Robert Lawson Vaught (April 4, 1926 – April 2, 2002) was a

mathematical logic Mathematical logic is the study of logic, 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 for ...

ian and one of the founders of

model theory In mathematical logic, model theory is the study of the relationship between formal theories (a collection of sentences in a formal language expressing statements about a mathematical structure), and their models (those structures in which the s ...

.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 liberal arts college in Claremont, California. It was established in 1887 by a group of Congregationalists who wanted to recreate a "college of the New England type" in Southern California. In 1925, it became ...

, at age 16. When

World War II World War II or the Second World War, often abbreviated as WWII or WW2, was a world war that lasted from 1939 to 1945. It involved the vast majority of the world's countries—including all of the great powers—forming two opposin ...

broke out, he enlisted into the

US Navy The United States Navy (USN) is the maritime service branch of the United States Armed Forces and one of the eight uniformed services of the United States. It is the largest and most powerful navy in the world, with the estimated tonnage of ...

, which assigned him to the

University of California The University of California (UC) is a public land-grant research university system in the U.S. state of California. The system is composed of the campuses at Berkeley, Davis, Irvine, Los Angeles, Merced, Riverside, San Diego, San Francisco, ...

'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* algebra In mathematics, specifically in functional analysis, a C∗-algebra (pronounced "C-star") is a Banach algebra together with an involution satisfying the properties of the adjoint. A particular case is that of a complex algebra ''A'' of continuou ...

s. In 1950, in response to

McCarthyite McCarthyism is the practice of making false or unfounded accusations of subversion and treason, especially when related to anarchism, communism and socialism, and especially when done in a public and attention-grabbing manner. The term origina ...

pressures, Berkeley required all staff to sign a

loyalty oath A loyalty oath is a pledge of allegiance to an organization, institution, or state of which an individual is a member. In the United States, such an oath has often indicated that the affiant has not been a member of a particular organization or ...

. Kelley declined and moved his career to

Tulane University Tulane University, officially the Tulane University of Louisiana, is a private university, private research university in New Orleans, Louisiana. Founded as the Medical College of Louisiana in 1834 by seven young medical doctors, it turned into ...

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 a ...

, completing in 1954 a thesis on

, 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, simply Washington, or informally U-Dub) is a public research university in Seattle, Washington. Founded in 1861, Washington is one of the oldest universities on the West Coast; it was established in Seattle a ...

, 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

. 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 ofte ...

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 often ...

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 In mathematical logic, and particularly in its subfield model theory, a saturated model ''M'' is one that realizes as many complete types as may be "reasonably expected" given its size. For example, an ultrapower model of the hyperreals is \al ...

. 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 finite o ...

stating that the

number A number is a mathematical object used to count, measure, and label. The original 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 c ...

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 was his paper "Invariant sets in topology and logic", introducing the

Vaught transform Vaught is a surname, and may refer to: * BC Vaught, drummer for Hed PE * DeAnn Vaught, member of the Arkansas House of Representatives * James B. Vaught, United States Army Lieutenant General * Johnny Vaught, American college football player * Lo ...

. He is known for the Tarski–Vaught test for elementary substructures, the

Feferman–Vaught theorem 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 first-order structures to the first-order theory of elemen ...

, the

Łoś–Vaught test In model theory, a branch of mathematical logic, the Łoś–Vaught test is a criterion for a theory to be complete, unable to be augmented without becoming inconsistent. For theories in classical logic Classical logic (or standard logic or Frege-R ...

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).

Notes

References

* Feferman, Anita Burdman, and

Solomon Feferman Solomon Feferman (December 13, 1928 – July 26, 2016) was an American philosopher and mathematician who worked in mathematical logic. Life Solomon Feferman was born in The Bronx in New York City to working-class parents who had immigrated to th ...

, 2004. ''Alfred Tarski: Life and Logic''. Cambridge Univ. Press. 24 index entries for Vaught, especially pp. 185–88.

External links

* * {{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

Life

Work

See also

Notes

References

External links