**Alfred Tarski** made important contributions in many areas of mathematics, including metamathematics, set theory, measure theory, model theory, and general algebra.

- The school gave Alfred a broader education than he would otherwise gave received.
- There were major changes in Poland during the years that Alfred Teitelbaum was growing up and we need to look briefly at the background in order to understand events.
- Alfred Teitelbaum (who had still not changed his name to Tarski) spent a short while in the Polish army after leaving school and then entered the University of Warsaw in 1918, beginning a course which he intended would lead to a degree in biology.
- These appointments were of great significance since Alfred took a course on logic given by Lesniewski who quickly saw his genius and persuaded him to change from biology to mathematics.
- It was a defining moment for Alfred who now came under the influence not only of Lesniewski but also of Łukasiewicz, Sierpiński, Mazurkiewicz, and the philosopher Kotarbinski.
- It was around 1923 that Alfred Teitelbaum changed his name to Alfred Tarski.
- There is no doubt that Tarski was strongly influenced by these feelings and wished to be a Pole and not a Jew.
- There was also the realisation that anti-Semitic views in the country made it almost impossible for a Jew to be appointed to a university post and Tarski, nearing the end of his doctoral studies, certainly wished to follow an academic career.
- Tarski's first paper was published in 1921 when he was only 19 years old.
- In this paper he investigated set theory questions, and in fact set theory would be a continuing research interest for Tarski throughout his life.
- In 1924 Tarski graduated with a doctorate, and became the youngest person ever to be awarded the degree by the University of Warsaw.
- Tarski's first major results were published in 1924 when he began building on the set theory results obtained by Cantor, Zermelo and Dedekind.
- He published a joint paper with Banach in that year on what is now called the Banach-Tarski paradox.
- Tarski taught logic at the Polish Pedagogical Institute in Warsaw from 1922 to 1925 then in that year he was appointed Docent in mathematics and logic at the University of Warsaw.
- Given Tarski's Polish patriotism which we mentioned above, it may be relevant to note that Maria was a Roman Catholic and she had worked as a courier for the army during Poland's fight for independence.
- This was a time when Tarski's international reputation continued to grow.
- Tarski was awarded a fellowship to allow him to return to Vienna in January 1935 and he worked with Menger's research group until June.
- Tarski presented his ideas on truth in a lecture at this meeting.
- for the thesis that Alfred Tarski's original definition of truth, together with its later elaboration in model theory, is an explication of the classical correspondence theory of truth.
- Tarski published On the concept of logical consequence in 1936.
- In 1939 Tarski applied for the chair of philosophy at Lwów but failed to be appointed.
- However, it is certainly fair to say that by 1939 Tarski had an outstanding international reputation but was still forced to support himself by teaching mathematics in a high school.
- In August 1939 Tarski travelled to Harvard University in the United States to attend another Unity of Science meeting.
- Tarski had been in the United States for two weeks at the time.
- He failed to achieve this but fortunately all three survived the war and were able to join Tarski in 1946.
- Certainly Tarski's life was saved by being in the United States but he still had to secure a job.
- Tarski held a number of temporary research positions: Harvard from 1939 to 1941; City College of New York in 1940; and the Institute for Advanced Study at Princeton in 1941-42 when he held a Guggenheim Fellowship.
- At Princeton Tarski met Gödel again for he had also fled from the Nazi threat.
- After these years of temporary jobs, Tarski obtained a permanent post after he joined the staff at the University of California at Berkeley in 1942.
- Tarski certainly did not lead a quite life in Berkeley, but rather took many opportunities to visit other places.
- Tarski is recognised as one of the four greatest logicians of all time, the other three being Aristotle, Frege, and Gödel.
- Of these Tarski was the most prolific as a logician and his collected works, excluding his books, runs to 2500 pages.
- Tarski made important contributions in many areas of mathematics: set theory, measure theory, topology, geometry, classical and universal algebra, algebraic logic, various branches of formal logic and metamathematics.
- We have looked briefly at some of Tarski's work and we shall examine a little more of his work but it is impossible in a biography of this length to give a proper view of the range of his contributions.
- Metamathematics, introduced by Hilbert in 1922 meaning "proof theory" as a part of his programme to establish the consistency of arithmetic, was transformed by Tarski when he introduced semantic methods leading to his development of model theory with its combination of semantic and syntactic relations.
- Tarski destroyed the borderline between metamathematics and mathematics.
- Tarski presented his paper The axiomatic method : with special reference to geometry and physics to the International Symposium held at the University of California at Berkeley from 26 December 1957 to 4 January 1958.
- In 1968 Tarski wrote another famous paper Equational logic and equational theories of algebras in which he presented a survey of the metamathematics of equational logic as it then existed as well as giving some new results and some open problems.
- The paper considers Gödel's incompleteness theorem as well as Tarski's undefinability theorem and look at their consequences for the axiomatic method in mathematics.
- Tarski wrote nineteen monographs in different areas of mathematics.
- In A decision method for elementary algebra and geometry Tarski showed that the first-order theory of the real numbers under addition and multiplication is decidable which is in contrast, in a way which is really surprising to non-experts, to the results of Gödel and Church who showed that the first-order theory of the natural numbers under addition and multiplication is undecidable.
- In Undecidable theories Tarski showed that group theory, lattices, abstract projective geometry, closure algebras and others mathematical systems are undecidable.
- In Ordinal algebras Tarski defines an algebra which captures the properties of the additive theory of order types.
- The collected papers of Tarski were produced in four volumes edited by Steven R Givant and Ralph N McKenzie.
- It is only when we see Tarski's papers collected in one place that we can begin to appreciate the scope and profundity of his influence on modern mathematical thought and, in particular, on modern mathematical logic.
- Mathematical logic as we know it today is almost inconceivable without Tarski's contributions.
- Tarski was honoured by being elected to the National Academy of Sciences, the Royal Netherlands Academy of Sciences, and the British Academy.

Born 14 January 1901, Warsaw, Russian Empire (now Poland). Died 26 October 1983, Berkeley, California, USA.

View full biography at MacTutor

Algebra, Origin Poland, Puzzles And Problems

**O’Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive