constructed by forcing are built using a cumulative hierarchy.
  • The well founded sets in a model of set theory (possibly not satisfying the axiom of foundation) form a cumulative hierarchy whose union satisfies the axiom of foundation.

    • References

    Cumulative hierarchy Wikipedia
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