Richard Laver Explained

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,^[1] with a thesis on Order Types and Well-Quasi-Orderings. The largest part of his career he spent as Professor and later Emeritus Professor at the University of Colorado at Boulder.

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,≤),...,(A_i,≤), are countable ordered sets, then for some i<j (A_i,≤) isomorphically embeds into (A_j,≤). 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: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^[13] that if V[''G''] is a (set-)forcing extension of V, then V is a class in V[''G''].

Notes and References

1. Ralph McKenzie has been a doctoral student of James Donald Monk, who has been a doctoral student of Alfred Tarski.
2. http://ests.wordpress.com/2012/09/22/in-emoriam-richard-laver-1942-2012/ Obituary, European Set Theory Society
3. R. Laver . 1971 . On Fraïssé's order type conjecture . . 93 . 1 . 89–111 . 1970754 . 10.2307/1970754.
4. R. Laver . 1973 . An order type decomposition theorem . Annals of Mathematics . 98 . 1 . 96–119 . 1970907. 10.2307/1970907 .
5. R. Laver . 1976 . On the consistency of Borel's conjecture . . 137 . 151–169 . 10.1007/bf02392416. free .
6. R. Laver . 1978 . Making the supercompactness of κ indestructible under κ-directed closed forcing . . 29 . 4 . 385–388 . 10.1007/BF02761175. 115387536 .
7. Collegium Logicum: Annals of the Kurt-Gödel-Society, Volume 9, Springer Verlag, 2006, p. 31.
8. R. Laver . S. Shelah . 1981 . The ℵ₂ Souslin hypothesis . . 264 . 411–417 . 10.1090/S0002-9947-1981-0603771-7. free .
9. R. Laver . 1984 . Products of infinitely many perfect trees . . 29 . 3 . 385–396 . 10.1112/jlms/s2-29.3.385.
10. R. Laver . 1992 . The left-distributive law and the freeness of an algebra of elementary embeddings . . 91 . 2 . 209–231 . 10.1016/0001-8708(92)90016-E. free. 10338.dmlcz/127389 . free .
11. R. Laver . 1995 . On the algebra of elementary embeddings of a rank into itself . . 110 . 2 . 334–346 . 10.1006/aima.1995.1014. free. 119485709 .
12. R. Laver . 1996 . Braid group actions on left distributive structures, and well orderings in the braid groups . . 108 . 81–98 . 10.1016/0022-4049(95)00147-6 . free. .
13. R. Laver . 2007 . Certain very large cardinals are not created in small forcing extensions . . 149 . 1–3 . 1–6 . 10.1016/j.apal.2007.07.002.