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.
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