On the Warsaw interactions of logic and mathematics in the years 1919–1939

Annals of Pure and Applied Logic 127 (2004) 289 – 301 www.elsevier.com/locate/apal

On the Warsaw interactions of logic and mathematics in the years 1919 –1939 Roman Duda Institute of Mathematics, Wroclaw University, P.L. Gruwaldzki 2/4, Wroclaw 50-384, Poland

Abstract The article recalls shortly the early story of cooperation between the already existing Lvov philosophical school, headed by Twardowski, and the just then establishing Warsaw mathematical school, headed by Sierpi0nski. After that recollection the article proceeds to contributions made by men in1uenced by the two schools. Most prominent of them was Alfred Tarski whose work in those times, concentrated mainly upon the theory of deduction, axiom of choice, cardinal arithmetic, and measure problem, has been described in some detail. c 2003 Elsevier B.V. All rights reserved.
MSC: 01A60; 01A72; 03E10; 03E25; 093E50 Keywords: Set theory; Mathematical logic; Deduction; Axiom of choice; Cardinals; Measure

The story goes back to the Lvov university before World War I. When Wac law Sierpi0nski (1882–1969) has been invited to Lvov to take one of the two mathematical chairs in the university there (another one has been occupied by J0ozef Puzyna (1856– 1912)), he soon began to read regular courses on the theory of sets [60]. In those times it was quite a novelty, as the very status of that theory seemed then rather uncertain. By some suspected, by the most neglected. And what was even more important, he also began to collect around himself young researchers in that area, among them Stefan Mazurkiewicz (1888–1945) and Zygmunt Janiszewski (1888–1920), and to publish his own and their original results in the country and abroad. But in those years, before World War I, there 1ourished already in Lvov and enjoyed a high regard a philosophical school founded by Kazimierz Twardowski (1866 –1938). The school has laid emphasis upon preciseness of notions and of reasonings. Consequently, quite soon its characteristic feature became an interest in logic. Although the two men, Sierpi0nski c 2003 Elsevier B.V. All rights reserved. 0168-0072/$ - see front matter
doi:10.1016/j.apal.2003.11.025

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and Twardowski, 1 never openly cooperated, both have strongly in1uenced the course of events in the times to come. During the war two successors of Twardowski, logicians Stanis law Le0sniewski (1886 –1939) and Jan Lukasiewicz (1878–1956), have settled down in Warsaw. Similarly, two students of Sierpi0nski, mathematicians Janiszewski and Mazurkiewicz, also came to that city and Sierpi0nski, who returned to Warsaw in 1918, soon has become a key Fgure in establishing the Warsaw school of mathematics, interests of which lay in the theory of sets (in the then understanding of that theory) and areas close to it, including mathematical logic. The two arising schools, 2 Warsaw continuation of the Lvov philosophical school and the new Warsaw mathematical school, were thus personally close to each other. And much more than that. A close collaboration and a mutual understanding of logicians and mathematicians has led to the unique combination of logic and mathematics into a rich current. A relative isolation of logicians, seen almost everywhere in those times, was never the case in Warsaw. The current started as early as 1915 when, after an evacuation of the Russian Imperial University from Warsaw to Rostov-upon-Don (where it exists since then), the Polish University in Warsaw, was opened, in the fall that year. In the faculty of Mathematics and Natural Sciences there were two philosophical chairs, oIered to logicians Le0sniewski and Lukasiewicz, while two mathematical chairs were oIered to Janiszewski and Mazurkiewicz. Since 1918 there was a third mathematical chair which was taken by Sierpi0nski who in that year has just returned from internment in Russia. Newly nominated young professors formed a friendly group with a zest for research and teaching. For instance, Lukasiewicz read courses in logic, history of philosophy, and some special topics related to the methodology of deductive sciences and to the foundations of mathematical logic. “Although Lukasiewicz was not a mathematician, he had an exceptionally good sense of mathematics and therefore his lectures found a particularly good response among mathematicians. I remember a lecture of his on the methodology of the deductive sciences in which he analysed among other things the principles which any system of axioms should satisfy (such as the consistency and independence of axioms). [...] Lukasiewicz’s ideas had as a by-product in our country the exact formulation of such notions as those of an ordered set and of an ordered pair [...]. This illustrates the in1uences wrought by Jan Lukasiewicz, philosopher and logician, on the development of mathematical concepts” [35, pp. 23–24]. The concept of an ordered pair has been soon afterwards deFned by Kuratowski himself [27]. Also Le0sniewski surrounded himself with students, most eminent of which became Alfred Tarski (1901–1983) and Adolf Lindenbaum (1904–1941). They attended seminars led by Lukasiewicz and later on by Tarski, and were active in intermediate areas between mathematics and logic [19, p. 222].

