Leon Henkin - Knowledge (XXG)

1794:". Between 1961 and 1964, he participated in a series of courses for elementary school teachers, organized by the Committee on the Undergraduate Program in Mathematics. Also around that time, he promoted the Activities to Broaden Opportunity initiative, which sought to provide opportunities for promising students from ethnic minority groups by offering them summer courses and scholarships. He took part in the SEED (Special Elementary Education for the Disadvantaged) program, which encouraged college students to participate in elementary education, as well as in SESAME (Special Excellence in Science and Mathematics Education), the interdisciplinary doctoral program created by members of various science departments, whose purpose was to research teaching and learning of science, engineering, and mathematics. Between 1960 and 1968 he participated in a series of conferences in mathematics schools, and was involved in the development of several films produced by the National Council of Teachers of Mathematics (NCTM). These films dealt with topics such as the integer system and the rational number system. He also participated in support courses for female calculus students and convinced the mathematics department to allow graduate students to receive the same financial support for working as elementary school teachers as they did for working as assistant teachers in college. " 1097:, in whose search Henkin invested many efforts. Finally, he realized that by means of the deductive calculus he could form equivalence classes of expressions whose equality could be derived by the calculus, and form with these classes a model isomorphic to the new hierarchy of types formed by the named elements. He had been focusing on the interpretations of the formal language, when the key to solving the problem lied on the deductive system. It remained to make the universe of the objects named by the propositions a set of two elements: the truth values. This could be achieved by expanding the axioms to form a maximally consistent set. Once this was achieved, it could be proved that every consistent set of formulas 1529:, that it is incomplete. Following the idea of identifying the namable elements in the hierarchy of types, Henkin proposed a change in the interpretation of the language, accepting as types hierarchies some that previously were not admitted. If it was asked from each level of the hierarchy not that there must be all the corresponding functions, but only those that are definable, then a new semantics is obtained, and with it a new logic. The resulting semantics is known as general semantics. In it the structures that are admissible as models are those known as 'general models'. These can be used not only in Type Theory, but also, for instance, to obtain complete (and compact) 56:, 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. 1537:

an immense expressive power, in exchange for which the power of deductive calculus is lost: the latter is not enough to produce the extense set of valid formulas of this logic (with standard semantics). Changing the calculus does not solve anything, since Gödel's incompleteness theorem ensures that no deductive calculus could achieve completeness. On the contrary, by changing the semantics, that is, by changing the sets that form the universes in which the predicative variables and constants are interpreted, the logic turns out to be complete, at the cost of losing expressive capacity.

747:, had been proved by Gödel in 1929, in his own doctoral thesis. Henkin's proof is more general, more accessible than Gödel's and more easily generalizable to languages of any cardinality. It approaches completeness from a new and fruitful perspective and its greatest quality is perhaps that its proof can be easily adapted to prove the completeness of other deductive systems. Other results central to model theory are obtained as corollaries of the strong completeness of the first-order logic proved by Henkin. From it follows, for example, the following result for a first order language 931:

systems. This method has continued to be used to give proofs of completeness in both classical and non-classical logics, and it has become the usual proof of completeness for first-order logic in Logic textbooks. When Henkin published this result in 1949, completeness was not even part of the canonical subjects covered by the textbooks; some twenty years later, this theorem, along with its proof and corollaries, was part of virtually every Logic textbook. As for non-classical logics, Henkin's method can be used, among other things, to extend the completeness of

229:, 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. 346:, which is still active today. As part of this project they created an interdisciplinary postgraduate program culminating in a Ph.D. Tarski and Henkin boosted the project by organizing important congresses and conferences on Logic, following Tarski's conception of "logic as a common basis for the whole of human knowledge". The intense activity that took place in Berkeley in the 1950s and 1960s on metalogic was largely due to the activity of Tarski and Henkin, both in teaching and research. Many results of what are today crucial to 964:

theory of "sense and reference". Motivated by Frege's ideas, Church wanted to put them into practice through a formal axiomatic theory. To do so, he took the simple Theory of Types he had published a few years earlier, and supplied it with a hierarchy of types, inspired by the idea of "sense" exposed by Frege. It was in this course that Henkin became acquainted with Church's Theory of Types, which he found of great interest. He immediately made a conjecture about it, whose proof he hoped could become his doctoral dissertation.

