From the lowest grades, Banach and Wilkosz shared a love of mathematics. During the so-called breaks I often saw them solving mathematical problems, which to me as a humanist were simply Chinese. Banach's friendship with Wilkosz was not limited to the school grounds; they also met after classes at Wilkosz's house on Zwierzyniecka Street or in the school building, and in the Kraków Planty [a large city park]. In later times, it would take half the night for the excited students - Banach and Wilkosz - to walk together through the streets of Kraków when some issue occupied their minds. I did not take part in these mathematical conversations, but I often had equally long discussions with Wilkosz himself, with whom I was more close. In secondary school and after the final exams, we were united by literary interests and ... a weakness for the same female students.

Wilkosz had an extraordinary talent for languages; not only did he master French and German very early on, he made extraordinary progress in Latin and Greek, mastered Esperanto in 3 hours, but he also studied Sanskrit, Hebrew and other Eastern languages ​​while still in secondary school.

... to promote all aspects of the knowledge of Asia and of the countries closely related to it ... the Society will deal not only with oriental literature but also with the history of these countries and the research of their situation in both earlier and more recent times.

... had to deal with the problem of composition of absolutely continuous functions. It turned out that the composition of two absolutely continuous functions does not have to be an absolutely continuous function, which may be surprising.

Even though Kraków was formally a fortress, one could take walks in Planty in the evenings. During such a walk I heard the words 'Lebesgue measure' - I went to the bench and introduced myself to two young adherents of mathematics. They told me that they had another companion, Witold Wilkosz, whom they highly praised. They were Stefan Banach and Otto Nikodym. Since then, we met regularly ...

At that time, the theory of absolutely continuous functions and Lebesgue integrals was a particularly important problem, because there was a lack of theorems on substitution in the Lebesgue integral, analogous to theorems for the Riemann integral. The work concerns only the Lebesgue integral on the real line. Its key concept is the concept of an absolutely continuous function. In all theorems on substitution proved in this work, the assumption of absolute continuity of the integrand is the most important, which is why the author precedes these theorems with a chapter devoted to the properties of absolutely continuous functions.

In 1918 there were three professors of mathematics at the Jagiellonian University: Stanisław Zaremba, Kazimierz Żorawski and Jan Sleszyński. Lectures were also given by Antoni Hoborski, Alfred Rosenblatt and Włodzimierz Stożek, soon joined by Leon Chwistek, Witold Wilkosz and Franciszek Leja. The standard curriculum, established before 1918, comprised mathematical analysis (differential and integral calculus), analytic geometry, introduction to higher mathematics, differential equations, differential geometry, theory of analytic functions. ... New subjects were added by Witold Wilkosz, who became an extraordinary professor in 1922: foundations of mathematics, set theory, theory of quadratic forms, theory of functions of a real variable, geometric topology, group theory. The course of studies could be concluded with either the teacher's examination or PhD examination. An additional subject was required at the examination (most commonly, physics was chosen).

When I came for my first year of studies at the university, there were no assistants. The so-called "proseminarium" [a pre-seminar], which preceded the recitations, was led by docent Leja. I was in this pre-seminar of his. He was a high school professor and had contract classes at the university. Only when I entered the second year, Wilkosz took care to have two deputy assistants nominated, that is Jan Józef Leśniak (1901-1980) and Irena Wilkosz (Wilkosz's wife, who was a mathematician). Then also Stanisław Turski (1906-1986) made it. Even later, there was Krystyn Zaremba (1903-1990) (Stanisław Zaremba's son).

The aim of this study is to present some theorems concerning non-measurable sets in the Lebesgue sense, and thus to pave the way for a general treatment.

The first investigations of properties belonging to the remarkable class of derivative functions are found in the work of Cauchy, the great founder of modern Differential Calculus. The study of them has been pursued by Duhamel and Dini, and has attained a very considerable development in our time especially by the work of Lebesgue and his successors. In the famous book of Lebesgue "Sur l'intégration" we can find nearly all results of the known properties of these functions and we are able to recognise how many questions are hitherto unsolved. The modest scope of my paper is to pursue in some points the considerations of the subject.

