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 κ is a supercompact cardinal, a Laver function is a function ƒ:κ&nbsp;→&nbsp;V_κ such that for every set x and every cardinal λ&nbsp;≥&nbsp;|TC(x)|&nbsp;+&nbsp;κ there is a supercompact measure U on [λ]^<κ such that if j_&nbsp;U is the associated elementary embedding then j_&nbsp;U(ƒ)(κ) = x. (Here V_κ denotes the κ-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 κ is supercompact, there is a κ-c.c. forcing notion (P,&nbsp;≤) such after forcing with (P,&nbsp;≤) the following holds: κ is supercompact and remains supercompact after forcing with any κ-directed closed forcing.

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

References