1594:. He demonstrated that although all recursive operations can be introduced in the Peano models, this is not the case in the Induction Models. Concretely, there are Induction Models in which the exponentiation operation cannot be defined. In this article, Henkin also presents the mathematical structure that Induction models can have, which is quite simple: they can either be the standard model, that is, isomorphic to natural numbers, or in two more ways; isomorphic to cycles –which correspond to the 1790:

American Mathematical Society –which had been interested for some years in the use of films and visual material for mathematics education– to produce experimental films for this purpose, accompanied by printed manuals with appendices that would go deeper into the content and problems to be solved. Henkin participated in this project with a film on mathematical induction, whose supplementary manual was printed by the American Mathematical Society. The film was broadcast in the series "

149:. 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 (as he later expressed in a personal letter). Although he dedicated himself to university academic life, he never abandoned his interest in teaching elementary mathematics, to which he later actively contributed. 166:", 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 (these were offered by the Philosophy department), 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 285:". 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 " 20: 1648:", which he wrote in response to two articles on nominalism, one by Quine and the other jointly written by Quine and Goodman. The discussions relevant to this philosophical doctrine arise naturally in the proofs of completeness given by Henkin, as well as in his proposal for a change in semantics through general models. Both from the content of his works and from his own statements it is considered that his position was nominalist. 3659: 1093:. Going up through the hierarchy, he tried to specify which functions over those elements were nameable. The set of them was supernumerable, so there had to be some without a name, since there is only a numerable number of expressions. How could be said which elements were the nameable ones? To make each expression correspond to the element it denotated, he needed a 1698:

slow –they could continue at their own pace with the readings. However, he also considered that what was easily learned was easily forgotten, so he sought a balance between making his classes accessible and challenging for students, so that they would make the effort to learn more deeply. About his own experience as a student, he commented in an interview: "

273:. 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. 216:. 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: " 237:. 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. 1558:

logical framework is useful for comparing different logics by comparing the theories that represent them. Although Henkin does not speak of formulae translation, nor does he make explicit a Many-Sorted Language or calculus, the ideas he uses in two of his articles serve as a basis for the approach to translation: "

1322:, systematically adding to this set every formula that doesn't make the resulting set inconsistent, adding also exemplifications of the existential formulas. Thus, an infinite chain of consistent and exemplified sets is built, whose union is a maximally consistent and exemplified set; this will be the required set 1756:

Waves of history wash over our nation, stirring up our society and our institutions. Soon we see changes in the way that all of us do things, including our mathematics and our teaching. These changes form themselves into rivulets and streams that merge at various angles with those arising in parts of

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One of the aspects of his lectures in which he put special care was in finding an appropriate pace, facing the constant dilemma of how to find the optimal speed for learning. He considered it important that the students could follow the rhythm of the class, even if this meant that some would found it

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be exemplified, this is, for all existential formula there is a constant that acts as a witness of it. On the other hand, since the nature of the objects that make up the model's universe is irrelevant, no objection arises against taking as individuals the terms of the language themselves –or classes

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Grant, Henkin spent a year in Israel, in Haifa, at the Department of Science Education of the Technion University. On this occasion he also visited two universities in Egypt. In 1982 he first visited Spain. He gave conferences at several universities, including those in Barcelona, Madrid and Seville.

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Henkin was always grateful to Tarski, as it was thanks to him that he was able to settle in Berkeley. After Tarski's death in 1983, he wrote in a personal letter: “I write to tell you that Alfred Tarski, who came to Berkeley in 1942 and founded our great Center for the Study of Logic and Foundations,

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Henkin was convinced that changes could be achieved through education and, true to his idea, he committed himself to both elementary mathematics education programs and to programs whose aim was to combat exclusion. He showed a political commitment to society, defending progressive ideas. He inspired

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In addition to his courses and supervision of graduate students, Henkin's role in the scholars education was significant. Tarski had invited him to Berkeley with a clear purpose. As a mathematician, Henkin had a key role in Tarski's project to make Berkeley a center of development of logic, bringing

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as a common framework for the translation of logics. The aims of this proposal can be synthesized into two: 1) to use a single deductive calculus for all of them; and 2) to use the metaproperties of Many-Sorted Logic in order to more easily proof metaproperties of other logics. In addition, having a

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Among the other theorems of completeness given by Henkin, the most relevant is perhaps that of the completeness of Church's Theory of Types, which is the first of the completeness theorems Henkin proved. Then, he adapted the method developed in that proof to prove the completeness of other deductive

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The first two years he took courses in logic -taught by Church-, 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

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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

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Leon Albert Henkin was born on April 19, 1921, in Brooklyn, New York, 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

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Some of the social projects he formed or participated in are the following. Between 1957 and 1959 he was part of the Summer Institutes, aimed at mathematics teachers and dedicated to improving high school and college education. In 1958 the National Science Foundation authorized the committee of the

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Leon was committed to work toward equity in society. He was able to see that profes- sional mathematicians could make a difference, particularly regarding racial inequities in the United States. He was one of the first people to say that one thing holding back racial minorities and poorer people in

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In second-order logic the set of valid formulas is so large because the concept of standard structure is too restrictive and there are not enough of them to find models that refute the formulas. By relaxing the conditions we ask of the structures on which the language is interpreted, there are more

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Obtaining complete higher-order logics by the use of general semantics meets the expected balance between the expressive power of a logic and the power of its deductive calculus. In second-order logic with standard semantics it is known that quantifying over predicative variables gives the language

