# Andrzej Mostowski

### Born: 1 November 1913 in Lemberg, Austrian Empire (now Lviv, Ukraine)

Died: 22 August 1975 in Vancouver, Canada

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**Andrzej Mostowski**'s mother, Zofia Kramstyk, worked in a bank while his father, Stanisław Mostowski, was an assistant in the Department of Physical Chemistry of the University of Lvov. Andrzej was born one year before the outbreak of World War I and his father Stanisław Mostowski joined the army in 1914. Sadly, however, Stanisław died in the year after his son Andrzej was born. The family went to Zakopane, a winter-sports and health-resort centre in the Tatras Mountains, for a vacation in the winter of 1914. However they were forced to remain there due to World War I and it was not until 1920 that they left Zakopane and moved to Warsaw.

Mostowski entered the Stefan Batory Gymnasium in Warsaw in 1923. There he showed that he was an excellent pupil with particular strengths in mathematics and physics. However he developed a serious illness in 1930 but was still able to complete his studies at the Gymnasium in 1931. He entered Warsaw University after graduating from the Stefan Batory Gymnasium and it was at this time that he became especially interested in the foundations of mathematics, particularly mathematical logic and set theory. He was influenced by his lecturers Kuratowski, Lesniewski, Mazurkiewicz, Łukasiewicz and Sierpiński but an even stronger influence came from Lindenbaum and Tarski. His Ph.D. was awarded in February 1939 for his thesis

*On the Independence of Finitenesss Definitions in a System of Logic*, officially directed by Kuratowski but in practice directed by Tarski who was a young lecturer at that time. His research at Warsaw had been broken by studies in Vienna during the summer semester of 1937 (where he attended a course by Gödel whose ideas strongly influenced his research) and Zürich (where he attended courses by Polya, Weyl and Bernays). In fact he had decided to train as an actuary when in Zürich and, as a first step, he had begun to study statistics. However this subject did not please him so he attended the mathematics courses instead and undertook research on recursion theory and the axiom of choice.

He hoped to find a position at the University of Warsaw after the award of his doctorate but these months prior to the start of World War II were difficult and university positions were essentially unobtainable. He took a job at the National Meteorological Institute while hoping that he would soon have the chance of an academic appointment. However the deepening crisis during 1938 with the German invasion of Austria in March, then of the Sudetenland in October, and next Czechoslovakia in March 1939, became even more acute in May 1939 when Hitler declared himself ready to attack Poland. Mostowski's illness while at high school prevented him serving in the army and he became an accountant after the German invasion of Poland which began on 1 September 1939. As well as working as an accountant for a small firm which manufactured roofing from 1939 to 1944, he also taught in the 'Underground Warsaw University' from 1942 to 1944. Academics took a risk teaching at the underground university and students also took a risk attending lectures. One of the students who took the risk was Maria Matuszewska; she married Mostowski in September 1944. Sierpiński taught there and he afterwards spoke of his colleagues who had lost their lives, in particular a professor who had greatly influenced Mostowski:-

Mostowski hoped to habilitate at the Underground University and was close to achieving this in July 1944 when events intervened. The Soviet authorities encouraged the Polish underground in Warsaw to stage an uprising against the Germans. They attacked the Germans on 1 August and within three days had control of the city. The Germans sent in reinforcements while the Soviets not only refused assistance, but even refused permission for Allied planes to use their bases to supply the Poles with food. The Poles held out for 63 days before being defeated. During this time Mostowski married Maria Matuszewska. The Germans then began deporting the what was left of the city's population. Mostowski was captured and was waiting to be sent to a concentration camp when two Polish nurses helped him to escape across the German lines to a hospital [1]:-In1942another student of mine Adolf Lindenbaum was murdered. He was an assistant professor at Warsaw University and a distinguished author of works on set theory.

Living in Poland in the last few months of the war was hard in the extreme. Mostowski had no job so he tried to make a little money by giving private lessons. Of course this only brings in money if there are those around with sufficient to pay for tuition and not many people in war-torn Poland were in that position. He had to sell some of his possessions to buy food. Things improved shortly after the war ended and Mostowski went to Cracow where he worked for a few months at the Silesian Polytechnic, then for a few months at the Jagellonian University. He submitted his habilitation thesisMany of Mostowski's wartime results - on the hierarchy of projective sets, on arithmetically definable sets of natural numbers, and on consequences of the axiom of constructibility in descriptive set theory - were lost when his apartment was destroyed during the uprising. He had to choose whether to flee with a thick notebook containing those results or with bread. He chose bread.

*Axiom of choice for finite sets*to the Jagellonian University and it was approved in 1945. In the thesis he examined the independence of various forms of the axiom of choice for finite sets. By the beginning of 1946 he was in Łódź where he was an acting professor from January to September. He then returned to Warsaw University where he was appointed as an acting professor. He was made an extraordinary professor in 1947, then an ordinary professor in 1951.

After spending the academic year 1948-49 at the Institute for Advanced Study at Princeton, where he renewed contacts with Kurt Gödel, he returned to Warsaw to positions of higher standing. He was appointed as head of the division for the foundations of mathematics at the Mathematics Institute of the Polish Academy of Sciences in 1948. He was appointed dean of the Faculty of Mathematics and Physics at the university in 1952, then from 1953 he was head of the Department of Algebra. He spent further years abroad - 1958-59 at the University of California at Berkeley and 1969-70 at All Souls College, Oxford. In 1968 he ended his position as head of the division for the foundations of mathematics at the Mathematics Institute of the Polish Academy of Sciences and took up a similar position at the university.

