mathematics

James Earl Baumgartner

James Earl Baumgartner 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 James Earl Baumgartner rather than just read about it. In short: James Earl Baumgartner (March 23, 1943 – December 28, 2011) was an American mathematician who worked in set theory, mathematical logic and foundations, and topology. Baumgartner was born in Wichita, Kansas, began his undergraduate study at the California Institute of Technology in 1960, then transferred to the University of California, Berkeley, from which he received his PhD in 1970 in mathematics for a dissertatio…

Key takeaways

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

Reference excerpt

James Earl Baumgartner (March 23, 1943 – December 28, 2011) was an American mathematician who worked in set theory, mathematical logic and foundations, and topology. Baumgartner was born in Wichita, Kansas, began his undergraduate study at the California Institute of Technology in 1960, then transferred to the University of California, Berkeley, from which he received his PhD in 1970 in mathematics for a dissertation titled Results and Independence Proofs in Combinatorial Set Theory. His advisor was Robert Vaught. He became a professor at Dartmouth College in 1969, and spent his entire career there. One of Baumgartner's results is the consistency of the statement that any two ℵ 1 {\displaystyle \aleph _{1}} -dense sets of reals are order isomorphic (a set of reals is ℵ 1 {\displaystyle \aleph _{1}} -dense if it has exactly ℵ 1 {\displaystyle \aleph _{1}} points in every open interval). With András Hajnal he proved the Baumgartner–Hajnal theorem, which states that the partition relation ω 1 → ( α ) n 2 {\displaystyle \omega _{1}\to (\alpha )_{n}^{2}} holds for α < ω 1 {\displaystyle \alpha <\omega _{1}} and n < ω {\displaystyle n<\omega } . He died in 2011 of a heart attack at his home in Hanover, New Hampshire. The mathematical context in which Baumgartner worked spans Suslin's problem, Ramsey theory, uncountable order types, disjoint refinements, almost disjoint families, cardinal arithmetics, filters, ideals, and partition relations, iterated forcing and Axiom A, proper forcing and the proper forcing axiom, chromatic number of graphs, a thin very-tall superatomic Boolean algebra, closed unbounded sets, and partition relations.

Illustrations

Worked examples

Example 1 — a first encounter with James Earl Baumgartner

Start with the simplest possible case. Write down what James Earl Baumgartner 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 James Earl Baumgartner 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 James Earl Baumgartner 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 James Earl Baumgartner

In research: James Earl Baumgartner 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 James Earl Baumgartner 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: James Earl Baumgartner is common in secondary-school and first-year university syllabi. It links to neighbouring topics 1943 births, 2011 deaths, 20th-century American mathematicians, so understanding it makes those chapters shorter.
In everyday life: Look for James Earl Baumgartner 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.

Affiliate

Preply — study more efficiently by working with a personal tutor. 50% off.

How to study James Earl Baumgartner in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what James Earl Baumgartner 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 James Earl Baumgartner out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is James Earl Baumgartner in simple terms?

Why does James Earl Baumgartner 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 James Earl Baumgartner?

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 James Earl Baumgartner.