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James Earl Baumgartner
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 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 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 he proved the Baumgartner–Hajnal theorem, which states that the partition relation \omega_1\to(\alpha)^2_n holds for \alpha<\omega_1 and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wichita, Kansas
Wichita ( ) is the largest city in the U.S. state of Kansas and the county seat of Sedgwick County, Kansas, Sedgwick County. As of the 2020 United States census, 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 south-central Kansas on the Arkansas River. Wichita began as a trading post on the Chisholm Trail in the 1860s and was incorporated as a city in 1870. It became a destination for Cattle drives in the United States, cattle drives traveling north from Texas to Kansas railroads, earning it the nickname "Cowtown".Miner, Prof. Craig (Wichita State Univ. Dept. of History), ''Wichita: The Magic City'', Wichita Historical Museum Association, Wichita, KS, 1988Howell, Angela and Peg Vines, ''The Insider's Guide to Wichita'', Wichita Eagle & Beacon Publishing, Wichita, KS, 1995 Wyatt Earp served as a police officer in Wichita for around one year before going to Dodge City, Kansas, Dodge City. In the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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. Recent developments concern combinatorics of the continuum and combinatorics on successors of singular cardinals.Todd Eisworth, ''Successors of Singular Cardinals'' Chapter 15 in Handbook of Set Theory, edited by Matthew Foreman and Akihiro Kanamori, Springer, 2010 Ramsey theory for infinite sets Write κ, λ for ordinals, ''m'' for a cardinal number and ''n'' for a natural number. introduced the notation :\kappa\rightarrow(\lambda)^n_m as a shorthand way of saying that every partition of the set ºsup>''n'' of ''n''-element subsets of \kappa into ''m'' pieces has a homogeneous set of order type λ. A homogeneous set is in this case a subset of κ such that every ''n''-element subset is in the same element of the partition. When ''m'' i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Set Theorists
Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electronics and computing *Set (abstract data type), a data type in computer science that is a collection of unique values ** Set (C++), a set implementation in the C++ Standard Library * Set (command), a command for setting values of environment variables in Unix and Microsoft operating-systems * Secure Electronic Transaction, a standard protocol for securing credit card transactions over insecure networks * Single-electron transistor, a device to amplify currents in nanoelectronics * Single-ended triode, a type of electronic amplifier * Set!, a programming syntax in the scheme programming language Biology and psychology * Set (psychology), a set of expectations which shapes perception or thought *Set or sett, a badger's den *Set, a small tuber ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
American Logicians
American(s) may refer to: * American, something of, from, or related to the United States of America, commonly known as the "United States" or "America" ** Americans, citizens and nationals of the United States of America ** American ancestry, people who self-identify their ancestry as "American" ** American English, the set of varieties of the English language native to the United States ** Native Americans in the United States, indigenous peoples of the United States * American, something of, from, or related to the Americas, also known as "America" ** Indigenous peoples of the Americas * American (word), for analysis and history of the meanings in various contexts Organizations * American Airlines, U.S.-based airline headquartered in Fort Worth, Texas * American Athletic Conference, an American college athletic conference * American Recordings (record label), a record label previously known as Def American * American University, in Washington, D.C. Sports teams Soccer * B ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
21st-century American Mathematicians
The 1st century was the century spanning AD 1 ( I) through AD 100 ( C) according to the Julian calendar. It is often written as the or to distinguish it from the 1st century BC (or BCE) which preceded it. The 1st century is considered part of the Classical era, epoch, or historical period. The 1st century also saw the appearance of Christianity. During this period, Europe, North Africa and the Near East fell under increasing domination by the Roman Empire, which continued expanding, most notably conquering Britain under the emperor Claudius ( AD 43). The reforms introduced by Augustus during his long reign stabilized the empire after the turmoil of the previous century's civil wars. Later in the century the Julio-Claudian dynasty, which had been founded by Augustus, came to an end with the suicide of Nero in AD 68. There followed the famous Year of Four Emperors, a brief period of civil war and instability, which was finally brought to an end by Vespasian, ninth Roman empero ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
1943 Births
