Jack Howard Silver, a mathematician and philosopher, spent his career analyzing the boundaries of set theory and the structure of large cardinals. His work at the University of California, Berkeley, provided essential tools for understanding the constructible universe L, influencing how logicians approach the singular cardinals problem and the consistency of various mathematical models.
Academic Foundations and Research
Born in 1942 in Missoula, Montana, Silver completed his doctoral degree at the University of California, Berkeley, in 1966. His advisor was Robert Vaught. He joined the faculty at the same institution the following year, remaining there throughout his academic career. Between 1970 and 1972, he held an Alfred P. Sloan Research Fellowship.
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Silver identified 0# in his 1966 Ph.D. thesis, a concept later detailed in Frank R. Drake's textbook, Set Theory: An Introduction to Large Cardinals. In his 1975 paper, "On the Singular Cardinals Problem," he proved that for a singular cardinal with uncountable cofinality, the generalized continuum hypothesis implies 2κ = κ+. This proof challenged the prevailing view that forcing arguments might lead to the negation of the theorem within ZFC. To facilitate his research, he introduced the notion of a master condition, which served as a technical instrument for forcing proofs involving large cardinals.
Methodological Innovations
His research expanded to include the Silver collapse, a variant of the Levy collapse, which he utilized to prove the consistency of Chang’s conjecture. He demonstrated that, given the consistency of a supercompact cardinal, one can construct a model where 2κ = κ++ for a measurable cardinal κ. By developing Silver machines, he produced a proof of Jensen's covering lemma that functioned without fine structure. Furthermore, he discovered Silver indiscernibles and generalized the notion of a Kurepa tree, a concept known as Silver's Principle.
Fast facts
- Born: 1942, Missoula
- Died: 22 December 2016
- Education: Ph.D., University of California, Berkeley, 1966
- Field: Set theory and logic
- Key Discovery: 0# (zero sharp)
- Notable Fellowship: Alfred P. Sloan Research Fellowship (1970–1972)
Questions readers ask
What is 0#?
0# (zero sharp) is a mathematical concept discovered by Jack Silver in his 1966 doctoral thesis.
Where did Jack Silver work?
He was employed at the University of California, Berkeley, where he also earned his Ph.D.
Achievements
- Held posts at University of California, Berkeley
- Fields: set theory