The authors of [5] set out Mostowski's main scientific achievements which are contained in around 120 works:

- The theory of models, especially models for set theory and for arithmetic; model products and their theories; families of models as topological spaces; models with indiscernible elements and the Ehrenfeucht-Mostowski theorem.

- The Fraenkel-Mostowski permutation method and its application to proving the independence of the well-ordering principle from the linear-ordering principle in the theory of sets with atoms (urelements).

- The logical classification of arithmetical concepts and the Kleene-Mostowski hierarchy.

- Mostowski's well-known and highly regarded exposition of Gödel's incompleteness result, its strengthenings and simplifications.

- Mostowski's investigation into infinitistic methods in mathematics;
*w*-models and*b*-models for arithmetic.

- Cohen's forcing; topological approach to generic sets.

*On undecidable propositions in formalized systems of mathematics*(Polish) (1946). Mostowski states in the paper that his aims are to give:-

Mostowski wrote an important Polish text... a popular, and at the same time completely rigorous, presentation of Gödel's ideas: his theory reaching the conclusion that in every formal system of mathematics, satisfying certain very general conditions, there exist statements such that neither their truth nor falsity can be established within the system.

*Mathematical Logic*which was published in 1948. It gained high praise from a reviewer:-

In 1952 Mostowski published a monographThis is an exceptionally good handbook of mathematical logic. It covers a great deal of material and shows many applications of mathematical logic to mathematical problems ... The author is eager to persuade a mathematician that logic can be useful in his work.

*Theory of sets*written jointly with Kuratowski. Ulam reviewed the text and wrote:-

A second edition of the monograph was published in 1966. In many ways this is a new work. Boris Schein writes:-The book is extremely readable, due to a system of development of set-theory which combines the two approaches: the "naive" method followed by Cantor himself, and the formalistic treatment developed on the axiomatic method. The symbolism of mathematical logic is used throughout, but with moderation, and ample motivation is given in the text appealing to the intuition on the infinite sets. This intuition will continue to be necessary as no one system of axioms now known can claim to express the full intuitive freedom of construction available to "naive" set-theory. It is this combination of methodologies that makes the book unique as a monograph on set-theory.

In 1953 the first of two volumes of a university textbookAlthough the main themes of the book remain the same, they are exposed in a far different manner than in the first edition. The second edition contains ten chapters(instead of six)... The book has all the merits of the first edition: it is very readable, modern, and useful for many categories of readers; axiomatic rigor is successfully combined with intuitive freedom, and the clarity and lucidity of style make the reading of the book pleasurable.

*Elements of higher algebra*written by Mostowski and Marceli Stark was published, with a second volume in the following year:-

Many of Mostowski's works are aimed at general mathematicians, not experts in his field. For exampleThe book is written clearly, with great stress on didactic principles of the presentation.

*Widerspruchsfreiheit und Unabhängigkeit der Kontinuumhypothese*(1964) is a:-

Perhaps the expository paper... clearly written article is intended for the general mathematical reader. The author opines that the proofs of the consistency and the independence of the continuum hypothesis constitute the most important mathematical discovery of the last25years. He gives reasons for his opinion, and discusses the philosophical problems(Platonism v constructivism)of the notion of "set".

*Thirty years of foundational studies. Lectures on the development of mathematical logic and the study of the foundations of mathematics in 1930-1964*which reports on a 16 lecture course given by Mostowski in the summer of 1964 in Vaasa, Finland, gives the best view of what he saw as the most important developments in his subject during his career. Curry writes:-

We should explain how Mostowski came to die in Vancouver, Canada. Still an extremely active researcher, he had spent the summer of 1975 in the United States, at Berkeley and Stanford. He left there to travel to a conference in Ontario, Canada, but stopped in Vancouver to deliver an invited lecture at Simon Fraser University. His death was sudden and unexpected.The topics include completeness, incompleteness, decidability, and undecidability theorems; computability, recursive functions, hierarchies, and functionals; intuitionistic logic and its interpretations; constructive mathematics, foundations of set theory, including Cohen's independence proofs; and finally model theory, ending in a special chapter on direct and reduced products. There are references to over240papers. In most cases the author discusses the principal features of these with great clarity and insight. It is a valuable survey, even for a specialist.

Mostowski received many honours: a Polish state prize (1952); corresponding member of the Polish Academy of Sciences (1956); full member: (1963); Irzykowski Foundation Prize (1972); and member of the Finnish Academy of Sciences (1973).

Mostowski undertook many editorial duties. He was the editor of the Mathematical, Astronomical and Physical series of the

*Bulletin of the Polish Academy of Sciences*, on the editorial board of several journals including

*Fundamenta Mathematicae, Dissertationes Mathematicae*, the

*Journal of Symbolic Logic*and

*Studia Logica*. He was one of the founders and editors of the

*Annals of Mathematical Logic*.

**Article by:** *J J O'Connor* and *E F Robertson*

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