Paul Cohen

Paul J. Cohen
Paul J. Cohen
Paul J. Cohen
Born	Paul Joseph Cohen 2 4, 1934
Birthplace	Long Branch, New Jersey, U.S.
Died	Template:Death date and age Stanford, California, U.S.
Nationality	American
Occupation	Mathematician
Employer	Stanford University
Known for	Cohen forcing, independence of the continuum hypothesis and the axiom of choice
Education	University of Chicago (M.S., Ph.D.)
Awards	Bôcher Memorial Prize (1964), Fields Medal (1966), National Medal of Science (1967)

Paul Joseph Cohen (April 2, 1934 – March 23, 2007) was an American mathematician who transformed the landscape of modern set theory and mathematical logic. He is best known for developing the technique of forcing, which he used to prove that the continuum hypothesis and the axiom of choice are independent of the standard Zermelo–Fraenkel axioms of set theory. This landmark achievement resolved a question that had stood since Georg Cantor first proposed the continuum hypothesis in 1878 and that David Hilbert had placed first on his famous list of unsolved problems in 1900. For this work, Cohen was awarded the Fields Medal in 1966, becoming the only mathematician to receive the Fields Medal for work in mathematical logic. He also received the Bôcher Memorial Prize in 1964 for contributions to mathematical analysis and the National Medal of Science in 1967. Cohen spent the majority of his career as a professor at Stanford University, where he was a member of the mathematics faculty for over four decades. His proof technique of forcing became one of the most important and widely used methods in set theory, opening entire new fields of mathematical investigation and fundamentally altering the understanding of the foundations of mathematics.

Early Life

Paul Joseph Cohen was born on April 2, 1934, in Long Branch, New Jersey, United States. He grew up in a Jewish family and was raised in the New York City area, where he attended school in Brooklyn. Cohen showed an early aptitude for mathematics and is reported to have developed a deep interest in the subject while still a teenager. He was intellectually precocious and pursued advanced mathematical topics at a young age.

Cohen's early intellectual development was shaped by the rich mathematical culture available in New York City during the mid-twentieth century. He was drawn to both pure mathematics and the foundational questions that would later define his career. His childhood and adolescence were marked by a determined focus on mathematical study, and he moved quickly through his early education, demonstrating the kind of independent thinking and ambition that would characterize his later work.

Growing up in a modest household, Cohen's path into mathematics was driven by talent and intellectual curiosity rather than family connections to the academic world. He was largely self-directed in his mathematical education, and his early encounters with advanced mathematical texts helped cultivate the deep intuition that would later allow him to tackle some of the most difficult open problems in the field.

Education

Cohen pursued his higher education at the University of Chicago, one of the leading centers for mathematical research in the United States during the 1950s. The university's mathematics department was home to a distinguished faculty, and it provided an intellectually stimulating environment for the young mathematician.

At Chicago, Cohen studied under Antoni Zygmund, a prominent Polish-American mathematician who specialized in harmonic analysis and trigonometric series. Under Zygmund's supervision, Cohen completed his doctoral dissertation, titled Topics in the Theory of Uniqueness of Trigonometrical Series, which he defended in 1958. He received both his Master of Science and Doctor of Philosophy degrees from the University of Chicago.

Cohen's doctoral work in the theory of trigonometric series was a significant contribution to mathematical analysis and demonstrated his ability to work at the highest level in a field quite different from the set theory for which he would later become famous. His training in analysis under Zygmund gave him a broad mathematical foundation and a facility with technical arguments that would prove essential in his later groundbreaking work on the continuum hypothesis.

Career

Early Academic Career and Work in Analysis

After completing his doctorate in 1958, Cohen held positions at several institutions before joining the faculty of Stanford University. His early career was marked by significant contributions to mathematical analysis, the field in which he had been trained. Cohen's work in this period demonstrated a breadth of mathematical ability that extended well beyond a single specialty.

Cohen's contributions to analysis were recognized in 1964, when he was awarded the Bôcher Memorial Prize by the American Mathematical Society. The Bôcher Prize is one of the most prestigious awards in mathematical analysis, and Cohen received it for his work on Littlewood's conjecture and related problems in the theory of trigonometric series and harmonic analysis. This recognition established Cohen as a major figure in pure mathematics even before his most famous achievement in set theory.

