Richard Joseph Laver (October 20, 1942 – September 19, 2012) was an American mathematician, working in

set theory Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory – as a branch of mathema ...

Biography

Laver received his PhD at the

University of California, Berkeley The University of California, Berkeley (UC Berkeley, Berkeley, Cal, or California), is a Public university, public Land-grant university, land-grant research university in Berkeley, California, United States. Founded in 1868 and named after t ...

in 1969, under the supervision of Ralph McKenzie, 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 The University of Colorado Boulder (CU Boulder, CU, or Colorado) is a Public university, public research university in Boulder, Colorado, United States. Founded in 1876, five months before Colorado became a Federated state, state, it is the fla ...

. Richard Laver died in

Boulder, CO Boulder is a List of municipalities in Colorado#Home rule municipality, home rule city in Boulder County, Colorado, United States, and its county seat. With a population of 108,250 at the 2020 United States census, 2020 census, it is the most ...

, on September 19, 2012 after a long illness.

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 In mathematics, specifically order theory, a well-quasi-ordering or wqo on a set X is a quasi-ordering of X for which every infinite sequence of elements x_0, x_1, x_2, \ldots from X contains an increasing pair x_i \leq x_j with i x_2> \cdots) ...

), he proved Fraïssé's conjecture (now

Laver's theorem Laver's theorem, in order theory, states that order embeddability of countable total orders is a well-quasi-ordering. That is, for every infinite sequence of totally-ordered countable sets, there exists an order embedding from an earlier member of ...

): if (''A''₀,≤),(''A''₁,≤),...,(''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 Scattered may refer to: Music * ''Scattered'' (album), a 2010 album by The Handsome Family * "Scattered" (The Kinks song), 1993 * "Scattered", a song by Ace Young * "Scattered", a song by Lauren Jauregui * "Scattered", a song by Green Day from ...

ordered sets. * He proved the consistency of the

Borel conjecture In geometric topology, the Borel conjecture (named for Armand Borel) asserts that an aspherical closed manifold is determined by its fundamental group, up to homeomorphism. It is a rigidity conjecture, asserting that a weak, algebraic notion of ...

, i.e., the statement that every

strong measure zero set In mathematical analysis, a strong measure zero set is a subset ''A'' of the real line with the following property: :for every sequence (ε''n'') of positive reals there exists a sequence (''In'') of intervals such that , ''I'n'', < ε_{''n' ...}

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 Shelah may refer to: * Shelah (son of Judah), a son of Judah according to the Bible * Shelah (name), a Hebrew personal name * Shlach, the 37th weekly Torah portion (parshah) in the annual Jewish cycle of Torah reading * Salih, a prophet described i ...

to introduce proper and semiproper forcing. * He proved the existence of a Laver function for

supercompact cardinal In set theory, a supercompact cardinal is a type of large cardinal independently introduced by Solovay and Reinhardt. They display a variety of reflection properties. Formal definition If \lambda is any ordinal, \kappa is \lambda-supercompact m ...

s. 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, is used, for example, in the proof of the consistency of the

proper forcing axiom In the mathematical field of set theory, the proper forcing axiom (''PFA'') is a significant strengthening of Martin's axiom, where forcings with the countable chain condition (ccc) are replaced by proper forcings. Statement A forcing or partia ...

and variants. * Laver and

proved that it is consistent that the continuum hypothesis holds and there are no ℵ₂-

Suslin tree In mathematics, a Suslin tree is a tree of height ω1 such that every branch and every antichain is countable. They are named after Mikhail Yakovlevich Suslin. Every Suslin tree is an Aronszajn tree. The existence of a Suslin tree is independen ...

s. * 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'' 'G''is a (set-) forcing extension of ''V'', then ''V'' is a

class Class, Classes, or The Class may refer to: Common uses not otherwise categorized * Class (biology), a taxonomic rank * Class (knowledge representation), a collection of individuals or objects * Class (philosophy), an analytical concept used d ...

in ''V'' 'G''

Notes and references

External links

* {{DEFAULTSORT:Laver, Richard Set theorists 20th-century American mathematicians 21st-century American mathematicians University of Colorado Boulder faculty 1942 births 2012 deaths