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Among the research trips that Henkin did throughout the years are his visits to universities in Hanover, Princeton, Colorado, as well as to several European Universities, such as Oxford (in the United Kingdom), and others in Yugoslavia, Spain, Portugal and France. In 1979, with his second Fulbright

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published shortly before Henkin's birth. His family was sympathetic with 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

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Throughout his life, Leon Henkin showed a deep commitment to society and was often called a social activist. Many of his mathematics teaching projects sought to bring minority or socially disadvantaged groups closer to mathematics and related areas. He was aware that we are part of history and the

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One of the areas in which the foundations laid by Henkin's work have proved fruitful is in the search for a logic that works as a common framework for translation between logics. This framework is intended to be used as a metalogical tool; its purpose is not to choose "one logic" above the others,

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Despite being one of his best known results, Henkin got to the proof of the completeness of first-order logic "accidentally", trying to prove a completely different result. The order of publication of his articles and even the order of presentation of the theorems in his 1947 dissertation does not

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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 the semantics and syntactics, and the issue of completeness was not addressed at

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At the end of the war, Henkin returned to Princeton to complete his doctoral studies, for which he still had to write a dissertation containing an original research. As soon as he arrived at Princeton, he attended Church's course in logic that had begun one month earlier, which dealt with Frege's

395:: "I feel very fortunate to have been his graduate student since I learned from him much more than logic. It is his humanity that conquered my heart. I always wish I am not less kind to my graduate students and no less eager to follow their professional growth after graduation than he was to me". 420:

was to attract the interest of mathematicians to logic, convinced as they were that logic could provide unifying principles to mathematics: "In fact we would go so far as to venture a prediction that through logical research there may emerge important unifying principles which will help to give

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From 1953, most of Henkin's academic activity revolved around Berkeley, where he collaborated with a solid research group in Logic. He remained there for almost all his academic life, except for some periods in which he traveled abroad with scholarships and grants of diverse institutes, like the

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Having achieved to construct this maximally consistent and exemplified set, the model described by it can be constructed. Which individuals constitute the model's universe? In the case of first-order logic without equality, the elements of the domain will be the terms of the formal language. To

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for Henkin. However, Henkin did not want to accept it, as he was sympathetic to the protests recently raised by the controversial oath of allegiance that had been required of university professors since 1950. Once the oath requirement disappeared, Henkin accepted Tarski's offer and settled in

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After he retired, Henkin continued to work on math teaching projects. From 1991, he took part on a summer courses program at Mills College intended to give talented women from across the nation education in mathematics in order to prepare them for college. Finally, Ginette and Henkin moved to

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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

445:" with which Henkin received his Ph.D. degree at Princeton in 1947. One of Henkin's best known results is that of the completeness of first-order logic, published in the above-mentioned 1949 article, which appears as the first theorem of the 1947 dissertation. It states the following: 1715:". He also collaborated in organising important meetings and conferences that promoted interdisciplinary collaboration united by logic. The outcome was that in the 1950s and 1960s there was a vibrant development of logic in Berkeley, from which many advances in Model Theory emerged. 232:

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

1197:. Henkin's idea to build a suitable model relies on obtaining a sufficiently detailed description of such model using the sentences of the formal language, and to establish which objects could be the elements of such model. If it were known, for each formula of the language of 960:", published in 1996. In it, he describes the process of the development of his dissertation. He doesn't only explain the content of his work, but he also explains the ideas that led to it, from his first logic courses in College until the end of the writing of his thesis. 1783:

In these times when our traditionally trained mathematics Ph.D.’s are finding rough going in the marketplace, it seems to me that we on the faculty should particularly seek new realms wherein mathematics training can make a substantial contribution to the basic aims of

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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

361:, he held some administrative positions; he was director of the Department of Mathematics from 1966 to 1968, and subsequently from 1983 to 1985. One of the activities to which he devoted most energy was the teaching of mathematics, on which he also did some research. 1700:

That easy way in which ideas came made it too easy to forget them. I probably learned more densely condensed material in what we called the 'seminar for babies in conjunctive topology', conducted by Arthure Stone. I learned more because it forced us to do all the

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The Great Depression and World War II formed the background of my years of study; the Cold War and the Civil Rights Movement were the backdrop against which I began my career as a research mathematician, and later began to involve myself with mathematics

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Always kind to his students and colleagues, whom he frequently invited to his home to enjoy evenings with Ginette, he is remembered as a brilliant researcher, a teacher committed to his discipline and a person who showed solidarity with his community.

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Having obtained his Ph.D. degree, Henkin spent two more years at Princeton working on post-doctoral studies. During this time, in 1948, he met Ginette Potvin, during a trip to Montreal with his sister Estelle and Princeton mathematics graduate student

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adding an infinite collection of new individual constants, and then to order the formulas of the language (which are infinite). Once this is done, the aim is to inductively construct an infinite chain of consistent and exemplified sets: we start from

1446:. If the language includes equality, the domain of the model are classes of equivalence of the terms of the language instead. The equivalence relation is established by the formulas of the maximally consistent set: two terms are equal if there is in 157:

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

4282: 306:. Ginette would become his wife in 1950, a half year after Estelle married Harold. After completing his second year of postdoctoral studies at Princeton in 1949, Leon returned to California, where he entered the mathematics department at the 424:

According to Monk, Henkin's research on cylindrical algebra can be divided into the following parts: Algebraic Theory, Algebraic Set Theory, Representation Theorems, Non-representable Algebraic Constructions and Applications to Logic.

