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 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 mathematics, is mostly conce ...

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 land-grant research university in Berkeley, California. Established in 1868 as the University of California, it is the state's first land-grant u ...

in 1969, under the supervision of

Ralph McKenzie Ralph Nelson Whitfield McKenzie (born October 20, 1941) is an American mathematician, logician, and abstract algebraist. He received his doctorate from the University of Colorado Boulder in 1967. He is a fellow of the American Mathematical Socie ...

, 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 research university in Boulder, Colorado. Founded in 1876, five months before Colorado became a state, it is the flagship university of the University of Colorado sys ...

. Richard Laver died in

Boulder, CO Boulder is a home rule city that is the county seat and most populous municipality of Boulder County, Colorado, United States. The city population was 108,250 at the 2020 United States census, making it the 12th most populous city in Colora ...

, 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 is a quasi-ordering such that any 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) nor infinite sequenc ...

), 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 ordered sets. * He proved the consistency of the

Borel conjecture In mathematics, specifically 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 ...

, 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 ...

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

Laver function In set theory, a Laver function (or Laver diamond, named after its inventor, Richard Laver) is a function connected with supercompact cardinals. Definition If κ is a supercompact cardinal, a Laver function is a function ''ƒ'':κ → ''V ...

for

supercompact cardinal In set theory, a supercompact cardinal is a type of large cardinal. They display a variety of reflection properties. Formal definition If ''λ'' is any ordinal, ''κ'' is ''λ''-supercompact means that there exists an elementary ...

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 parti ...

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 at most countable. They are named after Mikhail Yakovlevich Suslin. Every Suslin tree is an Aronszajn tree. The existence of a Suslin tree is ind ...

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 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 differentl ...

in ''V'' 'G''

Notes and references

External links

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