Jack Silver
Encyclopedia
Jack Howard Silver 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=κ++ holds for some measurable cardinal
κ. With the introduction of the so called Silver machine
s he was able to give a fine structure free proof of Jensen's
covering lemma
. He is also credited with inventing Silver Indiscernibles and generalizing the notion of a Kurepa tree (called Silver's Principle). He invented 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. He earned his Ph.D.
in Mathematics at Berkeley in 1966 under Robert Vaught
. 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.
University of California, Berkeley
The University of California, Berkeley , is a teaching and research university established in 1868 and located in Berkeley, California, USA...
. He has made several deep contributions to set theory
Set theory
Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...
. In his 1975 paper On the Singular Cardinals Problem, Silver proved that if κ is singular with uncountable cofinality
Cofinality
In mathematics, especially in order theory, the cofinality cf of a partially ordered set A is the least of the cardinalities of the cofinal subsets of A....
and 2λ = λ+ for all infinite cardinals λ < κ, then 2κ = κ+. Prior to Silver's proof, many mathematicians believed that a forcing
Forcing (mathematics)
In the mathematical discipline of set theory, forcing is a technique invented by Paul Cohen for proving consistency and independence results. It was first used, in 1963, to prove the independence of the axiom of choice and the continuum hypothesis from Zermelo–Fraenkel set theory...
argument would yield that the negation of the theorem is consistent
Consistency
Consistency can refer to:* Consistency , the psychological need to be consistent with prior acts and statements* "Consistency", an 1887 speech by Mark Twain...
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
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 for a countable language has an elementary submodel of type . A model is of type if it is of cardinality α and a unary relation is represented by a subset of...
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
Measurable cardinal
- Measurable :Formally, a measurable cardinal is an uncountable cardinal number κ such that there exists a κ-additive, non-trivial, 0-1-valued measure on the power set of κ...
κ. With the introduction of the so called Silver machine
Silver machine
In set theory, Silver machines are devices used for bypassing the use of fine structure in proofs of statements holding in L. They were invented by set theorist Jack Silver as a means of proving global square holds in the constructible universe....
s he was able to give a fine structure free proof of Jensen's
Ronald Jensen
Ronald Björn Jensen is an American mathematician active in Europe, primarily known for his work in mathematical logic and set theory.-Career:...
covering lemma
Covering lemma
In mathematics, under various anti-large cardinal assumptions, one can prove the existence of the canonical inner model, called the Core Model, that is, in a sense, maximal and approximates the structure of V...
. He is also credited with inventing Silver Indiscernibles and generalizing the notion of a Kurepa tree (called Silver's Principle). He invented 0#
Zero sharp
In the mathematical discipline of set theory, 0# is the set of true formulas about indiscernibles in the Gödel constructible universe. It is often encoded as a subset of the integers , or as a subset of the hereditarily finite sets, or as a real number...
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. He earned his Ph.D.
Doctor of Philosophy
Doctor of Philosophy, abbreviated as Ph.D., PhD, D.Phil., or DPhil , in English-speaking countries, is a postgraduate academic degree awarded by universities...
in Mathematics at Berkeley in 1966 under Robert Vaught
Robert Lawson Vaught
Robert Lawson Vaught was a mathematical logician, and one of the founders of model theory.-Life:Vaught was a bit of a musical prodigy in his youth, in his case the piano. He began his university studies at Pomona College, at age 16. When World War II broke out, he enlisted US Navy which assigned...
. 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.
External links
- Jack Silver at Berkeley