Life
Childhood and first youth
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,Henkin, Leon (1996-06). «The Discovery of My Completeness Proofs». ''Bulletin of Symbolic Logic'' 2 (2): 127-158. ISSN 1079-8986. doi:10.2307/421107. 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'sThe 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.Henkin, Leon (1962). «Are Logic and Mathematics Identical?». ''Science'' 138 (3542): 788-794. ISSN 0036-8075. 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 byPostgraduate 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 a M.A. degree. He would then have another two years to write a doctoral dissertation containing an original research, after which he would get the degree of Ph.D. 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 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 theAfter the graduation
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 Harold Kuhn. 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 theHis life in Berkeley
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 one-year stay he had in Amsterdam or the one in Israel with the Fulbright Research Grants he was awarded (in 1954 and 1979 respectively).Manzano et al (Eds.) (2014). ''The Life and Work of Leon Henkin, Essays on His Contributions,'' Springer International Publishing. doi:10.1007/978-3-319-09719-0. 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, 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.” 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 ''Center for the Study of Logic and Foundations'' in Berkeley, but with Henkin's help he was able to bring together a group of logicians, mathematicians and philosophers who formed the ''Group in Logic and the Methodology of Science'', which is still active today.See Mancosu, Paolo (2018-01). «The Origin of the Group in Logic and the Methodology of Science». ''Journal of Humanistic Mathematics'' 8 (1): 371-413. doi:10.5642/jhummath.201801.19. 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 Model Theory came as a result of the academic activity in Berkeley that took place in those years. 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 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. 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 ''Group in Logic and the Methodology of Science'', 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. On some occasions Henkin attended to his children's schools to talk to elementary school children about maths, talking to them about "''the negative numbers''", or "''how to subtract by addition''". 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. In 1991 he was granted the title of Professor Emeritus at the University of Berkeley and retired.Retirement and death
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 Oakland, where Henkin died a few years later, in November 2006. 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. One of the phrases that best captures the sentiment expressed in various testimonies of his students is that given by Douglas Hofstadter: "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".Legacy
Algebra
Henkin's work on algebra focused onCompleteness Theorems
In 1949 "''The completeness of the first order functional calculus''" was published, as well as "''Completeness in the theory of types''" in 1950. Both presented part of the results exposed in the dissertation "''The completeness of formal systems''" 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:Any set of sentences of formally consistent in the deductive system of is satisfiable by a numerable structure .This theorem is nowadays called the 'completeness theorem', since from it the following easily follows:
If is a set of sentences of and is semantic consequence of , then is deducible from .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 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
Every set of well-formed formulas of that is satisfiable in a −structure is satisfiable in an infinite numerable structure.This result is known as the "downwards" Löwenheim-Skolem theorem. One other result obtained from the completeness theorem is:
A set of well-formed formulas of has a model if and only if each finite subset of it has a model.The latter is known as the "
The Discovery of the Completeness Theorems
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 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 "''The discovery of my completeness proofs''", 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. 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 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. One of the attributes that drew Henkin's attention to Church's Theory of Types was that the -operator allowed to name many objects in the type hierarchy. As he explains in "''The discovery of my completeness proofs''", 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 as the universe of individuals, and added a constant for each the number and the successor function , so that each element in the domain was named from and repeated occurrences of . 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 aHenkin's method
Henkin's method to give the completeness proofs consists on building a certain model: it starts with a set of formulas , of which the consistency is assumed. A model is then constructed, which satisfies exactly the formulas of . 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 , 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 containing 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 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 of equivalence of them–. The first step that must be taken is to extend the language of 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 , 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 . 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 construct the functions and relations of the model we follow thoroughly what dictates: if the language contains a -relator , its interpretation in the model will be a relationship formed by all the -tuples of terms in the model's universe such that the formula that says they are related belongs to . 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 a formula stating they are. Summarizing, the demonstration in the case of a numerable language has two parts: # Extending the set to a maximally consistent and exemplified set. # 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.General models
The simple Theory of Types, with the -calculus and the standard semantics is sufficiently rich to express arithmetic categorically, from where it follows, by Gödel's incompleteness theorem, 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) Higher-Order Logics. 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 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. 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 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.Manzano, María (1993). ''Extensions of First-Order Logic''. Cambridge University Press.Towards a translation between logics
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, 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. A research that takes Henkin's ideas in this direction is that of María Manzano, one of his students, whose proposal is to useMathematical Induction
The topic ofPhilosophical position
Of the articles published by Henkin, the most philosophical is "''Some Notes on Nominalism''", 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.Teaching
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 "''Mathematical Logic''", "''Metamathematics''" or "''Cylindric Algebra''", but others extended to a great diversity of areas, including, among others, "''Fundaments of Geometry''", "''Algebra and Trigonometry''", "''Finite Mathematics''", "''Calculus with Analytic Geometry''" or "''Mathematical Concepts for Elementary School Teachers''". His students agree that his explanations were extremely clear and caught the listener's attention.Resek, Diane (2014). «Lessons from Leon». In Manzano et at. (Eds.) ''The Life and Work of Leon Henkin, Essays on His Contributions''. Springer International Publishing. . doi:10.1007/978-3-319-09719-0_11. In the words of one of his students, "''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 audiences''." 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 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: "''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 work.''" 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 together mathematicians, logicians and philosophers. Henkin aided him to carry out the project, helping him in the creation of the interdisciplinary ''Group in Logic and the Methodology of Science'', 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 "''Logic, Methodology and Philosophy of Science''". 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. 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: "''Retracing Elementary Mathematics''", "''New directions in secondary school mathematics''" or "''The roles of action and of thought in mathematics education''". 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. Henkin liked to write expository articles, for some of which he received awards such as the Chauvenet Prize (1964), for the article "''Are Logic and Mathematics Identical?'' " or the Lester R. Ford Award, for the article "''Mathematical Foundations of Mathematics''".Social Projects
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 context around us, as one of his writings records:"''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 our society quite different from education, mathematics, or science. Rivers are formed, contributing powerful currents that will produce future waves of history.'' ''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 education''."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 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:
"''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 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.''"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: "''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 society.''" 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 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 "''Mathematics Today''". 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. "''He not only believed in equality, but also worked actively to see that it was brought about''."
