Jack Howard Silver (born 23 April 1942) is a set theorist and logician at the University of California, Berkeley. He has made several deep contributions to set theory. In his 1975 paper On the Singular Cardinals Problem, Silver 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, and the thesis is the main topic of the widely used graduate textbook Set Theory: An Introduction to Large Cardinals by Frank R. Drake.[1] He earned his Ph.D. in Mathematics at Berkeley in 1966 under Robert Vaught.[2] 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.


  1. ^ Drake, F. R. (1974). Set Theory: An Introduction to Large Cardinals (Studies in Logic and the Foundations of Mathematics ; V. 76). Elsevier Science Ltd. ISBN 0-444-10535-2. 
  2. ^ Jack Silver at the Mathematics Genealogy Project