Robert Lawson Vaught (April 4, 1926 – April 2, 2002) was a
mathematical logic
Mathematical logic is the study of Logic#Formal logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory (also known as computability theory). Research in mathematical logic com ...
ian and one of the founders of
model theory
In mathematical logic, model theory is the study of the relationship between theory (mathematical logic), formal theories (a collection of Sentence (mathematical logic), sentences in a formal language expressing statements about a Structure (mat ...
.
In Memoriam: Robert Lawson Vaught, U. C. Berkeley
Life
Vaught was a musical prodigy in his youth, in his case playing the piano. He began his university studies at Pomona College
Pomona College ( ) is a private university, private Liberal arts colleges in the United States, liberal arts college in Claremont, California. It was established in 1887 by a group of Congregationalism in the United States, Congregationalists ...
, at age 16. When World War II
World War II or the Second World War (1 September 1939 – 2 September 1945) was a World war, global conflict between two coalitions: the Allies of World War II, Allies and the Axis powers. World War II by country, Nearly all of the wo ...
broke out, he enlisted into the US Navy
The United States Navy (USN) is the naval warfare, maritime military branch, service branch of the United States Department of Defense. It is the world's most powerful navy with the largest Displacement (ship), displacement, at 4.5 millio ...
, which assigned him to the University of California
The University of California (UC) is a public university, public Land-grant university, land-grant research university, research university system in the U.S. state of California. Headquartered in Oakland, California, Oakland, the system is co ...
'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 supervision of 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
Loyalty is a Fixation (psychology), devotion to a country, philosophy, group, or person. Philosophers disagree on what can be an object of loyalty, as some argue that loyalty is strictly interpersonal and only another human being can be the obj ...
. Kelley declined and moved his career to Tulane University
The Tulane University of Louisiana (commonly referred to as Tulane University) is a private research university in New Orleans, Louisiana, United States. Founded as the Medical College of Louisiana in 1834 by a cohort of medical doctors, it b ...
for three years. Vaught then began afresh under the supervision of Alfred Tarski
Alfred Tarski (; ; born Alfred Teitelbaum;School of Mathematics and Statistics, University of St Andrews ''School of Mathematics and Statistics, University of St Andrews''. January 14, 1901 – October 26, 1983) was a Polish-American logician ...
, completing in 1954 a thesis on mathematical logic
Mathematical logic is the study of Logic#Formal logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory (also known as computability theory). Research in mathematical logic com ...
, titled ''Topics in the Theory of Arithmetical Classes and Boolean Algebras''. After spending four years at the University of Washington
The University of Washington (UW and informally U-Dub or U Dub) is a public research university in Seattle, Washington, United States. Founded in 1861, the University of Washington is one of the oldest universities on the West Coast of the Uni ...
, 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 on model theory
In mathematical logic, model theory is the study of the relationship between theory (mathematical logic), formal theories (a collection of Sentence (mathematical logic), sentences in a formal language expressing statements about a Structure (mat ...
. In 1957, he and Tarski introduced elementary submodel
In model theory, a branch of mathematical logic, two structures ''M'' and ''N'' of the same signature ''σ'' are called elementarily equivalent if they satisfy the same first-order ''σ''-sentences.
If ''N'' is a substructure of ''M'', one oft ...
s and the Tarski–Vaught test
In model theory, a branch of mathematical logic, two structures ''M'' and ''N'' of the same signature ''σ'' are called elementarily equivalent if they satisfy the same first-order ''σ''-sentences.
If ''N'' is a substructure of ''M'', one oft ...
characterizing them. In 1962, he and Michael D. Morley
Michael Darwin Morley (September 29, 1930 – October 11, 2020) was an American mathematician. At his death in 2020, Morley was professor emeritus at Cornell University. His research was in mathematical logic and model theory, and he is best know ...
pioneered the concept of a saturated structure. His investigations on countable models of first-order theories led him to the 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 finit ...
stating that the number
A number is a mathematical object used to count, measure, and label. The most basic examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual numbers can ...
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 two nonisomorphic countable models.
He considered his best work his paper "Invariant sets in topology and logic", introducing the Vaught transform. He is known for the Tarski–Vaught test for elementary substructures, the Feferman–Vaught theorem
The Feferman–Vaught theorem in model theory is a theorem by Solomon Feferman and Robert Lawson Vaught that shows how to reduce, in an algorithmic way, the first-order theory of a product of structures to the first-order theory of elements of t ...
, 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).
See also
* Łoś–Vaught test
Notes
References
* Feferman, Anita Burdman, and Solomon Feferman
Solomon Feferman (December 13, 1928July 26, 2016) was an American philosopher and mathematician who worked in mathematical logic. In addition to his prolific technical work in proof theory, computability theory, and set theory, he was known for h ...
, 2004. ''Alfred Tarski: Life and Logic''. Cambridge Univ. Press. 24 index entries for Vaught, especially pp. 185–88.
External links
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{{DEFAULTSORT:Vaught, Robert Lawson
1926 births
2002 deaths
20th-century American mathematicians
21st-century American mathematicians
Model theorists
People from Alhambra, California
Pomona College alumni