Laver function
Encyclopedia
In set theory
, a Laver function (or Laver diamond, named after its inventor, Richard Laver
) is a function connected with supercompact cardinals.
of x)
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
.
Set theory
Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...
, a Laver function (or Laver diamond, named after its inventor, Richard Laver
Richard Laver
Richard Laver is an American mathematician, working in set theory. He is a professor emeritus at the Department of Mathematics of the University of Colorado at Boulder.-His main results:Among Laver's notable achievements some are the following....
) is a function connected with supercompact cardinals.
Definition
If κ is a supercompact cardinal, a Laver function is a function ƒ:κ → Vκ 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 Vκ denotes the κ-th level of the cumulative hierarchy, TC(x) is the transitive closureTransitive set
In set theory, a set A is transitive, if* whenever x ∈ A, and y ∈ x, then y ∈ A, or, equivalently,* whenever x ∈ A, and x is not an urelement, then x is a subset of A....
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 (mathematics)
In the mathematical discipline of set theory, forcing is a technique invented by Paul Cohen for proving consistency and independence results. It was first used, in 1963, to prove the independence of the axiom of choice and the continuum hypothesis from Zermelo–Fraenkel set theory...
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
Proper Forcing Axiom
In the mathematical field of set theory, the proper forcing axiom is a significant strengthening of Martin's axiom, where forcings with the countable chain condition are replaced by proper forcings.- Statement :...
.