Richard Laver died in Boulder, CO, on September 19, 2012 after a long illness.[2]
Among Laver's notable achievements some are the following.
Using the theory of better-quasi-orders, introduced by Nash-Williams, (an extension of the notion of well-quasi-ordering), he proved[3]Fraïssé's conjecture (now Laver's theorem): if (A0,≤),(A1,≤),...,(Ai,≤), are countable ordered sets, then for some i<j (Ai,≤) isomorphically embeds into (Aj,≤). This also holds if the ordered sets are countable unions of scattered ordered sets.[4]
He proved[5] the consistency of the Borel conjecture, i.e., the statement that every strong measure zero set is countable. This important independence result was the first when a forcing (see Laver forcing), adding a real, was iterated with countable support iteration. This method was later used by Shelah to introduce proper and semiproper forcing.
He proved[6] the existence of a Laver function for supercompact cardinals. With the help of this, he proved the following result. If κ is supercompact, there is a κ-c.c.forcing notion (P,≤) such that after forcing with (P,≤) the following holds: κ is supercompact and remains supercompact in any forcing extension via a κ-directed closed forcing. This statement, known as the indestructibility result,[7] is used, for example, in the proof of the consistency of the proper forcing axiom and variants.
Laver and Shelah proved[8] that it is consistent that the continuum hypothesis holds and there are no ℵ2-Suslin trees.
Laver proved[9] that the perfect subtree version of the Halpern–Läuchli theorem holds for the product of infinitely many trees. This solved a longstanding open question.
Laver started[10][11][12] investigating the algebra that j generates where j:Vλ→Vλ is some elementary embedding. This algebra is the free left-distributive algebra on one generator. For this he introduced Laver tables.
He also showed[13] that if V[G] is a (set-)forcing extension of V, then V is a class in V[G].
