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.

• Using the theory of better quasi ordered sets, introduced by Nash-Williams, (an extension of the notion of well-quasi-ordering), he proved^[1] Fraïssé's conjecture: 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.^[2]
• He proved^[3] 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^[4] 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 is used, for example in the proof of the consistency of the proper forcing axiom and variants.
• Laver and Shelah proved^[5] that it is consistent that the continuum hypothesis holds and there are no ℵ₂-Suslin trees.
• Laver proved^[6] 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^[7][8][9] 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^[10] that if V[G] is a (set-)forcing extension of V, then V is a class in V[G].

External Links

• Richard Laver at the Mathematics Genealogy Project.

References

1. ^ R. Laver: On Fraïssé's order type conjecture, Ann. of Math. (2), 93(1971), 89–111.
2. ^ R. Laver: An order type decomposition theorem, Ann. of Math., 98(1973), 96–119.
3. ^ R. Laver: On the consistency of Borel's conjecture, Acta Math., 137(1976), 151–169.
4. ^ R. Laver: Making the supercompactness of κ indestructible under κ-directed closed forcing, Israel J. Math., 29(1978), 385–388.
5. ^ R. Laver, S. Shelah: The ℵ₂ Souslin hypothesis, Trans. Amer. Math. Soc., 264(1981), 411–417.
6. ^ R. Laver: Products of infinitely many perfect trees, Journal of the London Math. Soc., 29(1984), 385–396.
7. ^ R. Laver: The left-distributive law and the freeness of an algebra of elementary embeddings, Advances in Mathematics, 91(1992), 209–231.
8. ^ R. Laver: The algebra of elementary embeddings of a rank into itself, Advances in Mathematics, 110(1995), 334–346.
9. ^ R. Laver: Braid group actions on left distributive structures, and well orderings in the braid groups, Jour. Pure and Applied Algebra, 108(1996), 81–98.
10. ^ R. Laver: Certain very large cardinals are not created in small forcing extensions, Annals of Pure and Applied Logic, 149(2007) 1–6.