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See Andréka, H., Van Benthem, J., Bezhaishvili, N. y Német, I. (2014). Changing a Smantics: Opportunism or Courage? In Manzano et al. (Eds.), The Life and Work of Leon Henkin, Essays on His Contributions, pp.305-324. Birkhäuser.

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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

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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

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models in which the formulas must be true to be valid and therefore the set of valid formulas is reduced; it does so in such a way that it coincides with the set produced by a deductive calculus, giving rise to completeness.

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part of his magic was his elegant expression of the mathematics, but he also worked hard to engage his audience in conjecturing and seeing the next step or in being surprised by it. He certainly captured the interest of his

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Henkin's activity as a university professor was vigorous. He taught at all levels, putting the same care and dedication into each of them. Some of the courses he taught were directly related to his research area, such as

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The same method used to prove the completeness of Church's Theory of Types could easily be adapted to give a proof of (strong) completeness of first-order logic, and of others that followed later on. The ideas on the

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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.

1730:". From 1979 onwards he put special emphasis on this facet of his research and the last doctoral theses he directed are related to the teaching of mathematics or the integration of minority groups in research. 268:

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

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See the first section of Manzano, M., Movshovitz-Hadar, N., Resek, D. (2017). Leon Henkin: A Logician's view on Mathematics Educaction. In: Pinchinat, S., Schwarzentruber, F. (eds). (2017). Special Issue:

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reflect the evolution that followed the ideas that led him to his completeness results. However, Henkin simplifies the difficult task of tracing the development and shaping of his ideas by his article "

1217:, if it should be satisfied or not by the model, we would have a comprehensive description of the model that would allow its construction. This is exactly what is being looked for: a set of sentences 257:

mathematics; with this he would receive a 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.

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which would suppress the richness provided by the diversity of them, but to provide the adequate context to contrast them, understand them and thus make the best use of the qualities of each one.

109:, 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 240:

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

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Henkin had an active role in research and teaching, but his activities at the university went far beyond that. In addition to the dedication he put in his teaching as well as and in guiding the

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Although Henkin's first encounter with teaching mathematics was as a professor, later in life he began to do research in mathematics' teaching as well. Some of his writings in this field are: "

2831:. doi: 10.1007/978-3-319-09719-0_11 In this text a careful explanation can be found about the development and ideas incorporation that led to the completeness results, following the article 2648:

in Alonso, E. (2014). Henkin's Theorem in Textbooks. En Manzano et al. (Eds.), The Life and Work of Leon Henkin, Essays on His Contributions, pp. 135-148. Springer International Publishing.

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Aware of the contributions that mathematicians could make through teaching, Henkin defended that teaching should be valued in the academy environment, as he expressed in a personal letter: "

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Tarski not only offered Henkin a job opportunity, but also provided him with a very fertile interdisciplinary collaborative environment for the development of Logic. Tarski had founded the

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See Manzano, M., Movshovitz-Hadar, N., Resek, D. (2017). Leon Henkin: A Logician's view on Mathematics Educaction. In: Pinchinat, S., Schwarzentruber, F. (eds). (2017). Special Issue:

1137:–the elements of such model are the equivalence classes of the expressions themselves–. That is, he would have managed to give a proof of the completeness of the deductive calculus. 3399:

Manzano, M., Movshovitz-Hadar, N., Resek, D. (2017). Leon Henkin: A Logician's view on Mathematics Educaction. In: Pinchinat, S., Schwarzentruber, F. (eds). (2017). Special Issue:

991:", he set out to find out which elements had names in this theory. He began by exploring the elements that were named in the two domains at the base of the type hierarchy. He took 75:

in 1929). To prove the completeness of type theory, Henkin introduces new semantics, not equivalent to standard semantics, based on structures called general models (also known as

289:", 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 " 1711:, whose successful performance was largely due to Henkin's drive. Part of this project was the creation of an interdisciplinary university program that culminated in a Ph.D. in " 372:". Around that time (about 1960), Henkin began to alternate his research work in mathematics with research work in teaching mathematics; the latter became increasingly frequent. 3717: 2414:

See Feferman, S. (2014). A Fortuitous Year with Leon Henkin. In Manzano et al. (Eds.), The Life and Work of Leon Henkin, Essays on His Contributions, pp. 135-148. Birkhäuser.