Henkin's Main Articles
* Henkin, L. (1949). The completeness of the first-order functional calculus. ''The Journal of Symbolic Logic'', 14(3), 159-166. * Henkin, L. (1950). Completeness in the theory of types. ''The Journal of Symbolic Logic'', 15(2), 81-91. * Henkin, L. (1953). Banishing the Rule of Substitution for Functional Variables. ''The Journal of Symbolic Logic'', 18(3), 201-208. * Henkin, L. (1953). Some interconnections between modern algebra and mathematical logic. ''Transactions of the American Mathematical Society'', 74, 410-427. * Henkin, L. (1953). Some notes on nominalism, ''The Journal of Symbolic Logic'', 18(1), 19-29. * Henkin, L. (1954) A generalization of the concept of $\omega$-consistency. ''The Journal of Symbolic Logic''. 19(3), 183-196. * Henkin, L. (1955) The nominalistic interpretation of mathematical language. ''Bulletin of the Belgian Mathematical Society''. 7, 137-141. * Henkin, L. (1955) The representation theorem for cylindrical algebras. En Skolem, Th., Hasenjaeger, G., Kreisel, G., Robinson, A. (Eds.) ''Mathematical Interpretation of Formal Systems'', pp. 85–97. * Henkin, L. (1957) A generalization of the concept of -completeness. ''The Journal of Symbolic Logic''. 22(1), 1-14. * Henkin, L. (1960). On mathematical induction. ''The American Mathematical Monthly''. 67(4), 323-338. * Henkin, L. (1961). Mathematical Induction. En ''MAA Film Manual No.1'' The Mathematical Association of America, University of Buffalo, Nueva York. * Henkin, L., Tarski, A. (1961) Cylindric algebras. En Dilworth, R.P. (Ed.) ''Lattice Theory. Proceedings of Symposia in Pure Mathematics. American Mathematical Society'', 2, 83-113. * Henkin, L. Smith, W. N., Varineau, V. J., Walsh, M. J. (1962) ''Retracing Elementary Mathematics''. Macmillan, New York. * Henkin, L. (1962). Are logic and mathematics identical?, ''Science,'' vol.138, 788-794. * Henkin, L. (1963). New directions in secondary school mathematics. En Ritchie, R. W. (Ed.) ''New Directions in Mathematics'', 1-6. Prentice Hall, New York. * Henkin, L. (1963). An Extension of the Craig-Lyndon Interpolation theorem. ''The Journal of Symbolic Logic''. 28(3), 201-216. * Henkin, L. (1963). A theory of propositional types. ''Fundamenta mathematicae''. 52, 323-344. * Henkin, L. (1971). Mathematical foundations for mathematics. ''The American Mathematical Monthly''. 78(5), 463-487. * Henkin, L. (1975). Identity as a logical primitive. ''Philosophia'' 5, 31-45. * Henkin, L. (1977). The logic of equality.''The American Mathematical Monthly''. 84(8), 597-612. * 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.), ''Changing the Culture: Mathematics Education in the Research Community'', CBMS Issues in Mathematics Education, vol. 5, pp. 3–16. American Mathematical Society in cooperation with Mathematical Association of America, Providence. * Henkin, L. (1996). The discovery of my completeness proofs, ''Bulletin of Symbolic Logic'', vol. 2(2), 127-158.Awards received
*1964 — TheSee also
*References
Further reading
* * *Henkin, Leon (1949). "The Completeness of the First-Order Functional Calculus", ''External links
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