Leon Henkin
Leon Albert Henkin was an American logician, whose works played a strong role in the development of logic, particularly in the theory of types. He was an active scholar at the University of California, Berkeley, where he made great contributions as a researcher and teacher, as well as in administrative positions. At this university he directed, together with Alfred Tarski, the Group in Logic and the Methodology of Science, from which many important logicians and philosophers emerged. He had a strong sense of social commitment and was a passionate defender of his pacifist and progressive ideas. He took part in many social projects aimed at teaching mathematics, as well as projects aimed at supporting women's and minority groups to pursue careers in mathematics and related fields. A lover of dance and literature, he appreciated life in all its facets: art, culture, science and, above all, the warmth of human relations. He is remembered by his students for his great kindness, as well as for his academic and teaching excellence.
Henkin is mainly known for his completeness proofs of diverse formal systems, such as type theory and first-order logic. To prove the completeness of type theory, Henkin introduces new semantics, not equivalent to standard semantics, based on structures called general models. The change of semantics that he proposed permits to provide a complete deductive calculus for type theory and for second-order logic, amongst other logics. Henkin methods have aided in proving various model theory results, both in classical and non-classical logics. Besides logic, the other branch on which his investigations were centered was algebra; he specialized in cylindric algebras, in which he worked together with Tarski and Donald Monk. As for the philosophy of mathematics, although the works in which he explicitly approaches it are scarce, he can be considered to have a nominalist position.
Life
Childhood and first youth
Leon Albert Henkin was born on April 19, 1921, in Brooklyn, New York City, to a Jewish family that had emigrated from Russia a generation earlier. The first of the family to emigrate was Abraham Henkin, the eldest of the brothers of Leon's father. According to Leon, his father had been extremely proud of him since he was just a boy. His high expectations were evident in the name he gave him: he chose to name his son Albert after a series of articles on Einstein's theory of relativity that the New York Times published shortly before Henkin's birth. His family was sympathetic to pacifist and progressive ideas, and although he was not religious, he had deeply rooted Jewish traditions. Leon grew up surrounded by tight family ties; he was very close to his cousins, with whom he lived during his childhood in Brooklyn.Henkin studied primarily in New York City public schools; he attended Lincoln High School, where he graduated at age 16 to enter Columbia University. Both in college and high school he was a member of the chess teams; he always preferred games that involved rational thinking to games of chance. In the years of his high school education, Henkin considered becoming a math teacher and also came to desire to become a writer. Although he dedicated himself to university academic life, he never abandoned his interest in teaching elementary mathematics, to which he later actively contributed.
The first university studies
In 1937 Leon entered Columbia University as a mathematics student. It was during his time at this institution that he developed an interest in logic, which would determine the course of his academic career. His first contact with logic was through B. Russell's book Mysticism and Mathematics, which drew his interest during a visit to the library. This interest was increased and cultivated by some courses. Although the mathematics department of the University did not offer courses in Logic, Leon was one of the few mathematics students interested in that discipline and he decided to attend them. In the fall of 1938, in his second year as a Columbia University student, he participated in a first course in Logic taught by Ernest Nagel, who had contributed to the creation of the Association of Symbolic Logic two years earlier. This course brought him closer to Russell's book Principles of Mathematics, where he first encountered the axiom of choice; Russell's presentation made a strong impression on him and led him to explore the Principia Mathematica that Russell wrote with Whitehead a few years later. He was struck by the general ideas of Type Theory and by the mysterious axiom of reducibility. Both the axiom of choice and Type Theory later played an important role in his doctoral dissertation.The following year, in the fall semester of 1939, Henkin took a second course of Logic with Nagel, in which formal systems of propositional logic and first-order logic were addressed. These constituted his first experience with the mathematical treatment of deductive systems. The course did not go into metalogical results that established a relationship between semantics and syntactics, and the issue of completeness was not addressed at all. However, Nagel proposed to Henkin as an independent project the reading of the proof of the completeness of propositional logic given by Quine, which had appeared a few months before in the Journal of Symbolic Logic. This reading was highly significant for Henkin, not so much because of the content itself, but because with it he discovered that he could understand the research on logic and mathematics that was taking place at the time. According to Henkin, although he managed to follow Quine's demonstration, he did not manage to capture the idea of the proof: "I simply noted that the aim of the paper was to show that every tautology had a formal proof in the system of axioms presented, and I expended my utmost effort to check Quine's reasoning that this was so, without ever reflecting on why author and reader were making this effort. This strictly limited objective also kept me from wondering how the author thought of putting the steps of the proof together; the result was that I failed to get 'the idea of the proof', the essential ingredient needed for discovery."