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This is the strong version of the completeness theorem, from which the weak version is obtained as a corollary. The latter states the result for the particular case in which

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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

293:", which contains a detailed review of the contents of his dissertation. The procedures used in it have become frequent methods of proofs in various branches of logic. 1523: 985: 1491: 1464: 1444: 1364: 1340: 1320: 1299: 1275: 1255: 1235: 1215: 1195: 1175: 925: 905: 2315:

is published in Mancosu, P. (2014). The Adventure of Reason: Interplay Between Philosophy of Mathematics and Mathematical Logic 1900-1940. Oxford University Press.

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for which it holds that every sentence of the language or its negation belongs to Gamma. In the case of first-order logic one more thing is required: that the set

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Henkin, L. (1995). The roles of action and of thought in mathematics education -one mathematician's passage. Fisher, N.D., Keynes, H.B., Wagreich, Ph.D. (Eds.),

1951:, CBMS Issues in Mathematics Education, vol. 5, pp. 3–16. American Mathematical Society in cooperation with Mathematical Association of America, Providence. 1947:

Henkin, L. (1995). The roles of action and of thought in mathematics education –one mathematician's passage. Fisher, N.D., Keynes, H.B., Wagreich, Ph.D. (Eds.),

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America was their low participation rates in math/science careers. He believed that there were ways of teaching and new programs that could correct this problem.

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our society quite different from education, mathematics, or science. Rivers are formed, contributing powerful currents that will produce future waves of history.

4307: 4267: 4262: 1634: 1424: 1404: 1384: 1135: 1115: 1091: 1071: 1051: 1031: 885: 857: 837: 809: 789: 765: 741: 686: 615: 575: 555: 527: 507: 487: 467: 46:, Berkeley, where he made great contributions as a researcher, teacher, as well as in administrative positions. At this university he directed, together with 2124:

Gödel, Kurt (1929). «On the completeness of the calculus of logic». In Feferman, S., Dawson, J., Kleene, S., Moore, G., Solovay R., van Heijenoort, J., ed.

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Constructing the model described by the formulae of this set using the terms of the language –or its equivalence classes– as objects of the model's universe.

3311:, CBMS Issues in Mathematics Education, vol. 5, pp. 3-16. American Mathematical Society in cooperation with Mathematical Association of America, Providence. 1768:

many of his students to become involved in mathematics education. Diane Resek, one of his students with an affinity for teaching, described him as follows:

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Henkin, L. (1963). New directions in secondary school mathematics. En Ritchie, R. W. (Ed.) New Directions in Mathematics, pp. 1-6. Prentice Hall, New York.

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elements in the hierarchy of types underlying the discovery of Henkin's completeness proofs led to the successful introduction of new semantics, called

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are true. It is known as "compactness theorem" because it corresponds to the compactness of a certain topological space, defined from semantic notions.

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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

1972:, Mathematical Association of America award to the author of an outstanding expository article on a mathematical topic by a member of the Association. 1707:

together mathematicians, logicians and philosophers. Henkin aided him to carry out the project, helping him in the creation of the interdisciplinary

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is the empty set, this is to say, the deductive calculus of first-order logic is capable of deriving all valid formulas. The weak version, known as

412:, a subject he investigated together with Alfred Tarski and Donald Monk. Cylindric Algebra provides structures that are to first-order logic what 206:

all. However, Nagel proposed to Henkin as an independent project the reading of the proof of the completeness of propositional logic given by

3544: 3476: 3442: 3421: 3387: 3233: 3212: 3191: 3110: 2935: 2850: 2828: 2807: 2753: 2732: 2711: 2653: 2606: 2567: 2533: 2512: 2459: 2419: 2259: 2189: 2161: 2112: 2091: 2035: 1582:". This was Henkin's favorite article of his own, of which he even wrote that he considered it his best expository article. In it he defined 3648: 3643: 1689:". His students agree that his explanations were extremely clear and caught the listener's attention. In the words of one of his students, " 1526: 335:

died Wednesday night, at age 82 . It was he who brought me to Berkeley in 1953, so I owe much to him personally as well as scientifically.”

4277: 2558:

Cited in p.31 of Movshovitz-Hadar, N. (2014). Tracing Back "Logic in Wonderland'' to My Work with Leon Henkin. In Manzano et al. (Eds.),

1856:

Henkin, L. (1955) The representation theorem for cylindrical algebras. En Skolem, Th., Hasenjaeger, G., Kreisel, G., Robinson, A. (Eds.)

3703: 342:

in Berkeley, but with Henkin's help he was able to bring together a group of logicians, mathematicians and philosophers who formed the

364:

On some occasions Henkin attended to his children's schools to talk to elementary school children about maths, talking to them about "

314: 43: 1733:

Henkin liked to write expository articles, for some of which he received awards such as the Chauvenet Prize (1964), for the article "

744: 4297: 2690: 2320: 2133: 307: 52: 1578:

was frequently addressed in Henkin's activities on teaching. Probably his experience in this field was the result of his article "

4302: 145:

Henkin studied primarily in New York City public schools; he attended Lincoln High School, where he graduated at age 16 to enter

2115:. doi:10.1007/978-3-319-09719-0_11 diverse texts may be found on which the authors share their experiences as Henkin's students. 3985: 1553:

A research that takes Henkin's ideas in this direction is that of María Manzano, one of his students, whose proposal is to use

815:

This result is known as the "downwards" Löwenheim-Skolem theorem. One other result obtained from the completeness theorem is:

4207: 2835:. There is also explained the completeness proof of first-order logic given by Henkin in his lectures, which was not his own. 2782:

Manzano, Maria; Martins, Manuel; Huertas, Antonia (2019-12). «Completeness in Equational Hybrid Propositional Type Theory».

