**Jack Howard Silver** (23 April 1942 – 22 December 2016^{[1]}) 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^{[2]} 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*.

## Contributions

In his 1975 paper "On the Singular Cardinals Problem", Silver proved that if a cardinal κ is singular with uncountable cofinality and 2^{λ} = λ^{+} for all infinite cardinals λ < κ, then 2^{κ} = 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.^{[3]}

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 2^{κ}=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.^{[4]}

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.

## Selected publications

- Silver, Jack H. (1980). "Counting the number of equivalence classes of Borel and coanalytic equivalence relations".
*Annals of Mathematical Logic*18(1), pp. 1–28. - Silver, Jack (1975). "On the singular cardinals problem". In
*Proceedings of the International Congress of Mathematicians*1, pp. 265–268 - Silver, Jack H. (1974). "Indecomposable ultrafilters and 0#". In
*Proceedings of the Tarski Symposium*, Proceedings of Symposia in Pure Mathematics XXV, pp. 357–363 - Silver, Jack H. (1973). "The bearing of large cardinals on constructibility". In
*Studies in Model Theory*, MAA Studies in Mathematics 8, pp. 158–182. - Silver, Jack H. (1971). "Some applications of model theory in set theory".
*Annals of Mathematical Logic*3(1), pp. 45–110.

## References

**^**Group in Logic and the Methodology of Science, "Jack Howard Silver", University of California–Berkeley**^**Jack Silver at the Mathematics Genealogy Project**^**Cummings, James (2009). "Iterated Forcing and Elementary Embeddings". In*Handbook of Set Theory*, Springer, pp. 775-883. See pp. 814ff.**^**Drake, F. R. (1974).*Set Theory: An Introduction to Large Cardinals*. Studies in Logic and the Foundations of Mathematics 76, Elsevier. ISBN 0-444-10535-2

## External links

- Jack Silver at Berkeley