**One of the fathers of modern logic, German-born mathematician Abraham Fraenkel (1891-1965) first became widely known for his work on set theory. Long fascinated by the pioneering work in set theory of fellow German Ernst Zermelo (1871-1953), Fraenkel launched research to put set theory into an axiomatic setting that improved the definitions of Zermelo's theory and proposed its own system of axioms. Within that system, Fraenkel proved the independence of the axiom of choice. The Zermelo-Fraenkel axioms of set theory, known collectively as ZF, are the standard axioms of axiomatic set theory on which, together with the axiom of choice, all of ordinary mathematics is based. When the axiom of choice is included, the resulting system is known as ZFC.**

## Studied at Several Universities

Abraham Adolf Fraenkel was born on February 17, 1891, in Munich, Germany. The son of Sigmund and Charlotte (Neuberger) Fraenkel, he was strongly influenced by his orthodox Jewish heritage. B.H. Auerbach-Halberstadt, Fraenkel's great-grandfather, had been widely known for his rabbinical teachings. As a child, Fraenkel was enrolled in Hebrew school and was reading Hebrew by the time he was five. Raised in a family that set a high priority on education, Fraenkel advanced rapidly in his general studies and, like most German students of that era, studied at a number of universities. He began his higher studies at the University of Munich in his hometown and studied subsequently at the German universities of Marburg, Berlin, and Breslau. In 1914, at the age of 23, Fraenkel received his doctoral degree in mathematics from the University of Breslau.

World War I broke out in August 1914, shortly after Fraenkel had completed his studies at Breslau. For the next two years, he served in the German military as a sergeant in the medical corps. He also worked briefly for the German army's meteorological service. In 1916 Fraenkel accepted a position at the University of Marburg as an unsalaried lecturer, or privatdocent. It was at Marburg that Fraenkel began his most important research in mathematical theory. On March 28, 1920, he married Malkah Wilhemina Prins. The couple eventually had four children.

## Focused on Set Theory

Fraenkel's earliest research was on the p-adic numbers first described by Kurt Hensel in the late nineteenth century and on the theory of rings. Before long, however, he became deeply involved in the study of set theory, specifically the work of Ernst Zermelo, who in the early years of the twentieth century had published his controversial and innovative views on the subject. Zermelo had postulated that from any set of numbers, a single element could be selected and that definite properties of that element could be determined. This was known as the axiom of choice, but Zermelo offered no real proof for his theory, suggesting that the study of mathematics could only progress if certain axioms were simply accepted without question. For many mathematicians, Zermelo's lack of proof was unacceptable. Some, including French mathematician Jacques Hadamard, reluctantly agreed to accept Zermelo's theory until a better way could be found, while others, including Jules-Henri PoincarĂ©, adamantly opposed acceptance of Zermelo's theory.

Without either accepting or rejecting Zermelo's theory outright, Fraenkel set about to find ways to put Zermelo's work on a firmer foundation. In the case of finite sets of numbers, Fraenkel found, Zermelo's theory already worked quite well. However, for infinite sets, Zermelo's assumptions were more questionable. Fraenkel eventually substituted a notion of function for Zermelo's idea of determining a definite property of a number in a set. In so doing, he significantly clarified Zermelo's set theory and also rid it of its dependence on the axiom of choice, which had clearly been one of the most controversial elements of Zermelo's work.

Just as Fraenkel's research built on theories advanced earlier by Zermelo, others' refinements to the work of Zermelo and Fraenkel have buttressed their theories and advanced the mathematical community's understanding of set theory. Fraenkel's system of axioms was modified by Norwegian mathematician Thoralf Skolem in 1922 to create what is known today as the ZFS system, named for Zermelo, Fraenkel, and Skolem. Within the ZFS system, it is harder to prove the independence of the axiom of choice, a goal that was not achieved until the work of American Paul Joseph Cohen in the 1960s. Cohen used a technique called "forcing" to prove the independence in set theory of the axiom of choice and the generalized continuum hypothesis.

## Published Set Theory Findings

Fraenkel published his conclusions on set theory in two separate works—a popular introductory textbook published in 1919 and a 1922 research article determining the independence of the axiom of choice. The conclusions in the latter work were later included as part of the proof for a newly coined term, Ur-elements—infinite and distinct pairs of objects that do not in themselves define a set. A number of prominent mathematicians of the period questioned the validity of Ur-elements, but only three years later German physicist Wolfgang Pauli used them in his proof of the exclusion principle.

In 1922 Fraenkel was promoted to assistant professor of mathematics at the University of Marburg. His earlier work on set theory had propelled him to the forefront of set theory research, and over the next few years he published a number of articles on the subject while he continued to teach. In 1928 Fraenkel was offered a full professorship at the University of Kiel. He accepted but only a year later took a leave of absence to become a visiting professor at Jerusalem's Hebrew University. For the next two years he taught at Hebrew University, leaving in 1931 after a disagreement with the school's administration.