During this period, Cohen also developed interests in number theory and other areas of mathematics. Colleagues noted his remarkable mathematical range and his willingness to attack problems across different branches of the discipline. His confidence in tackling difficult, long-standing open questions would soon lead him toward the most celebrated work of his career.

The Independence of the Continuum Hypothesis

Cohen's most famous and consequential achievement was his proof, completed in 1963, that the continuum hypothesis and the axiom of choice are independent of the Zermelo–Fraenkel (ZF) axioms of set theory. This result, together with Kurt Gödel's earlier proof in 1940 that the continuum hypothesis is consistent with the ZF axioms, established that the continuum hypothesis can be neither proved nor disproved from the standard axioms of set theory. The problem had been open since Georg Cantor first formulated the continuum hypothesis in 1878, and it had been listed by David Hilbert as the first of his 23 unsolved problems at the International Congress of Mathematicians in 1900.

The continuum hypothesis states that there is no set whose cardinality is strictly between that of the integers and that of the real numbers — in other words, that the cardinality of the continuum (the set of real numbers) is the next cardinal number after the cardinality of the natural numbers. Cantor had conjectured that this was true but was unable to prove it. The question became one of the central open problems in the foundations of mathematics.

Cohen developed a new technique, which he called forcing, to construct models of set theory in which the continuum hypothesis fails. His method allowed him to extend a given model of set theory by adding new sets in a carefully controlled manner, thereby producing models in which certain statements — such as the continuum hypothesis — could be made false. This was a profoundly original contribution: prior to Cohen, no technique existed for constructing such models.

Cohen published his initial results in two landmark papers in the Proceedings of the National Academy of Sciences in 1963 and 1964. These papers demonstrated that in a suitably constructed model, the continuum hypothesis does not hold, thereby proving its independence from the ZF axioms (with or without the axiom of choice). The result was initially met with intense scrutiny from the mathematical community, given the magnitude and difficulty of the problem.

Before the proof was formally published, Cohen sought verification from Kurt Gödel himself. Gödel reviewed the work and confirmed its correctness, reportedly expressing admiration for the achievement. The endorsement by Gödel — widely considered one of the greatest logicians in history — cemented the mathematical community's acceptance of Cohen's result.

The Method of Forcing

The technique of forcing, which Cohen invented to carry out his proof, became one of the most important tools in modern set theory and mathematical logic. Forcing provides a systematic method for constructing new models of set theory from existing ones by adjoining "generic" sets that satisfy prescribed properties. The method allows mathematicians to show that a wide variety of mathematical statements are independent of the standard axioms, meaning they can neither be proved nor disproved from those axioms alone.

The impact of forcing on set theory and mathematical logic was transformative. After Cohen's initial papers, the technique was rapidly adopted and extended by other mathematicians, leading to a proliferation of independence results. Forcing became the standard method for demonstrating that statements about infinite sets are undecidable within ZF set theory, and it has been applied to problems across many areas of mathematics, including topology, measure theory, algebra, and combinatorics.

Cohen later elaborated on his methods in the book Set Theory and the Continuum Hypothesis, which provided a detailed and accessible account of his proofs and the forcing technique. The book became a standard reference for students and researchers working in set theory and mathematical logic.

Career at Stanford University

Cohen joined the faculty of Stanford University as a professor of mathematics and spent the rest of his career there. Stanford became his intellectual home for over four decades, and he was a prominent figure in the university's mathematics department.

At Stanford, Cohen supervised doctoral students and mentored younger mathematicians. Among his doctoral students was Peter Sarnak, who went on to become one of the leading mathematicians of his generation, known for contributions to number theory and mathematical physics. Cohen's influence on the next generation of mathematicians extended beyond his formal doctoral students, as his work inspired a wide community of researchers in set theory, logic, and analysis.

Throughout his career at Stanford, Cohen continued to pursue research in multiple areas of mathematics. While his work on the continuum hypothesis and forcing remained his most famous contribution, he also maintained active interests in number theory and analysis. He was known for his ambition to solve major open problems, and he reportedly spent significant time attempting to make progress on the Riemann hypothesis, one of the most important unsolved problems in mathematics.

Cohen was also known within the Stanford mathematics community for his engaging personality and his energetic approach to mathematical discussion. Colleagues recalled his intense intellectual curiosity and his ability to communicate deep mathematical ideas with clarity and enthusiasm.

