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Stanford Report, June 4, 2003

Feferman awarded Rolf Schock Prize in logic and philosophy

Solomon Feferman, the Patrick Suppes Family Professor in the School of Humanities and Sciences at Stanford, was awarded Sweden's Rolf Schock Prize in logic and philosophy on May 15. The Schock prizes are awarded every two years in the fields of logic and philosophy, mathematics, the visual arts and music. The prize citation lauded Feferman, who has made important contributions in key areas of logic, "for his works on the arithmetization of metamathematics, transfinite progressions of theories and predicativity."

Solomon Feferman is the recipient of the Rolf Schock Prize for his accomplishments in logic and philosophy. Photo: Anita Feferman

"Though the Schock Prize is of relatively recent vintage, it is a major honor; indeed, perhaps the highest prize currently awarded for philosophy," said Kenneth A. Taylor, professor and chair of the Department of Philosophy. "The [short] list of past winners of the logic and philosophy prize reads like an honor roll of the greatest philosophers and logicians of the mid to late 20th century. They include Willard Van Orman Quine, Dana Scott, Michael Dummett, John Rawls and Saul Kripke. Sol Feferman richly deserves to be counted as one among the true giants of 20th-century logic and philosophy."

Arithmetization is a method that was introduced by Austrian logician Kurt Gödel to prove his famous incompleteness theorems for axiomatic systems of mathematics. It is now a standard technique in logic, and it was Feferman who first systematically established its scope and limitations. Gödel's first incompleteness theorem states that any consistent and sufficiently expressive axiomatic theory cannot prove all true statements about the whole numbers. The method of arithmetization uses an assignment of "Gödel numbers" to sequences of symbols to translate logical properties of a theory of numbers into its own language. In that way, the consistency of the theory can be re-expressed as a statement about numbers. Feferman's work on arithmetization led him to establish a sharpened and very general formulation of Gödel's second incompleteness theorem, which says that the consistency of a theory -- if that theory is indeed consistent -- cannot be proved within the theory itself.

"In a progression of theories based on an initial theory of numbers, one attempts to overcome incompleteness by successively adding consistency statements or stronger principles reflecting on the systems in question," Feferman explained in an e-mail interview. "Not only can this procedure be repeated any finite number of times, it can also be iterated into the transfinite." Feferman investigated how far into the transfinite such progressions could be carried, and established fundamental completeness and incompleteness results for them, depending on the way that they are generated; these results made essential use of his work on arithmetization.

Feferman then turned to the study of predicativity, in which certain definitions ubiquitous in mathematics, such as that for the least upper bound of a set of real numbers, are restricted to avoid apparent circularities. Using a formulation of this idea in terms of a transfinite progression of theories, Feferman determined the exact limit of predicativity as measured by a certain transfinite ordinal number. In later work he greatly enlarged the understanding of the scope of predicativity, in part by demonstrating that practically all the mathematical principles needed for scientific applications can be carried out entirely according to predicative restrictions.

Feferman obtained his doctorate in mathematics from the University of California-Berkeley in 1957. Since 1956, he has taught at Stanford in the departments of Mathematics and Philosophy. He served as chair of the Mathematics Department from 1985 to 1992 and as president of the Association for Symbolic Logic from 1980 to 1982. He is a fellow of the American Academy of Arts and Sciences and editor-in-chief of Kurt Gödel's Collected Works, Volumes I-V. He and his wife, Anita Burdman Feferman, are currently working on a biography of the logician Alfred Tarski, who was Feferman's doctoral adviser at UC-Berkeley.