set theory Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concer ...

, a Laver function (or Laver diamond, named after its inventor,

Richard Laver Richard Joseph Laver (October 20, 1942 – September 19, 2012) was an American mathematician, working in set theory. Biography Laver received his PhD at the University of California, Berkeley in 1969, under the supervision of Ralph McKenzie, wi ...

) is a function connected with

supercompact cardinal In set theory, a supercompact cardinal is a type of large cardinal. They display a variety of reflection properties. Formal definition If ''λ'' is any ordinal, ''κ'' is ''λ''-supercompact means that there exists an elementary ...

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 �sup><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 In mathematics, specifically set theory, a cumulative hierarchy is a family of sets W_\alpha indexed by ordinals \alpha such that * W_\alpha \subseteq W_ * If \lambda is a limit ordinal, then W_\lambda = \bigcup_ W_ Some authors additionally ...

, TC(''x'') is the

transitive closure In mathematics, the transitive closure of a binary relation on a set is the smallest relation on that contains and is transitive. For finite sets, "smallest" can be taken in its usual sense, of having the fewest related pairs; for infinit ...

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 Forcing may refer to: Mathematics and science * Forcing (mathematics), a technique for obtaining independence proofs for set theory *Forcing (recursion theory), a modification of Paul Cohen's original set theoretic technique of forcing to deal with ...

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 In the mathematical field of set theory, the proper forcing axiom (''PFA'') is a significant strengthening of Martin's axiom, where forcings with the countable chain condition (ccc) are replaced by proper forcings. Statement A forcing or partia ...

References

* Set theory Large cardinals Functions and mappings {{settheory-stub