The purpose of this note is to demonstrate: If the series $f_{1}(x), f_{2}(x), .., f_{n}(x), ...$ which is convergent to a limited $f(x)$, has all the terms $f_{n}(x), n = 1, 2, ...$ Riemannian (R) then the necessary and sufficient condition for $f(x)$ to be (R) is the simply uniform convergence at "almost" every point of $[a, b]$ i.e. with a possible exception of a set of zero measure.

Witold Wilkosz was a very versatile mathematician. He published about 40 works in various branches of mathematics: set theory, logic, topology, measure theory, mathematical analysis, analytic functions, differential equations, mathematical physics and algebra. His favourite field was mathematical logic and the foundations of mathematics. ... He understood and appreciated the importance of applied mathematics well. B Średniawa [8] writes about his close cooperation with physicists. Many of W Wilkosz's works became the basis for further research by Polish and foreign mathematicians.

Witold Wilkosz was also an equally passionate and excellent lecturer. Working with young people gave him great joy. His lectures were very well attended. He taught a wide variety of topics in an innovative way: set theory, topology, theory of functions of real variables, vector analysis and algebra. He also gave lectures for physicists and a seminar on mathematical problems of atomic physics. He was the author of several university textbooks. He was passionate about teaching mathematics not only at the university level, but also at its earlier levels.

The systems of arithmetic discussed in this work are non-elementary theories. In this paper, natural numbers are characterised axiomatically in two different ways. We begin by recalling the classical set P of axioms of Peano's arithmetic of natural numbers proposed in 1889 (including such primitive notions as: set of natural numbers, zero, successor of natural number) and compare it with the set W of axioms of this arithmetic (including the primitive notions like: set of natural numbers and relation of inequality) proposed by Witold Wilkosz, a Polish logician, philosopher and mathematician, in 1932. The axioms W are those of ordered sets without largest element, in which every non-empty set has a least element, and every set bounded from above has a greatest element. We show that P and W are equivalent and also that the systems of arithmetic based on W or on P, are categorical and consistent.

From the moment the Kraków radio station was launched, i.e. from 15 February 1927, Wilkosz participated in its work. He became a member of the programme council. In the beginning of the station's activity, the programme was dominated by the simplest radio forms - lectures, readings and talks. Specialists from various fields were invited. The largest group of speakers were academics from the Jagiellonian University ...

From the moment it was launched, he supplied the Kraków Radio with talks on radio engineering, astronomy, and above all, he popularised mathematics in a great way. ... Witold Wilkosz, quicksilver in front of the microphone, direct and brilliant, conveyed a difficult subject, the intricate nature and beauty of mathematics in an intriguing way ...

His radio talks, conducted live with incredible eloquence and freedom, sometimes slightly improvised, won the special sympathy of the listeners. He never had the slightest stumble. He was allowed to appear in front of the microphone without a previously "read" text, which was a significant distinction.

Fascinated by radio engineering, Wilkosz worked on designing and improving receivers. Among other things, he designed a very simple and cheap single-tube receiver. This device went down in the history of radio engineering under the name of "Wilkosz's device" (see [12] for more details). Following its design, the first pre-war miniature receiver powered by pocket batteries was later designed, called "Vademecum", the prototype of the post-war "Szarotka". Wilkosz also built a two-tube receiver of his own design and published a brochure 'Miniaturowy odbiornik dwulampowy' Ⓣ. He founded a factory and at the same time a radio clinic called "Teleradio - radio equipment factory and research laboratory", located in Kraków at 18 Jana Street, where he personally gave all kinds of advice to radio amateurs, completely selflessly.

Fultography is the method of transmitting images at a distance using methods improved and brought to ideal simplicity by the English inventor Captain Otto Fulton.