# Richard Laver

> Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Richard_Laver
> Markdown URL: https://mediated.wiki/source/Richard_Laver.md
> Source: https://en.wikipedia.org/wiki/Richard_Laver
> Source revision: 1273757077
> License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/)

{{short description|American mathematician}}
thumb|220px|Richard Laver
'''Richard Joseph Laver''' (October 20, 1942 – September 19, 2012) was an American mathematician, working in [set theory](/source/set_theory).

==Biography==
Laver received his PhD at the [University of California, Berkeley](/source/University_of_California%2C_Berkeley) in 1969, under the supervision of [Ralph McKenzie](/source/Ralph_McKenzie),<ref>Ralph McKenzie has been a doctoral student of James Donald Monk, who has been a doctoral student of [Alfred Tarski](/source/Alfred_Tarski).</ref> 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](/source/University_of_Colorado_at_Boulder).

Richard Laver died in [Boulder, CO](/source/Boulder%2C_CO), on September 19, 2012 after a long illness.<ref>[http://ests.wordpress.com/2012/09/22/in-emoriam-richard-laver-1942-2012/ Obituary, European Set Theory Society]</ref>

==Research contributions<!--'Indestructibility' and 'Indestructibility result' redirect here-->==
Among Laver's notable achievements some are the following.

* Using the theory of [better-quasi-order](/source/better-quasi-order)s, introduced by [Nash-Williams](/source/Crispin_Nash-Williams), (an extension of the notion of [well-quasi-ordering](/source/well-quasi-ordering)), he proved<ref>{{cite journal |author1=R. Laver |date=1971 |title=On Fraïssé's order type conjecture |journal=[Annals of Mathematics](/source/Annals_of_Mathematics) |volume=93 |issue=1 |pages=89–111 |jstor=1970754 |doi=10.2307/1970754}}</ref> [Fraïssé's](/source/Roland_Fra%C3%AFss%C3%A9) conjecture (now [Laver's theorem](/source/Laver's_theorem)): if (''A''<sub>0</sub>,≤),(''A''<sub>1</sub>,≤),...,(''A''<sub>''i''</sub>,≤), are countable ordered sets, then for some ''i''<''j''  (''A''<sub>i</sub>,≤) isomorphically embeds into (''A''<sub>''j''</sub>,≤). This also holds if the ordered sets are countable unions of [scattered](/source/scattered_order) ordered sets.<ref>{{cite journal |author1=R. Laver |date=1973 |title=An order type decomposition theorem |journal=Annals of Mathematics |volume=98 |issue=1 |pages=96–119 |jstor=1970907|doi=10.2307/1970907 }}</ref>
* He proved<ref>{{cite journal |author1=R. Laver |date=1976 |title=On the consistency of Borel's conjecture |url=https://projecteuclid.org/download/pdf_1/euclid.acta/1485889934 |journal=[Acta Mathematica](/source/Acta_Mathematica) |volume=137 |pages=151–169 |doi=10.1007/bf02392416|doi-access=free }}</ref> the consistency of the [Borel conjecture](/source/Borel_conjecture_(set_theory)), i.e., the statement that every [strong measure zero set](/source/strong_measure_zero_set) is countable. This important independence result was the first when a [forcing](/source/Forcing_(mathematics)) (see [Laver forcing](/source/List_of_forcing_notions)), adding a real, was iterated with countable support iteration. This method was later used by [Shelah](/source/Saharon_Shelah) to introduce proper and semiproper forcing.