1

When the name of a man appears for the Frst time, we provide it fully together with the life-span, but in successive instances we use a shortened version. 2 For the rise of modern mathematics in Poland and emergence of the Warsaw mathematical school, see [11,34,35,37,67] and others. The general development of logic between the two World Wars is covered in [18,19,55], while the development of the Warsaw logical school is described in [24,46,89].

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There were joint unoNcial meetings of a group of logicians and mathematicians (in the alphabetic order: Janiszewski, Tadeusz Kotarbi0nski (1886 –1981), Le0sniewski, Lukasiewicz, Mazurkiewicz, Sierpi0nski) together with some students including Bronis law Knaster (1893–1980), Kazimierz Kuratowski (1896 –1980), Tarski, and others. In the meetings they Frst discussed the book Principia Mathematica by Bertrand Russell (1872–1970) and Alfred North Whitehead (1861–1947), but quite soon the group became more active. Participants have started to read new papers, to comment upon them, and to present original results of their own, both mathematical and logical [34, p. 14]. An illustration of that atmosphere of joint work is a translation of Couturat’s logic by Knaster, a student of mathematics [9]. An extremely interesting feature of those years was a program for the Polish mathematical community, advanced by Janiszewski, with the aim “to distinguish ourselves and to win an independence” [22]. After 123 years of partitions, the Polish nation had been just winning a political independence and there was a common enthusiasm to establish a full-1edged state with a 1ourishing science. The essence of Janiszewski’s program consisted in concentrating all active Polish mathematicians in one research area, to work in an open and collaborative way, and to run an international journal devoted predominantly to the chosen area. The journal appeared in 1920 under the name of Fundamenta Mathematicae (cf. [10,38]). After Lvov times, when Sierpi0nski was a leader of’ a group consisting of Janiszewski, Mazurkiewicz and some others, the area was an obvious choice. It was the “theory of sets and its applications”, as the announcement of a few Frst issues of the new journal Fundamenta Mathematicae read, which meant the proper theory of sets, topology, and theory of real functions, all with a strong inclination to mathematical logic. As Sierpi0nski recalled, “When in 1919 all three of us—Janiszewski, Mazurkiewicz, and myself—found ourselves as the Frst professors of the revived Polish university in Warsaw, we decided to carry out Janiszewski’s idea of publishing in Warsaw a foreignlanguage periodical devoted to the set theory, topology, theory of real functions, and mathematical logic. This was the origin of Fundamenta Mathematicae” [67]. In those times all existing mathematical journals covered the whole of mathematics. A novel idea of restricting the area seemed risky and quite soon there came in friendly warnings, cf. [12,40]. The idea, however, has been supported by logicians, two of which—Le0sniewski and Lukasiewicz—have joined the editorial board and stayed there up to a break with Sierpi0nski in 1929. There even were deliberated propositions whether to publish alternately volumes devoted to mathematics and to logic, but mathematics soon prevailed and nothing came out of it. Friendly relationships between mathematicians and logicians were continued with the eIect that newcomers were well educated both mathematically and logically. Of those newcomers most known became Tarski 3 who received his Ph.D. from Le0sniewski in 1924 but there were also others like (in alphabetic order) Stanis law Ja0skowski (1906 –1965), Adolf Lindenbaum (1904 –1941), Andrzej Mostowski (1913–1975), Moj˙zesz Presburger (1904 –1943?), Jerzy S lupecki (1904 –1987). And outside Warsaw there were both logicians with a mathematical inkling, e.g., Leon Chwistek (1884 –1944) in 3

On the personality of Alfred Tarski, see [14,15].

