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 → V_κ such that for every set x and every cardinal λ ≥ |TC(x)| + κ there is a supercompact measure U on [λ]^<κ such that if j_U is the associated elementary embedding then j_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, ≤) such after forcing with (P, ≤) 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

Laver, Richard (1978). "Making the supercompactness of κ indestructible under κ-directed closed forcing". Israel Journal of Mathematics. 29: 385–388. Zbl 0381.03039. doi:10.1007/bf02761175.

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