Jack Howard Silver

Jack Howard Silver was born on April 23, 1942, in Missoula, Montana. After graduating with an A.B. from Montana State University (now the University of Montana) in 1961, he entered graduate school in mathematics at the University of California, Berkeley, earning his Ph.D. in 1966, at the age of 24. He joined the faculty of the Department of Mathematics in 1967. He was also a member of the Group in Logic and the Methodology of Science. He quickly rose through the ranks obtaining promotion to associate professor in 1970 and to full professor in 1975. From 1970 to 1972, he was an Alfred P. Sloan Research Fellow, and in 1974 he was an invited lecturer at the International Congress of Mathematicians. Professor Silver passed away in Oakland, California, on December 22, 2016.   

Silver was a reserved, private person, and this reserve manifested itself in his shy demeanor while lecturing. However, students found that his exposition in both lectures and written notes was the paragon of clarity and precision. At Berkeley he advised eighteen Ph.D. students in logic, mathematics, and philosophy. He retired in 2010.

Silver’s mathematical interests included set theory, model theory, and proof theory. His results were deep and influential. We summarize his results in set theory below.

Silver was a leading contributor to the explosive growth of set theory that took place immediately after Paul Cohen’s 1963 discovery of the method of forcing. In the period 1965 to 1975, Silver proved theorems and opened up lines of research that are still central to set theory today.

The standard system of axioms for mathematics is expressed through set theory and is called Zermelo-Fraenkel set theory with choice, or just “ZFC.” ZFC is known to be incomplete in significant ways. Perhaps the most famous instance of this incompleteness is the inability of ZFC to resolve Cantor's Continuum Hypothesis (CH) that every infinite set of real numbers is either countable or is equinumerous with the continuum itself. The Generalized Continuum Hypothesis (GCH) is a natural strengthening of CH that was also proposed by Cantor. It asserts that for every infinite cardinal κ, the collection of all subsets of κ has size κ⁺, where κ⁺ is the smallest cardinal number strictly greater than κ. In more compact notation, GCH asserts that for all infinite κ, 2^κ = κ⁺. As we discuss below, Silver did important work on the GCH.

Much of set theory investigates ways to remove some of this incompleteness. The most fruitful approach has been to strengthen the axiom of ZFC which asserts the existence of infinite sets. The resulting principles are often called large cardinal hypotheses. In his 1966 dissertation, written under Robert Vaught, Silver proved a number of basic results about one class of large cardinals, the partition cardinals, and then used them to extend earlier work of Gaifman and Rowbottom on Gödel’s universe L of constructible sets.

This led to the following landmark theorem: if is there is an ω₁-Erdős cardinal, then there is a unique closed, unbounded class C of indiscernibles for L such that every member of L is definable from points in C.

Leibniz’s Law, a classical philosophical principle, asserts the identity of indiscernibles, by which one would mean that two objects which have the same properties are, in fact, identical. However, when one is careful to say that a property is something expressible in a given formal language, it becomes possible to find in some structures infinite sequences (a_i) of elements which are indiscernible in the sense that for any two finite subsequences i₁ < … < i_n and j₁ < … < j_n of the same length, the sequences (a_i1, …, a_in) and (a_j1, …, a_jn) have the same properties expressible in that formal language.

C is now known as the class of Silver indiscernibles for L. The set of natural numbers coding formulae true in L of tuples from C is called 0^♯. It is a canonical, definable non-constructible set of natural numbers, and in many senses the simplest such set. C and 0^♯ are an important part of every pure set theorist’s conceptual scheme, and will remain so as long as large cardinals play a role in set theory.

In 1967, Silver developed a theory of canonical structure like C and 0^♯ that is further beyond L. Following a suggestion of Solovay, he showed that if there is a measurable cardinal, then there is a canonical minimal inner model called L[U] satisfying “there is a measurable cardinal”. 0^♯ and still more complicated sets are in L[U]; nevertheless, the model admits a theory similar to that of L. Silver took the first substantial steps in developing this theory, by showing that L[U] satisfies GCH and has well-order of its reals definable by a Δ¹₃ formula. His work on L[U] is the beginning of modern inner model theory, a subject which attempts to construct canonical minimal inner models for all the large cardinal hypotheses. Inner model theory has been the focus of much work in pure set theory since Silver’s 1967 and 1971 papers.