Broader Mathematical Contributions

Beyond the continuum hypothesis and forcing, Cohen contributed to several other areas of mathematics during his career. His early work on uniqueness of trigonometric series, completed under Zygmund's supervision, remained an important contribution to harmonic analysis. His research on Littlewood's conjecture contributed to the theory of approximation and number theory. Cohen's mathematical interests were notably broad, and he was among the few mathematicians of his era who made major contributions in both analysis and mathematical logic.

Cohen's willingness to move between different fields of mathematics and to attack problems that were considered extremely difficult or even intractable was a defining feature of his career. His success with the continuum hypothesis demonstrated the potential for individual insight and originality to resolve long-standing foundational questions.

Personal Life

Paul Cohen married Christina Cohen, and they had three children together. The family resided in the Stanford, California, area for much of Cohen's career.

Cohen was described by those who knew him as intellectually intense, deeply competitive, and driven by a desire to solve the most important problems in mathematics. His ambition to tackle fundamental questions — from the continuum hypothesis to the Riemann hypothesis — was a consistent theme throughout his life.

Cohen was diagnosed with a rare lung disease in later years. He died on March 23, 2007, at Stanford, California, at the age of 72. His death was reported widely in the mathematical community and in major newspapers, reflecting his stature as one of the most important mathematicians of the twentieth century.

Recognition

Cohen received some of the highest honors available in mathematics during his lifetime. His three major awards — the Bôcher Memorial Prize, the Fields Medal, and the National Medal of Science — were each received within a span of just four years, reflecting the extraordinary impact of his work during the 1960s.

Bôcher Memorial Prize (1964)

In 1964, Cohen was awarded the Bôcher Memorial Prize by the American Mathematical Society for his contributions to mathematical analysis. The prize recognized his work on problems in harmonic analysis and the theory of trigonometric series, predating the recognition of his set-theoretic achievements.

Fields Medal (1966)

Cohen received the Fields Medal at the International Congress of Mathematicians in Moscow in 1966. The Fields Medal, often described as the highest honor in mathematics, is awarded to mathematicians under the age of 40 for outstanding contributions. Cohen was recognized specifically for his proof of the independence of the continuum hypothesis and the axiom of choice from the Zermelo–Fraenkel axioms. He remains the only mathematician to have received the Fields Medal for work in mathematical logic.

National Medal of Science (1967)

In 1967, Cohen was awarded the National Medal of Science by the President of the United States, recognizing his contributions to mathematics. The National Medal of Science is the highest scientific honor bestowed by the United States government.

Honorary Doctorate

Cohen also received an honorary doctorate from Uppsala University in Sweden, further recognizing his contributions to mathematics.

Legacy

Paul Cohen's contributions to mathematics have had a lasting and profound impact on the field, particularly in set theory and mathematical logic. His invention of the forcing technique fundamentally changed the way mathematicians approach questions about the foundations of mathematics. Before Cohen's work, the independence of the continuum hypothesis from the standard axioms was an open question that many mathematicians had attempted without success for decades. After his proof, forcing became the primary tool for demonstrating the independence of mathematical statements from axiomatic systems, and it opened up vast new areas of research in set theory.

The broader significance of Cohen's achievement lies in its demonstration that certain fundamental questions about the nature of infinite sets cannot be resolved within the standard axiomatic framework of mathematics. This result has deep philosophical implications for the foundations of mathematics and has fueled ongoing debates about the nature of mathematical truth, the role of axioms, and the possibility of extending the standard axiom systems to settle questions like the continuum hypothesis.

Cohen's forcing technique has been extended and generalized by subsequent generations of mathematicians. Variants of forcing are now used routinely in set theory, and the method has found applications in other areas of mathematics and theoretical computer science. The community of set theorists who work with and build upon Cohen's methods remains active and productive decades after his original papers.

An account of Cohen's life and mathematical work was published in the London Mathematical Society Newsletter, which provided a detailed appreciation of his achievements and their significance for mathematics.

Among the mathematicians he influenced directly, his doctoral student Peter Sarnak became a leading figure in number theory and related fields, carrying forward the mathematical tradition that Cohen embodied. Cohen's impact extended well beyond his direct students, however, as his work inspired mathematicians around the world who never studied with him directly.

Cohen is widely remembered within the mathematical community as one of the most original and daring mathematicians of the twentieth century. His willingness to tackle problems that others considered impossible, his ability to invent entirely new mathematical methods, and the elegance and power of his results have secured his place among the most important figures in the history of mathematics.