Robert Lawson Vaught
Encyclopedia
Robert Lawson Vaught was a mathematical logic
ian, and one of the founders of model theory
.
, at age 16. When World War II
broke out, he enlisted US Navy which assigned him to the University of California
's V-12
program. He graduated in 1945 with an AB in physics.
In 1946, he began a Ph.D. in mathematics at Berkeley. He initially worked under the topologist John L. Kelley
, writing on C* algebras. In 1950, in response to McCarthyite
pressures, Berkeley required all staff to sign a loyalty oath
. Kelley declined and moved his career to Tulane University
for three years. Vaught then began afresh under Alfred Tarski
, completing in 1954 a thesis on mathematical logic
, titled Topics in the Theory of Arithmetical Classes and Boolean Algebras. After a four years at the University of Washington
, Vaught returned to Berkeley in 1958, where he remained until his 1991 retirement.
In 1957, Vaught married Marilyn Maca; they had two children.
. In 1957, he and Tarski introduced elementary submodels and the Tarski–Vaught test characterizing them. In 1962, he and Morley
pioneered the concept of a saturated structure. His investigation of countable models of first order theories led him to the Vaught conjecture
that the number of countable models of a complete first order theory (in a countable language) is always either finite, or countably infinite, or equinumerous with the real numbers. Vaught's "Never 2" theorem states that a complete first order theory cannot have exactly 2 nonisomorphic countable models.
He considered his best work was his paper "Invariant sets in topology and logic", introducing the Vaught transform. He is known for the Tarski–Vaught criterion for elementary extensionality, the Feferman–Vaught theorem, the Łoś–Vaught test for completeness and decidability, the Vaught two-cardinal theorem, and his conjecture on the nonfinite axiomatizability of totally categorical theories (this work eventually led to geometric stability theory).
Vaught was a capable teacher of undergraduates, and his writing was reputed for elegance and clarity. His Set Theory: An Introduction (2001, 2nd ed.) attests to his abilities in this regard.
Mathematical logic
Mathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...
ian, and one of the founders of model theory
Model theory
In mathematics, model theory is the study of mathematical structures using tools from mathematical logic....
.
Life
Vaught was a bit of a musical prodigy in his youth, in his case the piano. He began his university studies at Pomona CollegePomona College
Pomona College is a private, residential, liberal arts college in Claremont, California. Founded in 1887 in Pomona, California by a group of Congregationalists, the college moved to Claremont in 1889 to the site of a hotel, retaining its name. The school enrolls 1,548 students.The founding member...
, at age 16. When World War II
World War II
World War II, or the Second World War , was a global conflict lasting from 1939 to 1945, involving most of the world's nations—including all of the great powers—eventually forming two opposing military alliances: the Allies and the Axis...
broke out, he enlisted US Navy which assigned him to the University of California
University of California
The University of California is a public university system in the U.S. state of California. Under the California Master Plan for Higher Education, the University of California is a part of the state's three-tier public higher education system, which also includes the California State University...
's V-12
V-12 Navy College Training Program
The V-12 Navy College Training Program was designed to supplement the force of commissioned officers in the United States Navy during World War II...
program. He graduated in 1945 with an AB in physics.
In 1946, he began a Ph.D. in mathematics at Berkeley. He initially worked under the topologist John L. Kelley
John L. Kelley
John Leroy Kelley was an American mathematician at University of California, Berkeley who worked in general topology and functional analysis....
, writing on C* algebras. In 1950, in response to McCarthyite
McCarthyism
McCarthyism is the practice of making accusations of disloyalty, subversion, or treason without proper regard for evidence. The term has its origins in the period in the United States known as the Second Red Scare, lasting roughly from the late 1940s to the late 1950s and characterized by...
pressures, Berkeley required all staff to sign a loyalty oath
Loyalty oath
A loyalty oath is an oath of loyalty to an organization, institution, or state of which an individual is a member.In this context, a loyalty oath is distinct from pledge or oath of allegiance...
. Kelley declined and moved his career to Tulane University
Tulane University
Tulane University is a private, nonsectarian research university located in New Orleans, Louisiana, United States...
for three years. Vaught then began afresh under Alfred Tarski
Alfred Tarski
Alfred Tarski was a Polish logician and mathematician. Educated at the University of Warsaw and a member of the Lwow-Warsaw School of Logic and the Warsaw School of Mathematics and philosophy, he emigrated to the USA in 1939, and taught and carried out research in mathematics at the University of...
, completing in 1954 a thesis on mathematical logic
Mathematical logic
Mathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...
, titled Topics in the Theory of Arithmetical Classes and Boolean Algebras. After a four years at the University of Washington
University of Washington
University of Washington is a public research university, founded in 1861 in Seattle, Washington, United States. The UW is the largest university in the Northwest and the oldest public university on the West Coast. The university has three campuses, with its largest campus in the University...
, Vaught returned to Berkeley in 1958, where he remained until his 1991 retirement.
In 1957, Vaught married Marilyn Maca; they had two children.
Work
Vaught's work is primarily focused around the field of model theoryModel theory
In mathematics, model theory is the study of mathematical structures using tools from mathematical logic....
. In 1957, he and Tarski introduced elementary submodels and the Tarski–Vaught test characterizing them. In 1962, he and Morley
Michael D. Morley
Michael Darwin Morley is an American mathematician, currently professor emeritus at Cornell University.His research is in advanced mathematical logic and model theory, and he is best known for Morley's categoricity theorem, which he proved in his Ph.D. thesis "Categoricity in Power" in 1962.His...
pioneered the concept of a saturated structure. His investigation of countable models of first order theories led him to the Vaught conjecture
Vaught conjecture
The Vaught conjecture is a conjecture in the mathematical field of model theory originally proposed by Robert Lawson Vaught in 1961. It states that the number of countable models of a first-order complete theory in a countable language is finite or ℵ0 or 2ℵ0...
that the number of countable models of a complete first order theory (in a countable language) is always either finite, or countably infinite, or equinumerous with the real numbers. Vaught's "Never 2" theorem states that a complete first order theory cannot have exactly 2 nonisomorphic countable models.
He considered his best work was his paper "Invariant sets in topology and logic", introducing the Vaught transform. He is known for the Tarski–Vaught criterion for elementary extensionality, the Feferman–Vaught theorem, the Łoś–Vaught test for completeness and decidability, the Vaught two-cardinal theorem, and his conjecture on the nonfinite axiomatizability of totally categorical theories (this work eventually led to geometric stability theory).
Vaught was a capable teacher of undergraduates, and his writing was reputed for elegance and clarity. His Set Theory: An Introduction (2001, 2nd ed.) attests to his abilities in this regard.