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Cracow, later in Lvov, and mathematicians with a good understanding of logic, e.g., Stefan Banach (1892–1945) and Stanis law Ulam (1909–1984), both in Lvov. Such a numerous group, well versed both in mathematics and logic, has in1uenced the development of the two Warsaw schools, philosophical and mathematical, in many ways which I will now shortly try to describe. As a researcher, Lukasiewicz has been in those times occupied mostly with his many-valued logics. This, in turn, has arisen an interest in the methodology of deductive sciences and Ajdukiewicz wrote the Frst monograph devoted to that Feld [1]. As Tarski later recalled [84], the monograph has led him to the discovery of a deduction theorem in 1921. However, Tarski did not publish then his discovery and the theorem was later rediscovered by Herbrand in his thesis from 1930 [21]. Although it was published in the same year also by Tarski [73], the theorem is not, at the least not unanimously, named after him. Analogous delay in publishing has happened to him more than once. Comparative studies of many-valued logics and other deductive theories have thus led to the development of methodology and then to a general theory of formal deductive systems. Extending his ideas from the note [73], Tarski has published a larger treatise [74] in which he outlined his concept of metamathematics. The concept was wider than that of Hilbert in both raising the level of abstractness (Tarski considered entities, received by Hilbert in the process of formalization, as abstract ones) and neglecting Fnitary limitations. In the treatise a major role has been played by the notion of a logical consequence as well as by the deduction theorem. It should be noted that after Tarski and after the common practice of the Warsaw school the term “meta” has become commonly accepted as denoting all what is concerned with the study of formal objects of a formal system. Thus, e.g., a “metalogic” is a study of formal systems in logic. And when Tarski formalized metamathematics, also the latter term came into a common use. These were the years of the rapid growth of Tarski’s talent. Much of his work lay then in metatheory (metalogic, metamathematics), with a particular interest in semantic problems, apparently under in1uence of Le0sniewski, his teacher. His most celebrated work in this area was a treatise [78] (announced in [73]) in which a major role has been played again by the notion of a logical consequence and by the deduction theorem. The leading theme of his in1uential text-book [79], soon translated both into German and English, was the theory of deduction. Up to 1950 there appeared altogether 29 editions of that book, partial or complete, in diIerent languages, including Russian. The book has thus become an extraordinary success and it helped enormously both in popularizing metamathematics and in breaking the boundaries between mathematical logic and mathematics, wherever they existed [19, p. 247]. In Poland there never was a strict borderline between mathematical logic and mathematics but in the world outside Poland the picture was then quite diIerent. And the theory of deduction has been developed also by others, e.g. [44,45]. A characteristic feature and, as Tarski himself has remarked [83], “an essential contribution of the Polish school to the development of metamathematics” was the admission, from the very beginning, all fruitful methods, including inFnitary ones. “Restriction to Fnitary methods seems natural in certain parts of metamathematics, in particular in the

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discussion of consistency problems, though even here these methods may be inadequate” [ibidem]. That free admission of any fruitful method was a prevailing attitude within the Warsaw and Lvov logical and mathematical circles. One could say that for the two circles important were results and not the way of obtaining them. In particular, actual inFnite was for them one of the central and most natural objects of study. Prima facie logician, Tarski had also results in pure mathematics, some obtained jointly with his colleagues from the Warsaw and Lvov schools (see below). Dominating Fgure on the mathematical side and, in fact, an unquestioned leader of the Warsaw mathematical school, was Sierpi0nski. His scientiFc interests, results and opinions have largely shaped the school. And it was under his in1uence that much attention has been paid to foundational problems in the theory of sets and elsewhere. Much of that attention has been focused upon the Axiom of Choice ([17] is a good introduction to an early understanding of the Axiom). In his Paris lecture on this subject, Russell has shown [54] many uses of the Axiom in the ordinal arithmetic. Sierpi0nski, who about that time became intrigued by the Axiom (apparently under in1uence of that lecture), has started to uncover its implicit uses in other areas of mathematics, Frst of all in the real analysis. The idea has been unfolded in [56] but the real onset has been his extensive treatise [57] in which he enumerated a good number of propositions from diIerent domains, the only known proofs of which were implicitly based upon the Axiom of Choice. To give one example only, in 1916 Sierpi0nski has discovered [56], independently of Cipolla [8], that under the assumption of the Axiom of Choice continuity at a point and sequential continuity at a point are equivalent, and he formulated a weaker form of that Axiom which suNces to prove that equivalence. Continuing research about the Axiom, Sierpi0nski required cognizance of applications of the Axiom of Choice but also preferred avoidance of the Axiom, if only possible. Raising the question of what it signiFed when a proposition has been deduced from the Axiom, he demanded to interpret it as merely asserting the abstract existence of a choice function (an attitude shared by many supporters but rejected by most critics of the Axiom). According to him, Lusin accepted the Axiom only as a heuristic device. Nowadays it is a rather common opinion that Sierpi0nski can be granted with the initiating the trend that within mathematics one should study sine ira et studio the deductive strength of the Axiom of Choice and of various propositions relevant to it, disregarding any philosophical controversies. In regard to philosophy, he declared neutrality. Sierpi0nski posed himself to be independent of any philosophical presupposition. In the area of mathematics he strongly demanded eIectiveness and that at any cost, even at the cost of applying the Axiom of Choice, but always with a clear explanation whether the proof or construction is “eIective” (cf. [30,58,59]), i.e. without involving the Axiom, or “ineIective”. It is not my intention to tell the story of the Axiom of Choice in Warsaw in details (cf. [49; Chapter 4: The Warsaw school, widening applications, models of set theory (1918–1940)]) but it seems appropriate to recall some of the more important contributions of the Warsaw school. Relying upon the Axiom of Choice, Banach has proved an important theorem (now called the Hahn–Banach theorem) which says that each linear continuous functional
: E → K, deFned upon a linear subspace E of a Banach space X , can be extended

