James Earl Baumgartner (March 23, 1943 – December 28, 2011) was an American

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...

who worked 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 ...

mathematical logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal ...

and foundations, and

topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...

. Baumgartner was born in

Wichita, Kansas Wichita ( ) is the largest city in the U.S. state of Kansas and the county seat of Sedgwick County. As of the 2020 census, the population of the city was 397,532. The Wichita metro area had a population of 647,610 in 2020. It is located in ...

, began his undergraduate study at the

California Institute of Technology The California Institute of Technology (branded as Caltech or CIT)The university itself only spells its short form as "Caltech"; the institution considers other spellings such a"Cal Tech" and "CalTech" incorrect. The institute is also occasional ...

in 1960, then transferred to 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 ...

, from which he received his PhD in 1970 from for a dissertation entitled ''Results and Independence Proofs in Combinatorial Set Theory''. His advisor was Robert Vaught. He became a professor at

Dartmouth College Dartmouth College (; ) is a private research university in Hanover, New Hampshire. Established in 1769 by Eleazar Wheelock, it is one of the nine colonial colleges chartered before the American Revolution. Although founded to educate Native ...

in 1969, and spent his entire career there. One of Baumgartner's results is the consistency of the statement that any two

\aleph_1

-dense sets of reals are order isomorphic (a set of reals is

\aleph_1

-dense if it has exactly

\aleph_1

points in every open interval). With

András Hajnal András Hajnal (May 13, 1931 – July 30, 2016) was a professor of mathematics at Rutgers University and a member of the Hungarian Academy of Sciences known for his work in set theory and combinatorics. Biography Hajnal was born on 13 May 1931, ...

he proved the Baumgartner–Hajnal theorem, which states that the

partition relation In mathematics, infinitary combinatorics, or combinatorial set theory, is an extension of ideas in combinatorics to infinite sets. Some of the things studied include continuous graphs and trees, extensions of Ramsey's theorem, and Martin's axiom. R ...

\omega_1\to(\alpha)^2_n

holds for

\alpha<\omega_1

and

n<\omega

. He died in 2011 of a heart attack at his home in

Hanover, New Hampshire Hanover is a town located along the Connecticut River in Grafton County, New Hampshire, United States. As of the 2020 census, its population was 11,870. The town is home to the Ivy League university Dartmouth College, the U.S. Army Corps of En ...

. The mathematical context in which Baumgartner worked spans

Suslin's problem In mathematics, Suslin's problem is a question about totally ordered sets posed by and published posthumously. It has been shown to be independent of the standard axiomatic system of set theory known as ZFC: showed that the statement can neit ...

Ramsey theory Ramsey theory, named after the British mathematician and philosopher Frank P. Ramsey, is a branch of mathematics that focuses on the appearance of order in a substructure given a structure of a known size. Problems in Ramsey theory typically ask ...

, 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.Jean A. Larson: In memoriam: James Earl Baumgartner (1943-2011)
/ref>

Selected publications

* Baumgartner, James E., ''A new class of order types'', Annals of Mathematical Logic, 9:187–222, 1976 * Baumgartner, James E., ''Ineffability properties of cardinals I'', Infinite and Finite Sets, Keszthely (Hungary) 1973, volume 10 of Colloquia Mathematica Societatis János Bolyai, pages 109–130. North-Holland, 1975 * Baumgartner, James E.; Harrington, Leo; Kleinberg, Eugene, ''Adding a closed unbounded set'', Journal of Symbolic Logic, 41(2):481–482, 1976 * Baumgartner, James E., ''Ineffability properties of cardinals II'', Robert E. Butts and Jaakko Hintikka, editors, Logic, Foundations of Mathematics and Computability Theory, pages 87–106. Reidel, 1977 * Baumgartner, James E.; Galvin, Fred, ''Generalized Erdős cardinals and 0^#'', Annals of Mathematical Logic 15, 289–313, 1978 * Baumgartner, James E.; Erdős, Paul; Galvin, Fred; Larson, J., ''Colorful partitions of cardinal numbers'', Can. J. Math. 31, 524–541, 1979 * Baumgartner, James E.; Erdős, Paul; Higgs, D., ''Cross-cuts in the power set of an infinite set'', Order 1, 139–145, 1984 * Baumgartner, James E. (Editor), ''Axiomatic Set Theory'' (Contemporary Mathematics, Volume 31), 1990

References

{{DEFAULTSORT:Baumgartner, James Earl 1943 births 20th-century American mathematicians 21st-century American mathematicians American logicians Set theorists Mathematical logicians University of California, Berkeley alumni Dartmouth College faculty 2011 deaths People from Wichita, Kansas Mathematicians from Kansas

See also

Selected publications

References