3124: 3004: 2949: 4033: 3633: 3555: 3528: 2338: 1984: 1525:-calculus and the standard semantics is sufficiently rich to express arithmetic categorically, from where it follows, by 3875: 907:

there is a structure in which all of its formulas are true, then there is also a structure in which all the formulas of

1157:

Henkin's method to give the completeness proofs consists on building a certain model: it starts with a set of formulas

265:, as well as with the incompleteness of this axiomatic theory and the consequent incompleteness of second-order logic. 4287: 4215: 3682:

by John Addison, William Craig, Carolyn Kane, and Alan Schoenfeld (University of California Academic Senate memorial).

3523: 2765:

Areces, Carlos; Blackburn, Patrick; Huertas, Antonia; Manzano, María (2014-06). «Completeness in Hybrid Type Theory».

413: 391:

One of the phrases that best captures the sentiment expressed in various testimonies of his students is that given by

4173: 1980: 4049: 3897: 3745: 3289:

Henkin, L., Smith, W.N., Varineau, V.J., Walsh, M.J. (1962). Retracing Elementary Mathematics. Macmillan, New York.

2250:

Letter to María Manzano, in Manzano, María; Alonso, Enrique (2014). «Leon Henkin». In Manzano et al., María, ed.

212: 4167: 3490:"Yueh-Gin Gung and Dr. Charles Y. Hu Award for Distinguished Service | Mathematical Association of America" 1636:; or isomorphsicn to what Henkin called "spoons," which is a combination of a finite list followed by a cycle. 3669: 38:) was an American logician, whose works played a strong role in the development of logic, particularly in the 2597:

S. Feferman, quoted in Manzano, María; Alonso, Enrique (2014). «Leon Henkin». In Manzano et al., María, ed.

2422:. doi: 10.1007/978-3-319-09719-0\_11, and Henkin, Leon (1996-06). «The Discovery of My Completeness Proofs». 3467:

M. Manzano, quoted in Manzano, María; Alonso, Enrique (2014). «Leon Henkin». In Manzano et al., María, ed.

2503:

S. Feferman, cited in Manzano, María; Alonso, Enrique (2014). «Leon Henkin». En Manzano et al., María, ed.

1177:, of which the consistency is assumed. A model is then constructed, which satisfies exactly the formulas of 207: 197:

The following year, in the fall semester of 1939, Henkin took a second course of Logic with Nagel, in which

3378:

Henkin, L. 1995, quoted in Manzano, María; Alonso, Enrique (2014). «Leon Henkin». In Manzano et al.(Eds.)

4139: 4069: 1575: 187: 4191: 3518: 2744:

Parlamento, Franco (2014). «Henkin's Completeness Proof and Glivenko's Theorem». In Manzano et al., ed.

2618:

Feferman, S. In Feferman, S.: Tarski's conception of logic. Ann.Pure Appl. Log. 126, 5–13 (2004), pp.5-6.

4179: 4017: 3775: 3763: 1993:

1991 — Berkeley Citation — the highest honor/award bestowed by the University of California.

944: 194:. Both the axiom of choice and Type Theory later played an important role in his doctoral dissertation. 191: 183: 82: 4135: 4115: 4097: 3927: 3857: 936: 533:

This theorem is nowadays called the 'completeness theorem', since from it the following easily follows:

3653: 2524:

See pp.17-19 of Manzano, María; Alonso, Enrique (2014). «Leon Henkin». In Manzano et al., María, ed.

1587: 943:; and it allows one to test results of completeness in other non-classical logics, as in the cases of 619: 4257: 4252: 4231: 4201: 4091: 4087: 2007: 1976: 1591: 1426:-tuples of terms in the model's universe such that the formula that says they are related belongs to 940: 690: 241: 234: 132: 60: 4221: 4157: 3963: 887:

that is finitely satisfiable is satisfiable". This is to say, if for each of the finite subsets of

864: 282: 202: 146: 98: 77: 35: 4185: 4131: 4027: 2579:

See Henkin, L., Monk, J., Tarski, A. (1985). Cylindric Algebras Part I and Part II, North-Holland.

1597: 1583: 994: 375:

In 1991 he was granted the title of Professor Emeritus at the University of Berkeley and retired.