Silver also did important research on the Singular Cardinals Problem. A characteristic feature of infinite cardinals is that they are idempotent for multiplication. That is, if κ is an infinite cardinal, then κ.κ = κ whereas the only finite cardinals to satisfy this identity are 0 and 1.  This identity in cardinal arithmetic may be expressed by saying that if κ is an infinite cardinal, then a union of κ or fewer sets each of cardinality at most κ, itself has cardinality at most κ. The regular cardinals satisfy a near converse that the union of fewer than κ many sets each of cardinality less than κ must itself have cardinality less than κ. As the name suggests, these regular cardinals enjoy strong regularity properties. Those cardinals which are not regular are called singular. By 1970, work of Cohen, Solovay, and Easton had shown by forcing that the cardinal exponentiation function taking κ to 2^κ can have essentially any behavior at regular cardinals κ, subject to some elementary restrictions imposed by ZFC. The Singular Cardinals Problem, which asks for a description of the possible values of 2^κ at singular κ, became the focus of much research. In the early 1970s, Silver obtained the first substantial result, showing by forcing that it is consistent that for some strong limit cardinal κ of countable cofinality, 2^κ = κ⁺⁺. The proof was notable for starting with a ground model in which κ had the large cardinal property of being supercompact, and for introducing the important master condition method for preserving large cardinal properties in iterated forcing constructions. This method would have many applications in the future, especially in the formulations of, and consistency proofs for, forcing axioms.

In the early 1970s, the most common opinion among set theorists was that this use by Silver of supercompact cardinals was an ad hoc simplifying device, and that there would be a general theorem along the Cohen-Solovay-Easton lines for singular κ, provable by some complicated forcing construction that made no use of large cardinals. It was therefore a great surprise when Silver showed in 1974 that if κ is a singular cardinal of uncountable cofinality, and 2^α = α⁺ for all α < κ, then 2^κ = κ⁺. This was the first nontrivial theorem of ZFC about powers of singular cardinals to be proved after J. König’s 1905 result, and it showed that ZFC imposes much deeper restrictions than set theorists had been supposing.

Silver’s 1974 theorem opened up lines of research which are still active today. In inner model theory, it helped lead Jensen to his celebrated 1975 Covering Theorem, which has the consequence that Silver’s 1970 use of supercompact cardinals in the countable cofinality case was not completely superfluous–at least some large cardinals going beyond ZFC are needed in order to obtain models in which GCH fails at a singular strong limit cardinal. In forcing, it led to work by Gitik, Magidor, Mitchell, and Woodin that improved Silver’s 1970 result in various ways, for example by driving down its large cardinal hypothesis to what is now known to be the optimal one. And in infinitary combinatorics, it helped inspire a search for the deeper restrictions that ZFC imposes on the powers of singular cardinals, and on other structures associated to them. Shelah’s PCF theory is a major modern development in that area.

Silver also did work of importance in set-theoretic model theory. In 1971, he established the consistency of the Kurepa Conjecture and also of the Two-Cardinal Conjecture from model theory. About the same time, he used Ramsey cardinals to obtain the consistency of the model-theoretic Chang Conjecture.

In the quite different subject of descriptive set theory, Silver proved in 1970 that all analytic sets are Ramsey, and in 1974, he obtained the well-known Silver Dichotomy: if E is a Π¹₁ equivalence relation on some Polish space, then E has either countably many or perfectly many equivalence classes.

Silver became skeptical of the large cardinal hypotheses that underlay much of his previous work and searched for an inconsistency in them.  It is not clear how strongly he actually expected to find such an inconsistency – on at least one occasion, he described the search as having low probability of success, but high expected value. In any case, that pursuit did not bear fruit, and it is Silver’s earlier work which is his lasting contribution to mathematics.

Silver is survived by his partner Michiko Mori.

Paolo Mancosu
Thomas Scanlon
John Steel
2019

Photo credit: Photograph by Steven Givant - At the Tarski Symposium in 1971.