Jack Howard Silver (23 April 1942 – 22 December 2016) was a set theorist and

logician Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premises ...

at the

University of California, Berkeley The University of California, Berkeley (UC Berkeley, Berkeley, Cal, or California) is a public land-grant research university in Berkeley, California. Established in 1868 as the University of California, it is the state's first land-grant u ...

. Born in Montana, he earned his

Ph.D. A Doctor of Philosophy (PhD, Ph.D., or DPhil; Latin: or ') is the most common degree at the highest academic level awarded following a course of study. PhDs are awarded for programs across the whole breadth of academic fields. Because it is a ...

in Mathematics at Berkeley in 1966 under

Robert Vaught Robert Lawson Vaught (April 4, 1926 – April 2, 2002) was a mathematical logician and one of the founders of model theory.Alfred P. Sloan Research Fellowship The Sloan Research Fellowships are awarded annually by the Alfred P. Sloan Foundation since 1955 to "provide support and recognition to early-career scientists and scholars". This program is one of the oldest of its kind in the United States. ...

from 1970 to 1972. Silver made several contributions to set theory in the areas of

large cardinal In the mathematical field of set theory, a large cardinal property is a certain kind of property of transfinite cardinal numbers. Cardinals with such properties are, as the name suggests, generally very "large" (for example, bigger than the least � ...

s and the

constructible universe In mathematics, in set theory, the constructible universe (or Gödel's constructible universe), denoted by , is a particular class of sets that can be described entirely in terms of simpler sets. is the union of the constructible hierarchy . It w ...

''L''.

Contributions

In his 1975 paper "On the Singular Cardinals Problem", Silver proved that if a

cardinal Cardinal or The Cardinal may refer to: Animals * Cardinal (bird) or Cardinalidae, a family of North and South American birds **''Cardinalis'', genus of cardinal in the family Cardinalidae **''Cardinalis cardinalis'', or northern cardinal, the ...

''κ'' is

singular Singular may refer to: * Singular, the grammatical number that denotes a unit quantity, as opposed to the plural and other forms * Singular homology * SINGULAR, an open source Computer Algebra System (CAS) * Singular or sounder, a group of boar, ...

with

uncountable In mathematics, an uncountable set (or uncountably infinite set) is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal numb ...

cofinality In mathematics, especially in order theory, the cofinality cf(''A'') of a partially ordered set ''A'' is the least of the cardinalities of the cofinal subsets of ''A''. This definition of cofinality relies on the axiom of choice, as it uses the ...

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 In classical deductive logic, a consistent theory is one that does not lead to a logical contradiction. The lack of contradiction can be defined in either semantic or syntactic terms. The semantic definition states that a theory is consistent i ...

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 In model theory, a branch of mathematical logic, Chang's conjecture, attributed to Chen Chung Chang by , states that every model of type (ω2,ω1) for a countable language has an elementary submodel of type (ω1, ω). A model is of type (α,β) if i ...

using the Silver collapse (which is a variation of the

Levy collapse In mathematics, a collapsing algebra is a type of Boolean algebra sometimes used in forcing to reduce ("collapse") the size of cardinals. The posets used to generate collapsing algebras were introduced by Azriel Lévy in 1963. The collapsing alge ...

). He proved that, assuming the consistency of a

supercompact cardinal In set theory, a supercompact cardinal is a type of large cardinal. They display a variety of reflection properties. Formal definition If ''λ'' is any ordinal, ''κ'' is ''λ''-supercompact means that there exists an elementary ...

, it is possible to construct a model where 2^''κ'' = ''κ''⁺⁺ holds for some

measurable cardinal In mathematics, a measurable cardinal is a certain kind of large cardinal number. In order to define the concept, one introduces a two-valued measure on a cardinal , or more generally on any set. For a cardinal , it can be described as a subdivisio ...

''κ''. With the introduction of the so-called

Silver machine "Silver Machine" is a 1972 song by the UK rock group Hawkwind. It was originally released as a single on 9 June 1972, reaching number three on the UK singles chart. The single was re-issued in 1976, again in 1978 reaching number 34 on the UK s ...

s he was able to give a fine structure free proof of Jensen's

covering lemma In the foundations of mathematics, a covering lemma is used to prove that the non-existence of certain large cardinals leads to the existence of a canonical inner model, called the core model, that is, in a sense, maximal and approximates the struc ...

. He is also credited with discovering

Silver indiscernible In the mathematical discipline of set theory, 0# (zero sharp, also 0#) is the set of true formulae about indiscernibles and order-indiscernibles in the Gödel constructible universe. It is often encoded as a subset of the integers (using Gödel nu ...

s and generalizing the notion of a

Kurepa tree In set theory, a Kurepa tree is a tree (''T'', <) of height ω₁, each of whose levels is at most countable, and has at ...

(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.Drake, F. R. (1974). "Set Theory: An Introduction to Large Cardinals". ''Studies in Logic and the Foundations of Mathematics'' 76, Elsevier. 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. (1971). "Some applications of model theory in set theory". ''Annals of Mathematical Logic'' 3(1), pp. 45–110. *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. (1974). "Indecomposable ultrafilters and 0#". In ''Proceedings of the Tarski Symposium'', Proceedings of Symposia in Pure Mathematics XXV, pp. 357–363. *Silver, Jack (1975). "On the singular cardinals problem". In ''Proceedings of the International Congress of Mathematicians'' 1, pp. 265–268. *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.

References

External links

Jack Silver
at Berkeley {{DEFAULTSORT:Silver, Jack 1942 births 2016 deaths 20th-century American mathematicians 21st-century American mathematicians American logicians Set theorists University of California, Berkeley alumni University of California, Berkeley faculty