# A tribute to Solomon Feferman (1928–2016)

R. Lanier Anderson

## Solomon Feferman

**Professor of Mathematics and Philosophy, Em.
Patrick Suppes Professor of Humanities and Sciences, Em.**

With great sadness, the Philosophy Department notes the passing of our friend and colleague, Solomon Feferman, who died on Tuesday, July 26, 2016 at his Stanford home after an illness of about three months. He was 87.

Feferman, the Patrick Suppes Family Professor emeritus, was one of the leading mathematical logicians of the twentieth century, and he served as the central figure in the influential logic group at Stanford across a long career in the Philosophy and Mathematics departments. He arrived at Stanford in 1956 while in the last stages of completing his dissertation under Alfred Tarski at UC Berkeley. He held a B.S. in Mathematics from Cal Tech (1948) and received the Berkeley Ph.D., also in Mathematics, in 1957. He was the 2003 winner of the Rolf Schock Prize in Logic and Philosophy and a Fellow of the American Academy of Arts and Sciences. In addition to teaching at Stanford for 48 years as a full time faculty member (and for years afterward as emeritus), he served the University as Chair of the Mathematics Department from 1985 to 1992, and at various times he was a visiting fellow or professor at Princeton’s Institute for Advanced Study, MIT, Paris, Amsterdam, Oxford, Rome, and Berkeley. He was the President of the Association for Symbolic Logic from 1980-1982.

Feferman’s field-shaping body of work included major contributions to all of the main domains of mathematical logic (“the four pillars”): proof theory, set theory, recursion theory, and model theory. In the dissertation work, he obtained important results that sharpened and considerably extended the method of arithmetization of metamathematics that Kurt Gödel introduced in the 1930s to show the incompleteness of arithmetic. This launched Feferman on a long term project of exploring the limits of the incompleteness results and the extent to which they could be overcome. Pursuing this general research program led to Feferman’s important work in the 1960s and afterward about transfinite progressions of theories and about predicative analysis—including results which have served as the basis for much subsequent progress in proof theory. In the early 1960s, Feferman was also a constant sounding-board for his Mathematics colleague Paul Cohen while Cohen was working out his novel method of forcing and generic sets, which he used to solve the long outstanding problem about the independence of the Axiom of Choice and the Continuum Hypothesis. Feferman was then one of the first to build on those methods to achieve further results in set theory, including a negative result concerning a conjecture from Hilbert’s 1900 list of outstanding mathematical problems (Feferman showed that it is consistent with ZFC set theory together with the Generalized Continuum Hypothesis that there is no formula of set theory that can define a well-ordering of the continuum). By building on the Cohen methods in this way, Feferman helped to pioneer what became something of an industry in late twentieth century set theory. Later on, Feferman also made major contributions to the technical theory of truth, developing what has come to be known as the Kripke-Feferman (KF) theory of truth. Feferman showed that KF is proof-theoretically equivalent to the theory of ramified analysis up to certain limits, and he also devised a strengthening of KF that is as strong as full predicative analysis, or ramified analysis up to the Feferman-Schütte ordinal, thereby connecting the truth work back to his research program on transfinite progressions and predicative analysis.

Feferman’s style as a logician and thinker combined the best features of his two most important mentors in the field, Alfred Tarski and Georg Kreisel. From Tarski, Feferman acquired a lifelong devotion to certain mathematical virtues in logic—clarity and precision in the statement of results, full rigor in spelling out of proofs, and so on. From Kreisel, who was Feferman’s colleague in the logic group at Stanford from 1958 until 1985, Feferman came to appreciate the importance of articulating a philosophical framework for his logical work, and carefully explaining the significance of results up front.

Feferman was always a terrific and generous intellectual collaborator, and he achieved a number of important results through joint work, including landmark results in model theory with Bob Vaught, further applications of the method of forcing in set theory with Azriel Lévy, and the investigations of progressions with Clifford Spector. In one sense, the culmination of Feferman’s collaborative work in logic came with his major efforts as the lead general editor of the five volume Kurt Gödel’s Collected Works—a truly long-term project that spanned a period from 1986, when Volume I appeared, through the publication of the final volumes of correspondence in 2003.

Patras Logic Symposium 1980. Taken from Sol Feferman's website.

