122 Logic as a Tool Alfred Tarski (born Alfred Teitelbaum, 14.1.1901– 26.10.1983) was a Polish–American logician and mathematician who made seminal contributions to many areas of mathematics including set theory, universal algebra, topology, geometry, universal algebra, and several branches of mathematical logic, including theories of truth, logical consequence and definability, algebraic logic, model theory, and metamathematics (proof theory), as well as in methodology of science At age of 22 Tarski became the youngest student awarded a PhD from the University of Warsaw At that time he changed his religion from Jewish to Roman Catholic and his surname from Teitelbaum to Tarski to reaffirm his Polish identity, but also to make it possible to obtain an academic position which, at that time, was almost impossible for Jews in Poland Even though Tarski did not have a university position in Warsaw and taught mostly in high schools there until 1939, he made important contributions to logic and set theory which brought him international recognition In 1933 Tarski published his groundbreaking paper The concept of truth in formalised languages where he rigorously defined formal semantics of logical languages and the notion of truth of a logical formula In 1936 Tarski published another seminal work, On the concept of logical consequence, where he precisely defined logical consequence to hold when the conclusion is true in every model in which all premises are true Yet another work of fundamental importance was A decision method for elementary algebra and geometry, which he essentially completed in around 1930 but only published in the USA in 1948 In it Tarski showed, using the method of quantifier elimination, that the first-order theory of the real numbers under addition and multiplication is algorithmically decidable, one of the most important results on decidability of a mathematical theory In 1936 he also produced his very popular and influential undergraduate textbook on logic, Introduction to Logic and to the Methodology of Deductive Sciences During Tarski’s visit to the USA for a conference in 1939, the Nazi army invaded Poland and World War II began; Tarski could not return so remained in the USA, where he became a naturalized citizen in 1945 Since 1949 until his retirement in 1971 Tarski was a Professor of Mathematics at the University of California at Berkeley, where he organized his famous logic seminar and created a very strong research school Famous for his extreme productivity and precision, both in his teaching and his scientific writings, he was also an energetic and charismatic teacher whose lectures were meticulously prepared and presented Tarski was also an inspiring and demanding supervisor of 26 doctoral students, many of whom also became distinguished mathematicians Tarski was notorious for his all-night working sessions with his students in his smoke-filled study in the basement of his home in Berkeley He had a strong and colorful personality; he was a heavy smoker and drinker, a womanizer, very ambitious and self-righteous, yet very civilized and polite With his enormous legacy in scientific works, students, followers, and general scientific influence, Tarski is widely regarded as one of the greatest logicians of all times, along with Aristotle, Frege, and Gödel