Menachem Magidor
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Menachem Magidor (; born January 24, 1946) is an Israeli
mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...
who specializes 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 ...
, in particular
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 served as president of 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. ...
, was president of the Association for Symbolic Logic from 1996 to 1998 and as president of the Division for Logic, Methodology and Philosophy of Science and Technology of the International Union for History and Philosophy of Science (DLMPST/IUHPS) from 2016 to 2019. In 2016 he was elected an honorary foreign member of the American Academy of Arts and Sciences. In 2018 he received the Solomon Bublick Award.


Biography

Menachem Magidor was born in
Petah Tikva Petah Tikva (, ), also spelt Petah Tiqwa and known informally as Em HaMoshavot (), is a city in the Central District (Israel), Central District of Israel, east of Tel Aviv. It was founded in 1878, mainly by Haredi Judaism, Haredi Jews of the Old Y ...
, Israel. He received his Ph.D. in 1973 from 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. ...
. His thesis, ''On Super Compact Cardinals'', was written under the supervision of
Azriel Lévy Azriel Lévy (; born c. 1934) is an Israeli mathematician, logician, and a professor emeritus at the Hebrew University of Jerusalem. Biography Lévy obtained his Ph.D. at the Hebrew University of Jerusalem in 1958, under the supervision of Abr ...
. He served as president of 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. ...
from 1997 to 2009, following Hanoch Gutfreund and succeeded by Menachem Ben-Sasson. The Oxford philosopher Ofra Magidor is his daughter.


Mathematical theories

Magidor obtained several important consistency results on powers of
singular cardinal Singular may refer to: * Singular, the grammatical number that denotes a unit quantity, as opposed to the plural and other forms * Singular or sounder, a group of boar, see List of animal names * Singular (band), a Thai jazz pop duo *'' Singular ...
s substantially developing the method of forcing. He generalized the Prikry forcing in order to change the
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''. Formally, :\operatorname(A) = \inf \ This definition of cofinality relies o ...
of a
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 ...
to a predetermined
regular cardinal In set theory, a regular cardinal is a cardinal number that is equal to its own cofinality. More explicitly, this means that \kappa is a regular cardinal if and only if every unbounded subset C \subseteq \kappa has cardinality \kappa. Infinite ...
. He proved that the least
strongly compact cardinal In set theory, a strongly compact cardinal is a certain kind of large cardinal. An uncountable cardinal κ is strongly compact if and only if every κ-complete filter can be extended to a κ-complete ultrafilter. Strongly compact cardinals were ...
can be equal to the least
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 (mathematics), measure on a cardinal ''κ'', or more generally on any set. For a cardinal ''κ'', ...
or to the least supercompact cardinal (but not at the same time). Assuming consistency of huge cardinals he constructed models (1977) of set theory with first examples of nonregular
ultrafilter In the Mathematics, mathematical field of order theory, an ultrafilter on a given partially ordered set (or "poset") P is a certain subset of P, namely a Maximal element, maximal Filter (mathematics), filter on P; that is, a proper filter on P th ...
s over very small cardinals (related to the famous Guilmann– Keisler problem concerning existence of nonregular ultrafilters), even with the example of jumping cardinality of
ultrapower The ultraproduct is a mathematical construction that appears mainly in abstract algebra and mathematical logic, in particular in model theory and set theory. An ultraproduct is a quotient of the direct product of a family of structures. All fact ...
s. He proved consistent that \aleph_\omega is strong limit, but 2^=\aleph_. He even strengthened the condition that \aleph_\omega is strong limit to that
generalised 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 ...
holds below \aleph_\omega. This constituted a negative solution to the singular cardinals hypothesis. Both proofs used the consistency of very large cardinals. Magidor,
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 Univer ...
, and
Saharon Shelah Saharon Shelah (; , ; born July 3, 1945) is an Israeli mathematician. He is a professor of mathematics at the Hebrew University of Jerusalem and Rutgers University in New Jersey. Biography Shelah was born in Jerusalem on July 3, 1945. He is th ...
formulated and proved the consistency of Martin's maximum, a provably maximal form of Martin's axiom. Magidor also gave a simple proof of the Jensen and the Dodd-Jensen
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 ...
s. He proved that if 0# does not exist then every primitive recursive closed set of ordinals is the union of countably many sets in L.


Selected publications

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References

{{DEFAULTSORT:Magidor, Menachem 1946 births Living people Academic staff of the Hebrew University of Jerusalem 20th-century Israeli mathematicians 21st-century Israeli mathematicians Set theorists Presidents of universities in Israel