Saharon Shelah (; , ; born July 3, 1945) is an Israeli mathematician. He is a professor of

mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

at the

Hebrew University of Jerusalem The Hebrew University of Jerusalem (HUJI; ) is an Israeli public university, public research university based in Jerusalem. Co-founded by Albert Einstein and Chaim Weizmann in July 1918, the public university officially opened on 1 April 1925. ...

and

Rutgers University Rutgers University ( ), officially Rutgers, The State University of New Jersey, is a Public university, public land-grant research university consisting of three campuses in New Jersey. Chartered in 1766, Rutgers was originally called Queen's C ...

New Jersey New Jersey is a U.S. state, state located in both the Mid-Atlantic States, Mid-Atlantic and Northeastern United States, Northeastern regions of the United States. Located at the geographic hub of the urban area, heavily urbanized Northeas ...

Biography

Shelah was born in

Jerusalem Jerusalem is a city in the Southern Levant, on a plateau in the Judaean Mountains between the Mediterranean Sea, Mediterranean and the Dead Sea. It is one of the List of oldest continuously inhabited cities, oldest cities in the world, and ...

on July 3, 1945. He is the son of the

Hebrew Hebrew (; ''ʿÎbrit'') is a Northwest Semitic languages, Northwest Semitic language within the Afroasiatic languages, Afroasiatic language family. A regional dialect of the Canaanite languages, it was natively spoken by the Israelites and ...

poet and Canaanist political activist Yonatan Ratosh. He received his PhD for his work on stable theories in 1969 from the Hebrew University. Shelah is married to Yael, and has three children. His brother, magistrate judge Hamman Shelah was murdered along with his wife and daughter by an Egyptian soldier in the Ras Burqa massacre in 1985. Shelah planned to be a scientist while at primary school, but initially was attracted to physics and biology, not mathematics. Later he found

mathematical beauty Mathematical beauty is the aesthetic pleasure derived from the abstractness, purity, simplicity, depth or orderliness of mathematics. Mathematicians may express this pleasure by describing mathematics (or, at least, some aspect of mathematics) as ...

in studying geometry: He said, "But when I reached the ninth grade I began studying geometry and my eyes opened to that beauty—a system of demonstration and theorems based on a very small number of axioms which impressed me and captivated me." At the age of 15, he decided to become a mathematician, a choice cemented after reading Abraham Halevy Fraenkel's book ''An Introduction to Mathematics''. He received a B.Sc. from

Tel Aviv University Tel Aviv University (TAU) is a Public university, public research university in Tel Aviv, Israel. With over 30,000 students, it is the largest university in the country. Located in northwest Tel Aviv, the university is the center of teaching and ...

in 1964, served in the Israel Defense Forces Army between 1964 and 1967, and obtained a M.Sc. from the Hebrew University (under the direction of Haim Gaifman) in 1967. He then worked as a teaching assistant at the Institute of Mathematics of the Hebrew University of Jerusalem while completing a Ph.D. there under the supervision of Michael Oser Rabin, on a study of stable theories. Shelah was a lecturer at Princeton University during 1969–70, and then worked as an assistant professor at the University of California, Los Angeles during 1970–71. He became a professor at Hebrew University in 1974, a position he continues to hold. He has been a visiting professor at the following universities: the

University of Wisconsin A university () is an institution of tertiary education and research which awards academic degrees in several academic disciplines. ''University'' is derived from the Latin phrase , which roughly means "community of teachers and scholars". Uni ...

(1977–78), the

University of California, Berkeley The University of California, Berkeley (UC Berkeley, Berkeley, Cal, or California), is a Public university, public Land-grant university, land-grant research university in Berkeley, California, United States. Founded in 1868 and named after t ...

(1978 and 1982), the

University of Michigan The University of Michigan (U-M, U of M, or Michigan) is a public university, public research university in Ann Arbor, Michigan, United States. Founded in 1817, it is the oldest institution of higher education in the state. The University of Mi ...

(1984–85), at

Simon Fraser University Simon Fraser University (SFU) is a Public university, public research university in British Columbia, Canada. It maintains three campuses in Greater Vancouver, respectively located in Burnaby (main campus), Surrey, British Columbia, Surrey, and ...