* He proved<ref>{{cite journal |author1=R. Laver |date=1978 |title=Making the supercompactness of κ indestructible under κ-directed closed forcing |journal=[Israel Journal of Mathematics](/source/Israel_Journal_of_Mathematics) |volume=29 |issue=4 |pages=385–388 |doi=10.1007/BF02761175|doi-access=|s2cid=115387536 }}</ref> the existence of a [Laver function](/source/Laver_function) for [supercompact cardinal](/source/supercompact_cardinal)s. With the help of this, he proved the following result. If κ is supercompact, there is a κ-[c.c.](/source/Chain_condition) [forcing](/source/forcing_(mathematics)) notion (''P'',&nbsp;≤) such that after forcing with (''P'',&nbsp;≤) 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'''<!--boldface per WP:R#PLA-->,<ref>''Collegium Logicum: Annals of the Kurt-Gödel-Society'', Volume 9, Springer Verlag, 2006, p. 31.</ref> is used, for example, in the proof of the consistency of the [proper forcing axiom](/source/proper_forcing_axiom) and variants.
* Laver and [Shelah](/source/Saharon_Shelah) proved<ref>{{cite journal|author1=R. Laver |author2=S. Shelah |date=1981 |title=The ℵ<sub>2</sub> Souslin hypothesis |journal=[Transactions of the American Mathematical Society](/source/Transactions_of_the_American_Mathematical_Society) |volume=264 |pages=411–417 |doi=10.1090/S0002-9947-1981-0603771-7|doi-access=free }}</ref> that it is consistent that the continuum hypothesis holds and there are no ℵ<sub>2</sub>-[Suslin tree](/source/Suslin_tree)s.
* Laver proved<ref>{{cite journal |author=R. Laver |date=1984 |title=Products of infinitely many perfect trees |journal=[Journal of the London Mathematical Society](/source/Journal_of_the_London_Mathematical_Society) |volume=29 |issue=3 |pages=385–396 |doi=10.1112/jlms/s2-29.3.385}}</ref> that the perfect subtree version of the [Halpern–Läuchli theorem](/source/Halpern%E2%80%93L%C3%A4uchli_theorem) holds for the product of infinitely many trees. This solved a longstanding open question.
* Laver started<ref>{{cite journal |author=R. Laver |date=1992 |title=The left-distributive law and the freeness of an algebra of elementary embeddings |journal=[Advances in Mathematics](/source/Advances_in_Mathematics) |volume=91 |issue=2 |pages=209–231 |doi=10.1016/0001-8708(92)90016-E|doi-access=free|hdl=10338.dmlcz/127389 |hdl-access=free }}
</ref><ref>{{cite journal |author=R. Laver |date=1995 |title=On the algebra of elementary embeddings of a rank into itself |journal=[Advances in Mathematics](/source/Advances_in_Mathematics) |volume=110 |issue=2 |pages=334–346 |doi=10.1006/aima.1995.1014|doi-access=free|s2cid=119485709 }}</ref><ref>{{cite journal |author=R. Laver |date=1996 |title=Braid group actions on left distributive structures, and well orderings in the braid groups |journal=[Journal of Pure and Applied Algebra](/source/Journal_of_Pure_and_Applied_Algebra) |volume=108 |pages=81–98 |doi=10.1016/0022-4049(95)00147-6 | doi-access=free}}.
</ref> investigating the algebra that ''j'' generates where ''j'':''V''<sub>λ</sub>→''V''<sub>λ</sub> is some elementary embedding. This algebra is the free left-distributive algebra on one generator. For this he introduced [Laver table](/source/Laver_table)s.
* He also showed<ref>{{cite journal |author=R. Laver |date=2007 |title=Certain very large cardinals are not created in small forcing extensions |journal=[Annals of Pure and Applied Logic](/source/Annals_of_Pure_and_Applied_Logic) |volume=149 |issue=1–3 |pages=1–6 |doi=10.1016/j.apal.2007.07.002|doi-access=}}</ref> that if ''V''[''G''] is a (set-)[forcing](/source/Forcing_(mathematics)) extension of [''V''](/source/Von_Neumann_universe), then ''V'' is a [class](/source/Class_(set_theory)) in ''V''[''G''].

==Notes and references==
{{Reflist|2}}

==External links==
* {{MathGenealogy|id=13350}}

{{authority control}}

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

---
Adapted from the Wikipedia article [Richard Laver](https://en.wikipedia.org/wiki/Richard_Laver) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Richard_Laver?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.