3969: 3945: 3939: 3739: 3160: 3152: 3101:

Letter to María Manzano, quoted in Manzano, M. (2014). April the 19th. In Manzano et al. (Eds.),

3084: 3040: 3032: 2985: 2977: 2487:

See Mancosu, Paolo (2018-01). «The Origin of the Group in Logic and the Methodology of Science».

2363: 2355: 1530: 392: 226: 136: 86: 31: 4063: 2866:

of Manzano, M., (1996). Extensions of First-Order Logic, Cambridge University Press, Cambridge.

1990:

1990 — First recipient of the Gung and Hu Award for Distinguished Service to Mathematics.

4195: 4163: 4045: 3845: 3827: 3805: 3793: 3769: 3540: 3472: 3438: 3417: 3383: 3272: 3229: 3208: 3187: 3144: 3106: 3024: 2969: 2931: 2846: 2824: 2803: 2749: 2728: 2707: 2686: 2649: 2602: 2563: 2529: 2508: 2455: 2415: 2316: 2255: 2185: 2157: 2129: 2108: 2087: 2031: 1554: 1508: 970: 421:

coherence to a mathematics which sometimes seems in danger of becoming infinitely divisible".

409: 328: 270: 106: 68: 2875:

Manzano, M., Kurucz, A., Sain, I. (1998). The little mermaid. In Martínez, C., et al. (Eds.)

1476: 1449: 1429: 1349: 1325: 1305: 1284: 1260: 1240: 1220: 1200: 1180: 1160: 910: 890: 4103: 3989: 3979: 3915: 3885: 3612: 3574: 3560: 3262: 3224:

L. Henkin, quoted in Manzano, M., Alonso, E. (2014). Leon Henkin, In Manzano et al. (Eds.),

3136: 3076: 3016: 2961: 2665:

Dawson, John W. (1993-01). «The Compactness of first-order logic: from Gödel to Lindström».

2397:

Henkin, Leon (1953). «Some interconnections between modern algebra and mathematical logic».

2347: 967:

One of the attributes that drew Henkin's attention to Church's Theory of Types was that the

262: 159: 2026:

Wells, Benjamin (2014). «Leon Henkin and a Life of Service». In María Manzano et al., ed.

1905:

Henkin, L. (1963). New directions in secondary school mathematics. En Ritchie, R. W. (Ed.)

1796:

He not only believed in equality, but also worked actively to see that it was brought about

651: 580: 190:

a few years later. He was struck by the general ideas of Type Theory and by the mysterious

19: 4005: 3973: 3851: 3811: 3726: 1969: 1886:

Lattice Theory. Proceedings of Symposia in Pure Mathematics. American Mathematical Society

1094: 867:" of first-order logic, which can also be phrased as: "Any set of well formed formulas of 417: 179: 128: 94: 2588:

See also Monk, D., Bonnet, R. (Eds.) (1989). Handbook of Boolean Algebras. North-Holland.

1828:

Henkin, L. (1953). Some interconnections between modern algebra and mathematical logic.

4225: 4127: 4109: 4081: 3957: 3933: 3891: 3757: 2723:

Novák, Vilém (2014). «From Classical to Fuzzy Type Theory». In Manzano et al., (Eds.)

1619: 1409: 1389: 1369: 1120: 1100: 1076: 1056: 1036: 1016: 870: 842: 822: 794: 774: 750: 726: 671: 600: 560: 540: 512: 492: 472: 452: 3685: 3203:

D. Resek, quoted in Resek, Diane (2014). «Lessons from Leon». In Manzano et at., ed.

2082:

Manzano, María; Alonso, Enrique (2014). «Leon Henkin». In Manzano et al., María, ed.

350:

came as a result of the academic activity in Berkeley that took place in those years.

4246: 4145: 4075: 4055: 3993: 3951: 3250: 278: 245: 198: 72: 64: 47: 3164: 3044: 2989: 2367: 2152:

Monk, Donald (2014). «Leon Henkin and Cylindric Algebras.». In Manzano et al., ed.

182:; Russell's presentation made a strong impression on him and led him to explore the 4121: 4023: 3879: 3863: 3823: 3412:

D. Resek, quoted in Resek, D. (2014). Lessons from Leon. In Manzano et al. (Eds.),

2545:

See the diverse texts compiled in Manzano, M., Sain, I., Alonso, E. (Eds.) (2014).

2336:

Leon Henkin (Sep 1949). "The completeness of the first-order functional calculus".

347: 167: 90: 3638: 1469:

Summarizing, the demonstration in the case of a numerable language has two parts:

3616: 2702:

Alonso, Enrique (2014). «Henkin's Theorem in Textbooks». En Manzano et al., ed.

2683:

Compacidad en la lógica de primer orden y su relación con el teorema de completud

987:-operator allowed to name many objects in the type hierarchy. As he explains in " 416:

is to propositional logic. One of the purposes of Henkin and Tarski in promoting

4011: 3999: 3909: 3903: 3833: 3799: 3781: 3751: 3553:

Henkin, Leon (1949). "The Completeness of the First-Order Functional Calculus",

3433:

Letter quoted in Resek, D. (2014). Lessons from Leon. In Manzano et al. (Eds.),

1821:

Henkin, L. (1953). Banishing the Rule of Substitution for Functional Variables.