, Burnaby, British Columbia (1985), and

, New Jersey (1985). He has been a distinguished visiting professor at

since 1986.

Academic career

Shelah's main interests lie in

mathematical logic Mathematical logic is the study of Logic#Formal logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory (also known as computability theory). Research in mathematical logic com ...

model theory In mathematical logic, model theory is the study of the relationship between theory (mathematical logic), formal theories (a collection of Sentence (mathematical logic), sentences in a formal language expressing statements about a Structure (mat ...

in particular, and in

axiomatic set theory Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory – as a branch of mathema ...

. In

, he developed '' classification theory'', which led him to a solution of

Morley's problem In model theory, a branch of mathematical logic, the spectrum of a theory is given by the number of isomorphism classes of models in various cardinalities. More precisely, for any complete theory ''T'' in a language we write ''I''(''T'', ''κ'' ...

. In

set theory Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory – as a branch of mathema ...

, he discovered the notion of proper forcing, an important tool in iterated forcing arguments. With

PCF theory PCF theory is the name of a mathematical theory, introduced by Saharon , that deals with the cofinality of the ultraproducts of ordered sets. It gives strong upper bounds on the cardinalities of power sets of singular cardinals, and has many more a ...

, he showed that in spite of the undecidability of the most basic questions of cardinal arithmetic (such as the

continuum hypothesis In mathematics, specifically set theory, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets. It states: Or equivalently: In Zermelo–Fraenkel set theory with the axiom of choice (ZFC), this ...

), there are still highly nontrivial ZFC theorems about

cardinal Cardinal or The Cardinal most commonly refers to * Cardinalidae, a family of North and South American birds **''Cardinalis'', genus of three species in the family Cardinalidae ***Northern cardinal, ''Cardinalis cardinalis'', the common cardinal of ...

exponentiation. Shelah constructed a Jónsson group, an uncountable group for which every proper subgroup is countable. He showed that Whitehead's problem is independent of ZFC. He gave the first primitive recursive upper bound to van der Waerden's numbers V(C,N). He extended

Arrow's impossibility theorem Arrow's impossibility theorem is a key result in social choice theory showing that no ranked-choice procedure for group decision-making can satisfy the requirements of rational choice. Specifically, Arrow showed no such rule can satisfy the ind ...

on voting systems. Maryanthe Malliaris and Shelah studied Keisler's order, a construction from model theory, and in the process proved equality between two cardinal characteristics of the continuum, 𝖕 and 𝖙, resolving a problem that had been open for fifty years. This work earned them the 2017 Hausdorff Medal of the European Set Theory Society. Shelah's work has had a deep impact on model theory and set theory. The tools he developed for his classification theory have been applied to a wide number of topics and problems in model theory and have led to great advances in stability theory and its uses in algebra and algebraic geometry as shown for example by

Ehud Hrushovski Ehud Hrushovski (; born 30 September 1959) is a mathematical logician. He is a Merton Professor of Mathematical Logic at the University of Oxford and a Fellow of Merton College, Oxford. He was also Professor of Mathematics at the Hebrew Univers ...

and many others. Classification theory involves deep work developed in many dozens of papers to completely solve the spectrum problem on classification of first order theories in terms of structure and number of nonisomorphic models, a huge tour de force. Following that he has extended the work far beyond first order theories, for example for

abstract elementary class In model theory, a discipline within mathematical logic, an abstract elementary class, or AEC for short, is a class of models with a partial order similar to the relation of an elementary substructure of an elementary class in first-order model ...

es. This work also has had important applications to algebra by works of

Boris Zilber Boris Zilber (, born 1949) is a Soviet-British mathematician who works in mathematical logic, specifically model theory. He is a emeritus professor of mathematical logic at the University of Oxford. He obtained his doctorate (Candidate of Scien ...

Awards

* Three times speaker at the

International Congress of Mathematicians The International Congress of Mathematicians (ICM) is the largest conference for the topic of mathematics. It meets once every four years, hosted by the International Mathematical Union (IMU). The Fields Medals, the IMU Abacus Medal (known before ...