Just before Henkin began his second year at Columbia, World War II broke out. This had several repercussions on his life. One of them had a positive effect on his education. Days before the war broke out, the Polish mathematician and logician Alfred Tarski had come to Harvard, at Quine's invitation, to give a series of lectures on logic. With the invasion of Poland by Germany, Tarski found it impossible to return to Poland and he had to remain in the United States. Tarski visited several cities giving lectures on logic. One of these lectures was at Columbia, and Henkin, like the rest of the logic students, attended it with great enthusiasm. In it Tarski spoke of Gödel's work on undecidable propositions in Type Theory and on the existence of decision algorithms for formal systems, a subject that Henkin found extremely stimulating.
In his last year at Columbia, in 1941, Professor F. J. Murray, knowing that Henkin was a mathematics student interested in Logic, suggested that they review together the monograph by Gödel recently published at Princeton on the consistency of the axiom of choice with the generalized continuum hypothesis. Although the meetings they had to discuss it were scarce and Leon ended up revising this monograph practically alone, the experience was considered by him as the most enriching one in his formation at Columbia. According to Henkin, then began to take form some of the ideas that became the starting-point of his doctoral dissertation.
In 1940, Henkin decided to apply for admission to a doctoral program, without having fully defined what path to follow in his research. He was accepted to three universities, from which he chose Princeton, since the renowned logician Alonzo Church was there, although at the time Henkin was unaware of his work.
Postgraduate studies
Henkin began his graduate studies at Princeton in 1941, studying under the direction of Church. The Ph.D. program he attended consisted of two years of mathematics courses, after which he was to take a "qualifying" oral examination to show he was well educated in at least three branches of mathematics; with this, he would receive an M.A. degree. He would then have another two years to write a doctoral dissertation containing original research, after which he would get the degree of Ph.D.During the first two years, he took courses in logic, analysis and general topology. In the first logic course with Church were studied several formal systems of Propositional Logic and first-order logic; some proofs of completeness and discussed part of the Löwenheim-Skolem theorems were revised, as well as a presentation of Gödel's proof on the completeness of first-order logic. In the second one, they dealt in great detail with a Second-Order system for Peano Arithmetic, as well as with the incompleteness of this axiomatic theory and the consequent incompleteness of second-order logic.
In 1941, the United States entered the Second World War, altering Henkin's plans. He had to rush his oral qualification exam, with which he obtained the degree of M. A. and left Princeton to take part in the Manhattan Project. This interruption would last four years, during which he contributed his mathematical knowledge working on radar problems and in the design of a plant to separate uranium isotopes. Most of his work required numerical analysis to solve partial differential equations. During this period, all of his work and readings on logic were completely suspended.
Once the war was over, Henkin returned to Princeton in 1946, where he was still required to write a dissertation to complete his Ph.D. studies. Upon his return, he joined the logic course that Church had begun a month earlier on Frege's theory of "sense and reference". In this course, he discovered Church's theory of types, which he found extremely interesting. The questions he asked about it eventually led him to give his proof of the completeness of the theory of types, which he was able to adapt to also give a new proof of the completeness of first-order logic. These results, as well as others that other that emerged from the same ideas, came to take part in Henkin's doctoral dissertation, which was titled "The completeness of formal systems", with which he graduated in June 1947. The dissertation itself was not published, although parts of it were rewritten and published. Many years later, Henkin wrote the article "The discovery of my completeness proofs", which contains a detailed review of the contents of his dissertation. The procedures used in it have become frequent methods of proof in various branches of logic.