Alfred Tarski

Alfred Tarski was a Polish-American mathematician and logician who reshaped twentieth-century logic.

He was a prolific researcher whose work spanned many fields. These included model theory, metamathematics, and algebraic logic. He also contributed to set theory, topology, geometry, and measure theory. His work helped define how mathematicians understand the concept of truth. Alongside his contemporary, Kurt Gödel, Tarski fundamentally changed the study of logical systems. He was a central figure in the Lwów–Warsaw school of logic in Poland. This school made the University of Warsaw a world-leading center for mathematical research.

Tarski's mathematical journey began with a surprising change in direction. In 1918, he entered the University of Warsaw intending to study biology. However, the mathematician Stanisław Leśniewski recognized his immense potential. Leśniewski encouraged him to abandon biology for mathematics. Tarski became the only person to complete a doctorate under Leśniewski's supervision in 1924. His thesis, "On the Primitive Term of Logistic," was published in 1923. During this era, Alfred Teitelbaum and his brother Wacław changed their surname to Tarski. They also converted to Roman Catholicism, though Alfred remained an avowed atheist.

One of Tarski's most famous results is a surprising discovery in geometry. In 1924, Tarski and Stefan Banach studied the properties of a ball. They proved that if one accepts the Axiom of Choice, a ball can be cut into a finite number of pieces. These pieces can then be reassembled into a single ball of a larger size. Alternatively, the pieces can form two balls, each equal in size to the original. This result is known as the Banach–Tarski paradox. It demonstrates how mathematical logic can produce results that seem impossible in the physical world.

Tarski also made major breakthroughs in the study of decidability. In 1930, he showed that the first-order theory of real numbers is decidable. This means there is a method to determine the truth of statements within that system. He achieved this through a process called quantifier elimination. This was a significant finding because other systems are not so simple. For example, Alonzo Church proved in 1936 that Peano arithmetic is not decidable. Similarly, Gödel's incompleteness theorem showed that Peano arithmetic is also incomplete. Tarski's work helped map the boundaries between what math can and cannot solve.

His career was deeply impacted by the political turmoil in Europe. Because of his Jewish heritage, Tarski faced systemic discrimination in Poland. In 1937, a university chair in Poznań was abolished specifically to avoid appointing him. In August 1939, Tarski left Poland on the last ship to sail for the United States before the German and Soviet invasion. He left his wife, Maria Witkowska, and their two children in Warsaw. He did not see them again until 1946. During the war, nearly all of his Jewish extended family were murdered by German occupiers.

After arriving in the United States, Tarski held several research positions. He worked at Harvard University and the City College of New York. He also received a Guggenheim Fellowship to work at the Institute for Advanced Study at Princeton. In 1942, he joined the University of California, Berkeley. He remained at Berkeley for the rest of his life and became an American citizen in 1945.

At Berkeley, he became a legendary and demanding teacher. He supervised twenty-four Ph.D. dissertations, including students like Julia Robinson and Solomon Feferman. He was also notable for supervising five women at a time when most graduate students were men.

Tarski's mathematical interests remained exceptionally broad throughout his life. He published papers on relation algebra, which occupied much of his later career. He and his students also developed cylindric algebras. These are used to study first-order logic in a way similar to how Boolean algebra is used for sentential logic. His collected papers total about 2,500 pages. His work connects various branches of math, such as linking geometry to the theory of real numbers. Tarski's legacy lives on through the many mathematicians he trained and the complex logical systems he defined.