Laver function

Laver function

In set theory, a Laver function (or Laver diamond, named after its inventor, Richard Laver) is a function connected with supercompact cardinals.

Definition

If &kappa; is a supercompact cardinal, a Laver function is a function "&fnof;":&kappa; &rarr; "V"&kappa; such that for every set "x" and every cardinal &lambda; &ge; |TC("x")| + &kappa; there is a supercompact measure "U" on [&lambda;] <&kappa; such that if "j" "U" is the associated elementary embedding then "j" "U"("&fnof;")(&kappa;) = "x". (Here "V"&kappa; denotes the &kappa;-th level of the cumulative hierarchy, TC("x") is the transitive closure of "x")

Applications

The original application of Laver functions was the following theorem of Laver. If &kappa; is supercompact, there is a &kappa;-c.c. forcing notion ("P", &le;) such after forcing with ("P", &le;) the following holds: &kappa; is supercompact and remains supercompact after forcing with any &kappa;-directed closed forcing.

There are many other applications, for example the proof of the consistency of the proper forcing axiom.

Reference

* R. Laver: Making the supercompactness of κ indestructible under &kappa;-directed closed forcing, "Israel Journal of Mathematics", 29(1978), 385&ndash;388.


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