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Robert Martin Solovay (born December 15, 1938) is an American mathematician specializing in set theory.


Biography

Solovay earned his Ph.D. from the University of Chicago in 1964 under the direction of Saunders Mac Lane, with a dissertation on ''A Functorial Form of the Differentiable Riemann–Roch theorem''. Solovay has spent his career at the University of California at Berkeley, where his Ph.D. students include W. Hugh Woodin and
Matthew Foreman Matthew Dean Foreman is an American mathematician at University of California, Irvine. He has made notable contributions in set theory and in ergodic theory. Biography Born in Los Alamos, New Mexico, Foreman earned his Ph.D. from the Uni ...
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Work

Solovay's theorems include: *
Solovay's theorem In the mathematical field of set theory, the Solovay model is a model constructed by in which all of the axioms of Zermelo–Fraenkel set theory (ZF) hold, exclusive of the axiom of choice, but in which all sets of real numbers are Lebesgue measur ...
showing that, if one assumes the existence of an inaccessible cardinal, then the statement "every set of real numbers is Lebesgue measurable" is consistent with Zermelo–Fraenkel set theory without the axiom of choice; * Isolating the notion of 0#; * Proving that the existence of a real-valued measurable cardinal is
equiconsistent In mathematical logic, two theories are equiconsistent if the consistency of one theory implies the consistency of the other theory, and vice versa. In this case, they are, roughly speaking, "as consistent as each other". In general, it is not ...
with the existence of a measurable cardinal; * Proving that if \lambda is a strong limit
singular cardinal 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, s ...
, greater than a
strongly compact cardinal In set theory, a branch of mathematics, a strongly compact cardinal is a certain kind of large cardinal. A cardinal κ is strongly compact if and only if every κ-complete filter can be extended to a κ-complete ultrafilter. Strongly compact cardi ...
then 2^\lambda=\lambda^+ holds; * Proving that if \kappa is an uncountable regular cardinal, and S\subseteq\kappa is a
stationary set In mathematics, specifically set theory and model theory, a stationary set is a set that is not too small in the sense that it intersects all club sets, and is analogous to a set of non-zero measure in measure theory. There are at least three close ...
, then S can be decomposed into the union of \kappa disjoint stationary sets; * With Stanley Tennenbaum, developing the method of iterated forcing and showing the consistency of
Suslin's hypothesis In mathematics, Suslin's problem is a question about totally ordered sets posed by and published posthumously. It has been shown to be independent of the standard axiomatic system of set theory known as ZFC: showed that the statement can neither ...
; * With
Donald A. Martin Donald Anthony Martin (born December 24, 1940), also known as Tony Martin, is an American set theorist and philosopher of mathematics at UCLA, where he is an emeritus professor of mathematics and philosophy. Education and career Martin rece ...
, showed the consistency of Martin's axiom with arbitrarily large
cardinality of the continuum In set theory, the cardinality of the continuum is the cardinality or "size" of the set of real numbers \mathbb R, sometimes called the continuum. It is an infinite cardinal number and is denoted by \mathfrak c (lowercase fraktur "c") or , \mathb ...
; * Outside of set theory, developing (with Volker Strassen) the Solovay–Strassen primality test, used to identify large natural numbers that are prime with high probability. This method has had implications for cryptography; *Regarding the P versus NP problem, he proved with T. P. Baker and J. Gill that relativizing arguments cannot prove \mathrm \neq \mathrm. * Proving that GL (the normal modal logic which has the instances of the schema \Box(\Box A\to A)\to\Box A as additional axioms) completely axiomatizes the logic of the provability predicate of
Peano arithmetic In mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are axioms for the natural numbers presented by the 19th century Italian mathematician Giuseppe Peano. These axioms have been used nearly u ...
; * With Alexei Kitaev, proving that a finite set of quantum gates can efficiently approximate an arbitrary unitary operator on one qubit in what is now known as
Solovay–Kitaev theorem In quantum information and computation, the Solovay–Kitaev theorem says, roughly, that if a set of single-qubit quantum gates generates a dense subset of SU(2), then that set can be used to approximate any desired quantum gate with a relatively ...
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Selected publications

* * *


See also

* Provability logic


References


External links

* * {{DEFAULTSORT:Solovay, Robert M. American logicians Members of the United States National Academy of Sciences 20th-century American mathematicians Set theorists 1938 births Living people