Jack Howard Silver, 1942-2016

It is with great sadness that we announce that Professor Jack Howard Silver died on Thursday, December 22, 2016. Professor Silver was born April 23, 1942 in Missoula, Montana. After earning his A.B. at Montana State University (now the University of Montana) in 1961, he entered graduate school in mathematics at UC Berkeley. His thesis, Some Applications of Model Theory in Set Theory, completed in 1966, was supervised by Robert Vaught. In 1967 he joined the mathematics department at UC Berkeley where he also became a member of the Group in Logic and the Methodology of Science. He quickly rose through the ranks, obtaining promotion to associate professor in 1970 and to full professor in 1975. From 1970 to 1972 he was an Alfred P. Sloan Research Fellow. Silver retired in 2010. At UC Berkeley he advised sixteen students, three of whom were in the Group in Logic (Burgess, Ignjatovich, Zach).

His mathematical interests included set theory, model theory, and proof theory. His production was not extensive but his results were deep. Professor Silver was skeptical of the consistency of ZFC and even of third-order number theory. As Prof. Robert Solovay recently put it: “For at least the last 20 years, Jack was convinced that measurable cardinals (and indeed ZFC) was inconsistent. He strove mightily to prove this. If he had succeeded it would have been the theorem of the century (at least) in set theory.”

He will be greatly missed.

Jack’s contributions to set theory, according to Wikipedia (used under CC-BY-SA):

Silver has made several deep contributions to set theory. In his 1975 paper “On the Singular Cardinals Problem,” he proved that if κ is singular with uncountable cofinality and 2λ = λ+ for all infinite cardinals λ < κ, then 2κ = κ+. Prior to Silver’s proof, many mathematicians believed that a forcing argument would yield that the negation of the theorem is consistent with ZFC. He introduced the notion of master condition, which became an important tool in forcing proofs involving large cardinals. Silver proved the consistency of Chang’s conjecture using the Silver collapse (which is a variation of the Levy collapse). He proved that, assuming the consistency of a supercompact cardinal, it is possible to construct a model where 2κ++ holds for some measurable cardinal κ. With the introduction of the so-called Silver machines he was able to give a fine structure free proof of Jensen’s covering lemma. He is also credited with discovering Silver indiscernibles and generalizing the notion of a Kurepa tree (called Silver’s Principle). He discovered 0# in his 1966 Ph.D. thesis. Silver’s original work involving large cardinals was perhaps motivated by the goal of showing the inconsistency of an uncountable measurable cardinal; instead he was led to discover indiscernibles in L assuming a measurable cardinal exists.

[Photo ©Steven Givant. Picture taken on the occasion of the Tarski Symposium at UC Berkeley in 1971.]