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to a linear continuous functional   X ∗ (X ∗ is a conjugate space) in a way such that 
 =  [3]. The meaning of the theorem can be illustrated by the fact that later it became the main topic of one of Bourbaki’s books [7]. The success of that theorem and other results of Banach, obtained with the help of the Axiom of Choice, justiFes an opinion that also he did much to the eventual acceptance of the Axiom. In1uenced by Sierpi0nski’s interest in the Axiom of Choice, Tarski also entered the Feld and did it with a great power. One of his Frst works here was an extensive article [71] in which he systematized the theory of Fnite sets while also considering the role of the Axiom of Choice with respect to various propositions concerning Fnite sets. Another topic taken by Tarski were propositions in cardinal arithmetic which are equivalent to the Axiom of Choice. In [31] the author explained why the theory of sets needs large cardinals like (weak) inaccessible cardinals deFned by HausdorI. Tarski made in the years 1924 –1926 his most signiFcant contributions to the subject by studying cardinal equivalents to the Axiom of Choice. It might came as a surprise that several simple equalities here are as strong as the Axiom of Choice itself, e.g., if m and n are inFnite cardinals, then each of the following equalities is equivalent to the Axiom of Choice [70] m + n = m · n;

m2 = m;

if m2 = n2 ; then m = n; etc:

A detailed account of propositions in the cardinal arithmetic, which are equivalent to the Axiom of Choice, can be found in [53]. Most of those propositions come from Tarski. Related to Sierpi0nski’s interest in the Axiom of Choice was the Continuum Hypothesis. Also here he did not work alone. E.g., Lindenbaum and Tarski have aNrmed, in a joint paper [43], that the General Continuum Hypothesis implies the Axiom of Choice, cf. also [65]. Sierpi0nski’s numerous articles on it culminated in the book [61], but even after its publication he occasionally returned to the subject (cf. [62,64,66]). Around 1930 Tarski has introduced strongly inaccessible cardinals. He and Sierpi0nski [69] proved that strong inaccessibility implies weak inaccessibility and, under assumption of the General Continuum Hypothesis, vice versa [81]. Tarski considered also relations between inaccessible cardinals and the Axiom of Choice. The area of large cardinals has been explored also after War World II, cf. [85]. An interesting point is that problems concerning large (inaccessible, measurable etc.) cardinals have consequences in abstract parts of algebra and topology, cf. [26]. Beginning in 1926, Tarski conducted a seminar at the Warsaw university on set theoretic models. Later he claimed [49, p. 257] that he had worked on what was to be called the LVowenheim–Skolem–Tarski theorem: if a countable set of sentences has an inFnite model, then it has a model of every inFnite cardinality. In other words, if a countable set of sentences has an inFnite model, it has many models and thus is not categorical. Here we meet another instance of Tarski’s delay in publishing. The years 1920s were a period of a fast development of the Lebesgue measure theory, where the Axiom of Choice also played a central role: it has been used to establish the countable additivity of that measure and to construct a Lebesgue nonmeasurable set. No wonder then that the measure problem has intrigued also Polish mathematicians and logicians.

