Richard Laver

Biography

Research contributions

• Using the theory of better-quasi-orders, introduced by Nash-Williams,, he proved Fraïssé's conjecture : if,,...,, are countable ordered sets, then for some i<j isomorphically embeds into. This also holds if the ordered sets are countable unions of scattered ordered sets.
• He proved 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, adding a real, was iterated with countable support iteration. This method was later used by Shelah to introduce proper and semiproper forcing.
• He proved 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 such that after forcing with 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, is used, for example, in the proof of the consistency of the proper forcing axiom and variants.
• Laver and Shelah proved that it is consistent that the continuum hypothesis holds and there are no ℵ₂-Suslin trees.
• Laver proved 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 investigating the algebra that j generates where j:V_λ→V_λ is some elementary embedding. This algebra is the free left-distributive algebra on one generator. For this he introduced Laver tables.
• He also showed that if V is a forcing extension of V, then V is a class in V.