Nationality American Fields Mathematics | Name Jack Silver Known for Silver forcing | |
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Institutions University of California, Berkeley Alma mater University of California, Berkeley Doctoral students Jeremy Avigad
Randall Dougherty
William Mitchell
Karel Prikry Education University of California, Berkeley | ||
Doctoral advisor Robert Lawson Vaught Notable students Randall Dougherty | ||
2012 silver grulla filly sire jack silver dust
Jack Howard Silver (23 April 1942 – 22 December 2016) was a set theorist and logician at the University of California, Berkeley. Born in Montana, he earned his Ph.D. in Mathematics at Berkeley in 1966 under Robert Vaught before taking a position at the same institution the following year. He held a Alfred P. Sloan Research Fellowship from 1970 to 1972. Silver made several contributions to set theory in the areas of large cardinals and the constructible universe L.
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Contributions
In his 1975 paper "On the Singular Cardinals Problem", Silver proved that if a cardinal 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 a 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=κ++ 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# ("zero sharp") in his 1966 Ph.D. thesis, discussed in the graduate textbook Set Theory: An Introduction to Large Cardinals by Frank R. Drake.
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.