Richard Laver died in Boulder, CO, on September 19, 2012 after a long illness.[2]
Research contributions
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].
Notes and references
^Ralph McKenzie has been a doctoral student of James Donald Monk, who has been a doctoral student of Alfred Tarski.
^Obituary, European Set Theory Society
^R. Laver (1971). "On Fraïssé's order type conjecture". Annals of Mathematics. 93 (1): 89–111. doi:10.2307/1970754. JSTOR 1970754.
^R. Laver (1973). "An order type decomposition theorem". Annals of Mathematics. 98 (1): 96–119. doi:10.2307/1970907. JSTOR 1970907.
^R. Laver (1976). "On the consistency of Borel's conjecture". Acta Mathematica. 137: 151–169. doi:10.1007/bf02392416.
^R. Laver (1978). "Making the supercompactness of κ indestructible under κ-directed closed forcing". Israel Journal of Mathematics. 29 (4): 385–388. doi:10.1007/BF02761175. S2CID 115387536.
^Collegium Logicum: Annals of the Kurt-Gödel-Society, Volume 9, Springer Verlag, 2006, p. 31.
^R. Laver (1992). "The left-distributive law and the freeness of an algebra of elementary embeddings". Advances in Mathematics. 91 (2): 209–231. doi:10.1016/0001-8708(92)90016-E. hdl:10338.dmlcz/127389.
^R. Laver (1995). "On the algebra of elementary embeddings of a rank into itself". Advances in Mathematics. 110 (2): 334–346. doi:10.1006/aima.1995.1014. S2CID 119485709.
^R. Laver (1996). "Braid group actions on left distributive structures, and well orderings in the braid groups". Journal of Pure and Applied Algebra. 108: 81–98. doi:10.1016/0022-4049(95)00147-6..
^R. Laver (2007). "Certain very large cardinals are not created in small forcing extensions". Annals of Pure and Applied Logic. 149 (1–3): 1–6. doi:10.1016/j.apal.2007.07.002.