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Recall that the general measure problem asks [39, p. 110] whether there exists a function m (called a measure) which assigns a non-negative real number to every bounded subset of an n-dimensional Euclidean space Rn in a way that (a) The n-dimensional unit cube has measure 1. (b) Congruent sets have the same measure.
∞ (c) If the sets A1 ; A2 ; : : : ; Ak ; : : : are pairwise disjoint, then m(∪∞ k=1 Ak ) = k=1 m(Ak ). In 1914, Felix HausdorI (1868–1942) replaced the condition (c) of countable additivity by a weaker one, that of Fnite additivity, and proved that the weaker problem has a negative answer for n ¿ 3. To that eIect, he used the Axiom of Choice to partition a sphere into sets A; B; C; D such that A; B; C and B ∪ C were all congruent, while D was countable [20, p. 469]. Using the Axiom of Choice, Banach extended HausdorI’s result to the two missing spaces, that is, to the line R1 and to the plane R2 [2], and then turned to the HausdorI paradoxical partition. In a joint paper with Tarski, the two authors have shown that in Rn provided n ¿ 3, any two bounded sets with non-empty interiors are equivalent by Fnite decomposition [6]. More speciFcally, the unit ball can be decomposed into a Fnite number of pieces and reassembled into two unit balls, or to put it somewhat less seriously, “it is possible to cut an orange into a Fnite number of pieces that can be then reassembled to produce two oranges, each having the same size and volume as the Frst one” [13]. This is the famous Banach–Tarski paradox. By the way, one can ask about money: can a banknote produce two of its kind? It is a problem in applied mathematics, but the answer is, unfortunately negative: no bounded set in the plane can have such a paradoxical decomposition [41, footnote 1 on p. 218]. Measure problem has still occupied minds and in 1929 Lvov mathematicians, joined then by Kuratowski (who in 1928–1932 was a professor in the Lvov Polytechnic and tutored Ulam) and Tarski (a sort of guest performance), begun to reconsider it under diIerent conditions. They didn’t limit themselves to Euclidean spaces and so the problem changed its character from analytical to foundational, in due time involving large cardinals. Assuming Continuum hypothesis, Banach and Kuratowski [5] have shown that if we consider a set of the continuum cardinality, e.g., the real line, and resign from the invariance by translations, replacing the condition (b) above by the following one (suggested by Banach) (b ) Measure of each single point is 0, then the answer to the measure problem is negative: there is no measure upon continuum satisfying the conditions (a), (b ), and (c). This result has been then extended by Banach for all cardinals, assuming General Continuum Hypothesis [4]. Next year Banach asked whether there exist “measurable” cardinals, that is, whether all subsets of a set having such a “measurable” cardinality do support countably

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additive nontrivial measure [5]. Ulam has shown that if such a cardinal does not exceed continuum, then the measure has only two values 0 and 1 [87,88]. Almost instantly the problem of the existence of measurable cardinals turned out to belong to the study of large cardinals, for if measurable cardinals do exist, then they are “very large”, in any case larger than the Frst strongly inaccessible cardinal (the theorem of Tarski and Ulam, see [87, p. 150]). The Axiom of Choice is a very strong too1. Tarski has shown [75,76] that on every inFnite set there exists a two-valued additive measure such that each point has measure 0. More speciFcally, if one considers measure problem with the condition (a) replaced by (a ) Measure m is non-trivial, that is, m(A) ¿ 0 for some A, with the condition (b) replaced by (b ) above, and with the condition (c) of countable additivity replaced by that of Fnite additivity, (c ) If sets A1 , A2 are disjoint, then m (A1 ∪ A2 ) = m(A1 ) + m(A2 ), then the answer is positive, although the measure has two values only, 0 and 1. Extending Tarski’s result, Sierpi0nski proved that the existence of such a measure, even on N, yields the existence of a set which is not Lebesgue-measurable [63]. I cannot help quoting here an opinion of Lebesgue from 1938: “When 35 years ago the powder keg had been exploded through the match lighted by Zermelo, an interesting fact emerged: those who up to that moment had utilized set theory, who had worked with sets since long, were opposed to the axiom of choice, while those who had not used sets were ready to accept the axiom. It is an interesting fact and one can raise the question: were we reactionary by an inveterate habit of getting along without openly formulated axiom of choice or there was some other reason?” ([16], quoted after [48, p. 272–273]). Being close to mathematics, several logicians investigated logical problems arising in the sententional calculus, e.g. [45]. One important instance was a sententional calculus for intuitionism. In 1932, GVodel has shown that such a calculus cannot be based upon Fnitely-valued matrices and in 1935 Ja0skowski has provided an adequate inFnitely-valued matrix [23]. Two years later Tarski has found an interesting topological interpretation of the intuitionistic calculus of propositions [82]. Proposing a method to eliminate ordinal numbers and transFnite induction, Kuratowski stated a general principle on which it was based [28, p. 113]. Kuratowski’s maximal principle has become one of the most convenient forms of the Axiom of Choice. Rediscovered later by M. Zorn [90], it is now commonly called Zorn’s Lemma or, more properly, Kuratowski–Zorn’s Lemma. Another Warsaw mathematician, Edward (Szpilrajn)-Marczewski (1907–1976), employed Kuratowski’s reformulation of Zorn’s Lemma to establish the Order Extension Principle [47]. Skilful use of logical notation helped Kuratowski, Mazurkiewicz, Sierpi0nski, Tarski and others to deFne some speciFc sets in the framework of Baire classiFcation and to prove their properties (cf. [36,32]) with the important consequences in the theory of analytic and projective sets.