Events Below, the events of World War II have the "WWII" prefix. January * January 1 – WWII: The Soviet Union announces that 22 German divisions have been encircled at Stalingrad, with 175,000 killed and 137,650 captured. * January 4 – WWII: Greek-Polish athlete and saboteur Jerzy Iwanow-Szajnowicz is executed by the Germans at Kaisariani. * January 11 ** The United States and United Kingdom revise previously unequal treaty relationships with the Republic of China (1912–1949), Republic of China. ** Italian-American anarchist Carlo Tresca is assassinated in New York City. * January 13 – Anti-Nazi protests in Sofia result in 200 arrests and 36 executions. * January 14 – January 24, 24 – WWII: Casablanca Conference: Franklin D. Roosevelt, President of the United States; Winston Churchill, Prime Minister of the United Kingdom; and Generals Charles de Gaulle and Henri Giraud of the Free French forces meet secretly at the Anfa Hotel in Casablanca, Morocco, to plan the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Paul Erdős
Paul Erdős ( hu, Erdős Pál ; 26 March 1913 – 20 September 1996) was a Hungarian mathematician. He was one of the most prolific mathematicians and producers of mathematical conjectures of the 20th century. pursued and proposed problems in discrete mathematics, graph theory, number theory, mathematical analysis, approximation theory, set theory, and probability theory. Much of his work centered around discrete mathematics, cracking many previously unsolved problems in the field. He championed and contributed to Ramsey theory, which studies the conditions in which order necessarily appears. Overall, his work leaned towards solving previously open problems, rather than developing or exploring new areas of mathematics. Erdős published around 1,500 mathematical papers during his lifetime, a figure that remains unsurpassed. He firmly believed mathematics to be a social activity, living an itinerant lifestyle with the sole purpose of writing mathematical papers with other mathem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zero Sharp
In the mathematical discipline of set theory, 0# (zero sharp, also 0#) is the set of true formulae about indiscernibles and order-indiscernibles in the Gödel constructible universe. It is often encoded as a subset of the integers (using Gödel numbering), or as a subset of the hereditarily finite sets, or as a real number. Its existence is unprovable in ZFC, the standard form of axiomatic set theory, but follows from a suitable large cardinal axiom. It was first introduced as a set of formulae in Silver's 1966 thesis, later published as , where it was denoted by Σ, and rediscovered by , who considered it as a subset of the natural numbers and introduced the notation O# (with a capital letter O; this later changed to the numeral '0'). Roughly speaking, if 0# exists then the universe ''V'' of sets is much larger than the universe ''L'' of constructible sets, while if it does not exist then the universe of all sets is closely approximated by the constructible sets. Definition ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fred Galvin
Frederick William Galvin is a mathematician, currently a professor at the University of Kansas. His research interests include set theory and combinatorics. His notable combinatorial work includes the proof of the Dinitz conjecture. In set theory, he proved with András Hajnal that if ℵω1 is a limit cardinal, strong limit cardinal, then : 2^<\aleph_ holds. The research on extending this result led Saharon Shelah to the invention of PCF theory. Galvin gave an elementary proof of the Baumgartner–Hajnal theorem (). The original proof by James Earl Baumgartner, Baumgartner and Hajnal used Forcing (mathematics), forcing and absoluteness. Galvin and Shelah also proved the square bracket partition relations and . Galvin also proved the partition relation where η denotes the order type of the set ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Leo Harrington
Leo Anthony Harrington (born May 17, 1946) is a professor of mathematics at the University of California, Berkeley who works in recursion theory, model theory, and set theory. Having retired from being a Mathematician, Professor Leo Harrington is now a Philosopher. His notable results include proving the Paris–Harrington theorem along with Jeff Paris, showing that if the axiom of determinacy holds for all analytic sets then ''x''# exists for all reals ''x'', and proving with Saharon Shelah that the first-order theory of the partially ordered set of recursively enumerable Turing degrees In computer science and mathematical logic the Turing degree (named after Alan Turing) or degree of unsolvability of a set of natural numbers measures the level of algorithmic unsolvability of the set. Overview The concept of Turing degree is fund ... is undecidable. References External linksHome page * Living people American logicians 20th-century American mathematicians 21st-cent ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Baumgartner's Axiom
In mathematical set theory, Baumgartner's axiom (BA) can be one of three different axioms introduced by James Earl Baumgartner. A subset of the real line is said to be \aleph_1-dense if every two points are separated by exactly \aleph_1 other points, where \aleph_1 is the smallest uncountable cardinality. This would be true for the real line itself under the continuum hypothesis. An axiom introduced by states that all \aleph_1-dense subsets of the real line are order-isomorphic, providing a higher-cardinality analogue of Cantor's isomorphism theorem that countable dense subsets are isomorphic. Baumgartner's axiom is a consequence of the proper forcing axiom. It is consistent with a combination of ZFC, Martin's axiom, and the negation of the continuum hypothesis In mathematics, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets. It states that or equivalently, that In Zermelo–Fraenkel set theory with the axiom of choice ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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