nLab
C(n)-extendible cardinal

C(n)C(n)-extendible cardinals

Idea

A C(n)C(n)-extendible cardinal is a type of large cardinal which generalizes extendible cardinals. They come up in formulating refined versions of Vopěnka's principle.

Definition

We work in ZFC. Let C(n)C(n) denote the class of infinite cardinal numbers κ\kappa such that V κV_\kappa (see von Neumann hierarchy) is a Σ n\Sigma_n-elementary submodel of the universe VV, i.e. such that the embedding preserves and reflects the truth of Σ n\Sigma_n-formulas. In particular:

  • κ\kappa is C(0)C(0) if and only if it is infinite, and

  • κ\kappa is C(1)C(1) if and only if it is uncountable and V κV_\kappa coincides with the hereditarily κ\kappa-sized sets H(κ)H(\kappa), i.e. those whose transitive closure has cardinality <κ\lt\kappa.

Definition

If κ<λ\kappa\lt\lambda and both are in C(n)C(n), we say κ\kappa is λ\lambda-C(n)C(n)-extendible if there is an elementary embedding j:V λV μj:V_\lambda \to V_\mu with critical point κ\kappa, such that μC(n)\mu\in C(n), j(κ)>λj(\kappa)\gt \lambda, and j(κ)C(n)j(\kappa)\in C(n).

We say κC(n)\kappa\in C(n) is C(n)C(n)-extendible if it is λ\lambda-C(n)C(n)-extendible for all λC(n)\lambda\in C(n) with λ>κ\lambda\gt\kappa.

In particular:

  • κ\kappa is C(0)C(0)-extendible if and only if it is extendible.

References

Created on September 24, 2012 06:15:34 by Mike Shulman (192.16.204.218)