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Axiomatization of mathematical theories was another topic often considered within the Warsaw group. Kuratowski has axiomatized the concept of a general topological space in terms of closed sets [29,33]. In 1927 Tarski has proposed a very ingenious axiomatization of geometry, based upon mereology of Le0sniewski (see [72] for a summary). Papers [75] and [77] were a continuation of that topic. In the treatise [80] the author considered Boolean algebra from the viewpoint of mereology and in ([82], formula (11) on p. 111) he showed that regular open sets form a Boole algebra. At the end of the considered period there appeared Andrzej Mostowski (1913–1975), later one of the most prominent logicians of his generation, whose Frst research in the Feld has been in1uenced by Lindenbaum’s ideas on the Axiom of Choice [42], see also [50–52]. Looking in retrospect, one sees an extraordinarily rich mutual in1uence. An in1uence of mathematics upon logic can be seen in the theory of deduction for which logical and mathematical theories were the most natural objects to study, in elevating metamathematics to an autonomous branch, in model theory, later on developed to a major Feld, in applying reFned mathematical tools, like inFnite matrices, to logical constructions, and last but not least, in the whole atmosphere of a feeling of importance of the work upon logic, intellectual courage of logicians and security feeling provided by footing in the fast developing and enjoying high reputation Warsaw mathematical school. And the other way round, an in1uence of logic upon mathematics can be seen in a free and skillful use of logical symbolism, and in the signiFcance of foundational problems, most conspicuous of which were the Axiom of Choice, Continuum Hypothesis and General Continuum Hypothesis, maximal principles, and axiomatization. Following Sierpi0nski, mathematicians have in cold blood investigated the scope of the Axiom of Choice developing, in particular, a theory of Fnite sets or cardinal arithmetic but also proving important theorems from the theory of sets proper like the Hahn–Banach theorem or the Banach–Tarski paradox. With a similar interest were met Continuum Hypothesis and maximal principles, where we can note a monograph by Sierpi0nski [61], the Kuratowski–Zorn Lemma or the Order Extension Principle by (Szpilrajn)Marczewski. Equally natural was a tendency towards axiomatization of new and old theories, an example of which are Kuratowski axioms for closed sets, used to this day, cf. [33]. A characteristic feature of Polish mathematics in the interwar period, to some extent indebted to a mutual cooperation with logicians, was a free use of non-constructive methods like the Axiom of Choice, Baire category and probabilistics (equivalently, Lebesgue measure). Treated with a suspicion by many, perhaps even a majority of contemporary mathematicians, the methods enjoy nowadays a rather common recognition. In an attempt to evaluate Polish mathematics in the period 1918–1939, Jean–Pierre Kahane has expressed an opinion that its characteristic feature was a free use of the Axiom of Choice in non-constructive demonstrations of existence and, following to that, a free use of other non-measurable methods, based, e.g., upon Baire theory or upon probability (equivalently, Lebesgue measure). In a somewhat aIectionate manner he called Polish mathematics of that period “a monument of great concern and eternal beauty” [25] but there seems to be no doubt that its strength relied, to a great

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