From left to right: Peter Aczel, Sol Feferman, Robin Gandy, John and Margaret Shepherdson, Gert Müller

Feferman was also a gifted teacher and a dynamic leader of the logic group at Stanford. He was the constant presence anchoring an ever shifting cast of brilliant colleagues in logic, who made Stanford a world leading center for research in mathematical logic throughout the post-war period. In addition to Kreisel, these colleagues included John Myhill, Dana Scott, Harvey Friedman, Bill Tait, Dagfinn Føllesdal, and Jaakko Hintikka, during the first half of Feferman’s Stanford tenure, and figures like Jon Barwise, Grigori Mints, John Etchemendy, and Johan van Bentham later on. Feferman trained a number of graduate students at Stanford who have gone on to shape developments in the field, including Barwise himself, as well as Paolo Mancosu, Wilfried Sieg, Carolyn Talcott, Jeffery Zucker, and many others. In Feferman’s early time at the University, he taught a wide program of courses in Mathematics, as well as all levels of logic for the Philosophy Department. He eventually developed a yearlong sequence in Metamathematics, notes from which survive in the Stanford Mathematics Library, and that course was later replaced (as the subject continued its dramatic twentieth century expansion) by a rotating set of courses covering the four pillars of mathematical logic. In later years, Feferman expanded his teaching program to include theories of truth and the philosophy of mathematics, and he taught for the Philosophy Department in an emeritus capacity until 2015.

Feferman remained active and continued to branch out through the last year of his life and into his final illness. During the last year, for example, he took up an entirely new project to extend some of his ideas about model theory into applications in systems biology. His main efforts, however, focused on three book projects. Together with Gerhard Jäger and Thomas Strahm, he was developing a treatment of his distinctive approach to the foundations of mathematics—what he called Explicit Mathematics. The second project was a collection of essays by leading scholars in the areas of Feferman’s contributions, for which he was to write comments and a reflective intellectual biography; that volume (Feferman on Foundations, in the Springer series Outstanding Contributions to Logic) will now necessarily remain incomplete, but it will be seen through the press under the editorship of Jäger and Wilfried Sieg. Finally, Feferman had also contracted with Oxford for a second volume collecting previously published papers, which was to have complemented In the Light of Logic (1998). In late April, he gave a marvelous lecture at a Workshop in honor of Charles Parsons at Columbia, which reassessed his connections to Parsons’ work. That paper will have been the last contribution drafted by Feferman’s hand, and it is to appear in The Journal of Philosophy.

Aside from his technical work in mathematical logic, Feferman also made important contributions to the history of logic. These include his editorial work on the Gödel project and on the papers of Julia Robinson, and also a major co-authored biography of Tarski (Alfred Tarski: Life and Logic, 2004), which was joint work with his wife, Anita Burdman Feferman (1927-2015). The Tarski biography not only explains the impact and significance of Tarski’s technical work, but provides a compelling and deeply insightful picture of the internal workings of the remarkable logic group Tarski built up at Berkeley and how its scientific developments were related to the broader lives of the members.

Solomon and Anita Feferman were dedicated collaborators not only in writing and scholarship, but in life. They were as tight a couple as one is likely to meet, and Anita’s boundless curiosity and deep engagement made her an indelible presence in the life of the Philosophy Department, just as Sol was. The two of them brought a special spirit to departmental gatherings—always on the lookout to engage younger members of our community, to learn new things, and to reconnect with colleagues and friends of long standing. They shared a wide-ranging set of interests, from art and music to cooking and travel, and they were always eager to share their passions and to forge new connections among their broad circle of friends. Many among us have important friendships with others that we owe to Sol and Anita. Sol’s gentle manner, his generosity of spirit, and his quiet, thoughtful wisdom will be sorely missed in the counsels of our Department, and we will all feel the loss of his towering intellect and relentless curiosity. Sol was preceded in death by Anita and by his older daughter Rachel; he is survived by his younger daughter Julie and her two daughters, Isabel and Gracie. They have the deepest sympathy of the members of the Philosophy Department, and we share their grief.

**Solomon Feferman (1928-2016)**

*"Sol’s gentle manner, his generosity of spirit, and his quiet, thoughtful wisdom will be sorely missed in the counsels of our Department,*

and we will all feel the loss of his towering intellect and relentless curiosity."

and we will all feel the loss of his towering intellect and relentless curiosity."