mathematics

Richard Laver

Richard Laver is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Richard Laver rather than just read about it. In short: 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, with a thesis on Order Types and Well-Quasi-Orderings.

Key takeaways

Richard Laver belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Richard Laver to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Richard Laver from memory before moving on to harder problems.

Reference excerpt

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, 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.

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 Fraïssé's conjecture (now Laver's theorem): if (A0,≤),(A1,≤),...,(Ai,≤), are countable ordered sets, then for some i<j (Ai,≤) isomorphically embeds into (Aj,≤). This also holds if the ordered sets are countable unions of scattered ordered sets. He proved 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 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, is used, for example, in the proof of the consistency of the proper forcing axiom and variants. Laver and Shelah proved that it is consistent that the continuum hypothesis holds and there are no ℵ2-Suslin trees. 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:Vλ→Vλ 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 in V[G].

Notes and references

External links Richard Laver at the Mathematics Genealogy Project

Illustrations

Worked examples

Example 1 — a first encounter with Richard Laver

Start with the simplest possible case. Write down what Richard Laver claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Richard Laver before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Richard Laver ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Richard Laver

In research: Richard Laver appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Richard Laver in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Richard Laver is common in secondary-school and first-year university syllabi. It links to neighbouring topics 1942 births, 2012 deaths, 20th-century American mathematicians, so understanding it makes those chapters shorter.
In everyday life: Look for Richard Laver outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Richard Laver in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Richard Laver means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Richard Laver out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Richard Laver in simple terms?

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, with a thesis on Order Types and Well-Quasi-Orderings.

Why does Richard Laver matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Richard Laver?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Richard Laver.