1669:", but others extended to a great diversity of areas, including, among others, " 932: 303: 101:. Besides logic, the other branch on which his investigations were centered was 39: 3267: 4211: 4151: 4059: 4039: 2888:

Manzano, Maria; Alonso, Enrique (2014). «Completeness: from Gödel to Henkin».

110: 3695: 3276: 3148: 3028: 2973: 1842:

Henkin, L. (1954) A generalization of the concept of $ \omega$ -consistency.

1346:

construct the functions and relations of the model we follow thoroughly what

3869: 1849:

Henkin, L. (1955) The nominalistic interpretation of mathematical language.

1807:

Henkin, L. (1949). The completeness of the first-order functional calculus.

3660:

Obituaries: Leon Henkin, 85: professor steered minorities and women to math

3629: 2430:(2): 127-158. ISSN 1079-8986. doi:10.2307/421107. Consultado el 2020-11-10. 1912:

Henkin, L. (1963). An Extension of the Craig-Lyndon Interpolation theorem.

1881:

The Mathematical Association of America, University of Buffalo, Nueva York.

1406:, its interpretation in the model will be a relationship formed by all the 441:" in 1950. Both presented part of the results exposed in the dissertation " 2798:

See Manzano, M. (2014). Henkin on Completeness. In Manzano et al. (Eds.),

1884:

Henkin, L., Tarski, A. (1961) Cylindric algebras. En Dilworth, R.P. (Ed.)

81:). The change of semantics that he proposed permits to provide a complete 4283:

University of California, Berkeley College of Letters and Science faculty

3839: 3787: 3458:

The Mathematical Association of America, University of Buffalo, New York.

2180:

Manzano, María (2014). «Henkin on Completeness». In Manzano et al., ed.

1013:

as the universe of individuals, and added a constant for each the number

71:(the completeness of the latter, in its weak version, had been proven by 3644:

An interview with Henkin and others about their experiences at Princeton

3489: 939:; it also offers a way to obtain results that link classical logic with 3156: 3088: 3064: 3036: 2981: 2359: 102: 2819:

Manzano, M. (2014). Henkin on Completeness. In Manzano et al. (Eds.),

2290:

Quine, W. V. (1938-03). «Completeness of the propositional calculus».

3309:

Changing the Culture: Mathematics Education in the Research Community

1949:

Changing the Culture: Mathematics Education in the Research Community

174:

two years earlier. This course brought him closer to Russell's book "

3656:

by Robert Sanders, UC Berkeley News press release, November 9, 2006.

3578: 3564: 3337: 3182:

Resek, Diane (2014). «Lessons from Leon». In Manzano et at. (Eds.)

3140: 3080: 3020: 2965: 2405:(3): 410-410. ISSN 0002-9947. doi:10.1090/S0002-9947-1953-0055287-X. 2351: 1863:

Henkin, L. (1957) A generalization of the concept of -completeness.

3654:

Leon Henkin, advocate for diversity in math & science, has died

2549:, Springer International Publishing. doi: 10.1007/978-3-319-09719-0 2234:

Henkin, Leon (1996-06). «The Discovery of My Completeness Proofs».

89:, amongst other logics. Henkin methods have aided to prove various 3584: 2926:

Manzano, María (2014). «April the 19th.». In Manzano et al., ed.

18: 3251:"The Origin of the Group in Logic and the Methodology of Science" 2475:

Springer International Publishing. doi: 10.1007/978-3-319-09719-0

859:

has a model if and only if each finite subset of it has a model.

3594:

Henkin, Leon (1996). «The Discovery of My Completeness Proofs».

3569:

Henkin, Leon (1949). "Fragments of the propositional calculus",

1644:

Of the articles published by Henkin, the most philosophical is "

384:

Oakland, where Henkin died a few years later, in November 2006.

331:

Research Grants he was awarded (in 1954 and 1979 respectively).

327:

one-year stay he had in Amsterdam or the one in Israel with the

310:. There he held the position of assistant professor until 1953. 248:

was there, although at the time Henkin was unaware of his work.

3699: 2627:

Henkin, Leon (1950-06). «Completeness in the theory of types».

2380:

Henkin, Leon (1950-06). «Completeness in the theory of types».

1281:

The first step that must be taken is to extend the language of

3343:

The American Mathematical Monthly, vol. 71 (1964), no. 1, p. 3

2789:(6): 1159-1198. ISSN 0039-3215. doi:10.1007/s11225-018-9833-5. 2773:(2-3): 209-238. ISSN 0022-3611. doi:10.1007/s10992-012-9260-4. 2452:

The Life and Work of Leon Henkin, Essays on His Contributions,

1891:

Henkin, L. Smith, W. N., Varineau, V. J., Walsh, M. J. (1962)

3435: