Azriel Lévy (
Hebrew
Hebrew (; ; ) is a Northwest Semitic language of the Afroasiatic language family. Historically, it is one of the spoken languages of the Israelites and their longest-surviving descendants, the Jews and Samaritans. It was largely preserved ...
: עזריאל לוי; born c. 1934) is an Israeli
mathematician,
logician, and a
professor emeritus at the
Hebrew University of Jerusalem
The Hebrew University of Jerusalem (HUJI; he, הַאוּנִיבֶרְסִיטָה הַעִבְרִית בִּירוּשָׁלַיִם) is a public university, public research university based in Jerusalem, Israel. Co-founded by Albert Einstein ...
.
Biography
Lévy obtained his Ph.D. at the
Hebrew University of Jerusalem
The Hebrew University of Jerusalem (HUJI; he, הַאוּנִיבֶרְסִיטָה הַעִבְרִית בִּירוּשָׁלַיִם) is a public university, public research university based in Jerusalem, Israel. Co-founded by Albert Einstein ...
in 1958, under the supervision of
Abraham Fraenkel
Abraham Fraenkel ( he, אברהם הלוי (אדולף) פרנקל; February 17, 1891 – October 15, 1965) was a German-born Israeli mathematician. He was an early Zionist and the first Dean of Mathematics at the Hebrew University of Jerusalem. ...
and
Abraham Robinson
Abraham Robinson (born Robinsohn; October 6, 1918 – April 11, 1974) was a mathematician who is most widely known for development of nonstandard analysis, a mathematically rigorous system whereby infinitesimal and infinite numbers were reincorpo ...
. Using
Cohen's method of
forcing, he proved several results on the
consistency of various statements contradicting the
axiom of choice
In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection ...
. For example, with
J. D. Halpern he proved that the
Boolean prime ideal theorem
In mathematics, the Boolean prime ideal theorem states that ideals in a Boolean algebra can be extended to prime ideals. A variation of this statement for filters on sets is known as the ultrafilter lemma. Other theorems are obtained by cons ...
does not imply the axiom of choice. He discovered the models ''L''
'x''used in
inner model theory In set theory, inner model theory is the study of certain models of ZFC or some fragment or strengthening thereof. Ordinarily these models are transitive subsets or subclasses of the von Neumann universe ''V'', or sometimes of a generic extensio ...
. He also introduced the notions of
Lévy hierarchy of the formulas of
set theory,
Levy collapse In mathematics, a collapsing algebra is a type of Boolean algebra sometimes used in forcing to reduce ("collapse") the size of cardinals. The posets used to generate collapsing algebras were introduced by Azriel Lévy in 1963.
The collapsing alge ...
and the
Feferman–Levy model.
His students include
Dov Gabbay
Dov M. Gabbay (; born October 23, 1945) is an Israeli logician. He is Augustus De Morgan Professor Emeritus of Logic at the Group of Logic, Language and Computation, Department of Computer Science, King's College London.
Work
Gabbay has author ...
,
Moti Gitik
Moti Gitik () is a mathematician, working in set theory, who is professor at the Tel-Aviv University. He was an invited speaker at the 2002 International Congresses of Mathematicians, and became a fellow of the American Mathematical Society in ...
, and
Menachem Magidor
Menachem Magidor (Hebrew: מנחם מגידור; born January 24, 1946) is an Israeli mathematician who specializes in mathematical logic, in particular set theory. He served as president of the Hebrew University of Jerusalem, was president of t ...
.
Selected works
*
* A. Lévy: ''A hierarchy of formulas in set theory'', Memoirs of the American Mathematical Society, 57, 1965.
* J. D. Halpern, A. Lévy: The Boolean prime ideal theorem does not imply the axiom of choice,
Axiomatic Set Theory
Set theory is the branch of mathematical logic that studies 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 mathematics, is mostly concer ...
, Symposia Pure Math., 1971, 83–134.
* A. Lévy: ''Basic Set Theory'', Springer-Verlag, Berlin, 1979, 391 pages;
reprinted by Dover Publications, 2003.
Notes
References
*
External links
*
1934 births
Living people
Israeli mathematicians
Set theorists
Hebrew University of Jerusalem faculty
Hebrew University of Jerusalem alumni
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