Leon Henkin

# Leon Henkin

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Leon Albert Henkin was a logician at the University of California, Berkeley
University of California, Berkeley
The University of California, Berkeley , is a teaching and research university established in 1868 and located in Berkeley, California, USA...

. He was principally known for the "Henkin's completeness proof": his version of the proof of the semantic completeness of standard systems of first-order logic
First-order logic
First-order logic is a formal logical system used in mathematics, philosophy, linguistics, and computer science. It goes by many names, including: first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate logic...

.

## The completeness proof

Henkin's result was not novel; it had first been proved by Kurt Gödel
Kurt Gödel
Kurt Friedrich Gödel was an Austrian logician, mathematician and philosopher. Later in his life he emigrated to the United States to escape the effects of World War II. One of the most significant logicians of all time, Gödel made an immense impact upon scientific and philosophical thinking in the...

in his doctoral dissertation which was completed in 1929. (See Gödel's completeness theorem
Gödel's completeness theorem
Gödel's completeness theorem is a fundamental theorem in mathematical logic that establishes a correspondence between semantic truth and syntactic provability in first-order logic. It was first proved by Kurt Gödel in 1929....

. Gödel published a version of the proof in 1930.) Henkin's 1949 proof is much easier to survey than Gödel's and has thus become the standard choice of completeness proof for presentation in introductory classes and texts.

The proof is non-constructive (a pure existence proof
Existence theorem
In mathematics, an existence theorem is a theorem with a statement beginning 'there exist ..', or more generally 'for all x, y, ... there exist ...'. That is, in more formal terms of symbolic logic, it is a theorem with a statement involving the existential quantifier. Many such theorems will not...

): while it guarantees that if a sentence α follows (semantically) from a set of sentences Σ, then there is a proof of α from Σ, it gives no indication of the nature of that proof.

Henkin originally proved the completeness of Church's
Alonzo Church
Alonzo Church was an American mathematician and logician who made major contributions to mathematical logic and the foundations of theoretical computer science. He is best known for the lambda calculus, Church–Turing thesis, Frege–Church ontology, and the Church–Rosser theorem.-Life:Alonzo Church...

higher-order logic
Higher-order logic
In mathematics and logic, a higher-order logic is a form of predicate logic that is distinguished from first-order logic by additional quantifiers and a stronger semantics...

, and then observed that the same methods of proof could be applied to first-order logic.
Henkin's proof for higher-order logic uses a variant of the standard semantics.
This variant uses general models
(also called Henkin models): the higher types need not be interpreted by the full space of functions; a subset of the function space may be used instead.

## Early life

He was born in Brooklyn
Brooklyn
Brooklyn is the most populous of New York City's five boroughs, with nearly 2.6 million residents, and the second-largest in area. Since 1896, Brooklyn has had the same boundaries as Kings County, which is now the most populous county in New York State and the second-most densely populated...

, into a Russian
Russians
The Russian people are an East Slavic ethnic group native to Russia, speaking the Russian language and primarily living in Russia and neighboring countries....

Jewish immigrant family. His father expressed his high expectations for him by giving him the middle name "Albert"; at the time, the New York Times ran a series of articles on Albert Einstein
Albert Einstein
Albert Einstein was a German-born theoretical physicist who developed the theory of general relativity, effecting a revolution in physics. For this achievement, Einstein is often regarded as the father of modern physics and one of the most prolific intellects in human history...

's theory of relativity. However, Henkin did not use his middle name in his mathematical publications.

His first degree was in mathematics and philosophy from Columbia College
Columbia College of Columbia University
Columbia College is the oldest undergraduate college at Columbia University, situated on the university's main campus in Morningside Heights in the borough of Manhattan in New York City. It was founded in 1754 by the Church of England as King's College, receiving a Royal Charter from King George II...

, in 1941. He took a master's degree there in 1942.

He then worked in the Signal Corps Radar Laboratory, Belmar, New Jersey
Belmar, New Jersey
Belmar is a borough in Monmouth County, New Jersey, United States. As of the 2010 United States Census, the borough population was 5,794. The Borough of Belmar is governed under the Faulkner Act system of municipal government....

. As participant in the Manhattan project
Manhattan Project
The Manhattan Project was a research and development program, led by the United States with participation from the United Kingdom and Canada, that produced the first atomic bomb during World War II. From 1942 to 1946, the project was under the direction of Major General Leslie Groves of the US Army...

, he worked on isotope diffusion, in New York, and Oak Ridge, Tennessee
Oak Ridge, Tennessee
Oak Ridge is a city in Anderson and Roane counties in the eastern part of the U.S. state of Tennessee, about west of Knoxville. Oak Ridge's population was 27,387 at the 2000 census...

.

He was a doctoral student of Alonzo Church
Alonzo Church
Alonzo Church was an American mathematician and logician who made major contributions to mathematical logic and the foundations of theoretical computer science. He is best known for the lambda calculus, Church–Turing thesis, Frege–Church ontology, and the Church–Rosser theorem.-Life:Alonzo Church...

at Princeton University
Princeton University
Princeton University is a private research university located in Princeton, New Jersey, United States. The school is one of the eight universities of the Ivy League, and is one of the nine Colonial Colleges founded before the American Revolution....

, receiving his Ph.D. in 1947. He became Professor
Professor
A professor is a scholarly teacher; the precise meaning of the term varies by country. Literally, professor derives from Latin as a "person who professes" being usually an expert in arts or sciences; a teacher of high rank...

of Mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

at the University of California, Berkeley
University of California, Berkeley
The University of California, Berkeley , is a teaching and research university established in 1868 and located in Berkeley, California, USA...

, where he had a position from 1953. He received the 1964 Chauvenet Prize
Chauvenet Prize
The Chauvenet Prize is the highest award for mathematical expository writing. It consists of a prize of \$1,000 and a certificate, and is awarded yearly by the Mathematical Association of America in recognition of an outstanding expository article on a mathematical topic. The prize is named in...

for exposition.http://www.maa.org/awards/chauvent.html He was a collaborator of 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...

, and an ally in promoting logic.

Henkin was also a social activist who since the 1960s worked to increase higher education opportunities for women and minorities. In 1964 he spearheaded the formation of the Special Scholarships Committee at UC Berkeley, which resulted in setting up Special Opportunity Scholarships and other outreach programs at Berkeley. The Berkeley program served as a model for the federal Upward Bound Program that was founded several years later and for many outreach and special opportunities programs at other U.S. universities.

• 2000 — Leon Henkin Citation — for Distinguished Service, which is presented to a (UC) faculty member for "exceptional commitment to the educational development of students from groups who are underrepresented in the academy."
• 1991 — Berkeley Citation — the highest honor/award bestowed by the University of California.
• 1990 — First recipient of the Gung and Hu Award for Distinguished Service to Mathematics.
• 1972 — Lester R. Ford Award — for Mathematical foundations for mathematics, American Mathematical Monthly
American Mathematical Monthly
The American Mathematical Monthly is a mathematical journal founded by Benjamin Finkel in 1894. It is currently published 10 times each year by the Mathematical Association of America....

78 (1971), 463–487.
• 1964 — The Chauvenet Prize
Chauvenet Prize
The Chauvenet Prize is the highest award for mathematical expository writing. It consists of a prize of \$1,000 and a certificate, and is awarded yearly by the Mathematical Association of America in recognition of an outstanding expository article on a mathematical topic. The prize is named in...

, Mathematical Association of America award to the author of an outstanding expository article on a mathematical topic by a member of the Association.