## Germany in Turmoil

Fraenkel's return to Germany proved to be a bittersweet occasion. His native country was in economic disarray, suffering through the effects of the worldwide economic depression and the brutal conditions imposed by the Treaty of Versailles that had ended World War I. The economic pressures on the German people had given rise to increasing intolerance, most notably a disturbing wave of anti-Semitism. For the next two years, Fraenkel resumed his teaching duties at Kiel, keeping a wary eye on the increasingly unsettled political situation in Germany. In January 1933 Adolf Hitler, leader of the National Socialist German Workers' Party, better known as Nazis, became Germany's chancellor. Fraenkel and his family left the country a month later, moving first to Amsterdam in the neighboring Netherlands.

Fraenkel and his family spent only two months in Amsterdam, closely monitoring the situation in their native Germany while there. Convinced that there would not be a quick turnaround under the Nazi regime, Fraenkel drafted a letter of resignation to the University of Kiel in April 1933 and returned to Jerusalem to teach once again at Hebrew University. Despite his earlier disagreement with the university's administration, he was warmly welcomed back to the school's faculty.

## Focus of Research Changed

Following his exile from Germany, Fraenkel changed the focus of his research. Although he continued to publish texts on set theory for the remainder of his career, Fraenkel began to concentrate his studies on the evolution of modern logic and the contributions made by Jewish mathematicians and scientists in their respective fields. Fraenkel had written a number of books about the history of mathematics. In 1920 he had published an overview of the work of Carl Friedrich Gauss, who in his doctoral dissertation had proved the fundamental theorem of algebra. As early as 1930 he had begun the work of chronicling the accomplishments of Jewish mathematicians with his biography of Georg Cantor, who was half-Jewish. Cantor at that time was of greater interest to Fraenkel for the nature of his research into set theory than for his ethnic background. However, once he had resumed teaching at Hebrew University in 1933, he began a much wider study into the work of Jewish scientists and mathematicians. In 1960 Fraenkel published Jewish Mathematics and Astronomy.

In his research into the origins of modern logic, Fraenkel looked closely at natural numbers, describing them in terms of modern concepts of logic and reasoning. Although his research underscored the need for continuity in consideration of the number line, Fraenkel also expressed interest in opposing points of view. During this period, Fraenkel had a conversation with physicist Albert Einstein, who suggested that the prevailing theory of continuity in mathematics might some day be overtaken by the atomistic concept of the number line. Although Fraenkel himself remained unconvinced, largely because he considered mathematical continuity necessary to the foundation of modern calculus, he did publish an article explaining the views of the intuitionists, as Einstein and others who believed similarly were known.

## Taught at Einstein Institute of Mathematics

Fraenkel was among the first professors at Hebrew University's Einstein Institute of Mathematics. Along with fellow professor Edmond Landau, Fraenkel taught mathematical logic and mathematical analysis. In 1958, while still teaching at Hebrew University, Fraenkel published an overview of his work on set theory, a textbook entitled Foundations of Set Theory. A year later, he retired as a professor at Hebrew University. To mark Fraenkel's 70th birthday in 1961, several members of the mathematical community put together a collection of essays and research articles related to Fraenkel's life work. The collection, Essays on the Foundations of Mathematics, contained contributions from mathematicians around the world. Sadly, Fraenkel never saw the book in its final form. He died in Jerusalem on October 15, 1965, only months before the book was published.

Fraenkel will be remembered for his research in set theory and modern logic. His refinements to the set theory conclusions of Ernst Zermelo, codified as the Zermelo-Fraenkel axioms, or ZF, are almost always what scientists and mathematicians mean today when they speak of "set theory." Further enhancing the value of Fraenkel's contributions to the body of mathematical theory are the clarity and precision of his writings, several of which continue to be taught in colleges and universities worldwide. In its review of Fraenkel's summation of his set theory research— Foundations of Set Theory —the British Journal for the Philosophy of Science was lavish in its praise. Its reviewer wrote that the book "is a masterly survey of its field. It is lucid and concise on a technical level, it covers the historical ground admirably, and it gives a sensible account of the various philosophical positions associated with the development of the subject … essential reading for any mathematician or philosopher."

## Books

Contemporary Authors Online, Gale Group, 2000.

Mathematical Expeditions: Chronicles by the Explorers, Springer-Verlag, 2001.

Notable Scientists: From 1900 to the Present, Gale Group, 2001.

## Online

"Adolf Abraham Halevi Fraenkel," Groups, Algorithms, and Programming, http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Fraenkel.html (March 5, 2003).

"Adolf Fraenkel," 201E: Mathematical Foundations, http://ergo.ucsd.edu/~movellan/courses/245/people/Fraenkel.html (March 9, 2003).

"Paul Joseph Cohen," Groups, Algorithms, and Programming, http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Cohen.html (March 9, 2003).