(1974 invited, 1983 plenary, 1986 plenary) * The first recipient of the

Erdős Prize The Anna and Lajos Erdős Prize in Mathematics is a prize given by the Israel Mathematical Union to an Israeli mathematician (in any field of mathematics and computer science), "with preference to candidates up to the age of 40." The prize was e ...

, in 1977 * The Karp Prize of the

Association for Symbolic Logic The Association for Symbolic Logic (ASL) is an international organization of specialists in mathematical logic and philosophical logic. The ASL was founded in 1936, and its first president was Curt John Ducasse. The current president of the ASL ...

in 1983 * The

Israel Prize The Israel Prize (; ''pras israél'') is an award bestowed by the State of Israel, and regarded as the state's highest cultural honor. History Prior to the Israel Prize, the most significant award in the arts was the Dizengoff Prize and in Israel ...

, for mathematics, in 1998 * The Bolyai Prize in 2000 * The

Wolf Prize in Mathematics The Wolf Prize in Mathematics is awarded almost annually by the Wolf Foundation in Israel. It is one of the six Wolf Prizes established by the Foundation and awarded since 1978; the others are in Agriculture, Chemistry, Medicine, Physics and Arts. ...

in 2001 *

The EMET Prize for Art, Science and Culture The EMET Prize for Art, Science and Culture is an Israeli prize awarded annually for excellence in academic and professional achievements that have far-reaching influence and make a significant contribution to society. Prizes are awarded in the fo ...

in 2011 * The Leroy P. Steele Prize, for Seminal Contribution to Research, in 2013 * Honorary member of the

Hungarian Academy of Sciences The Hungarian Academy of Sciences ( , MTA) is Hungary’s foremost and most prestigious learned society. Its headquarters are located along the banks of the Danube in Budapest, between Széchenyi rakpart and Akadémia utca. The Academy's primar ...

, in 2013 * Advanced grant of the

European Research Council The European Research Council (ERC) is a public body for funding of scientific and technological research conducted within the European Union (EU). Established by the European Commission in 2007, the ERC is composed of an independent Scientific ...

(2013) * Hausdorff Medal of the European Set Theory Society, joint with Maryanthe Malliaris, 2017 *

Schock Prize The Rolf Schock Prizes were established and endowed by bequest of philosopher and artist Rolf Schock (1933–1986). The prizes were first awarded in Stockholm, Sweden, in 1993 and, since 2005, are awarded every three years. It is sometimes consider ...

in Logic and Philosophy of the

Royal Swedish Academy of Sciences The Royal Swedish Academy of Sciences () is one of the Swedish Royal Academies, royal academies of Sweden. Founded on 2 June 1739, it is an independent, non-governmental scientific organization that takes special responsibility for promoting nat ...

, 2018 *

Honorary doctorate An honorary degree is an academic degree for which a university (or other degree-awarding institution) has waived all of the usual requirements. It is also known by the Latin phrases ''honoris causa'' ("for the sake of the honour") or '' ad hon ...

from the

Technische Universität Wien TU Wien () is a public research university in Vienna, Austria. The university's teaching and research are focused on engineering, computer science, and natural sciences. It currently has about 28,100 students (29% women), eight faculties, and ...

, 2019

Selected works

* ''Proper forcing'', Springer 1982 * ''Proper and improper forcing'' (2nd edition of ''Proper forcing''), Springer 1998 * ''Around classification theory of models'', Springer 1986 * ''Classification theory and the number of non-isomorphic models'', Studies in Logic and the Foundations of Mathematics, 1978
2nd edition 1990
Elsevier * ''Classification Theory for

Abstract Elementary Classes In model theory, a discipline within mathematical logic, an abstract elementary class, or AEC for short, is a class of models with a partial order similar to the relation of an elementary substructure of an elementary class in first-order logic, f ...

'', College Publications 2009 * ''Classification Theory for

'', Volume 2, College Publications 2009 * ''Cardinal Arithmetic'', Oxford University Press 1994

References

External links