... was a senior liaison assistant with the army (1925-1939), and at the same time he held the position of a visiting physician at the Dermatology Clinic of the Stefan Batory University and a senior liaison assistant with the army (1925-1939). At that time he was also the senior head physician and head of the Skin and Venereal Department of the Vilnius Fortified Area Hospital.

... for the non-German population of the East there can be no type of school above the four-grade rudimentary school. The job of these schools should be confined to the teaching of counting (no higher than up to 500), the writing of one's name, and the teaching that God's commandment means obedience to the Germans, honesty, industry and politeness. Reading I do not consider essential.

In the usual formulations of Peano's axioms for arithmetic, the axiom of induction must be formulated as a scheme containing an infinite number of proper axioms. An axiomatisation of arithmetic by means of a finite number of axioms can be achieved if one includes among the primitive notions of arithmetic e.g. the notion of sets or of propositional functions. In the present paper I shall discuss the question whether it is possible to obtain a finite axiomatisation of arithmetic, using only those primitive notions as are admitted ordinarily in Peano's system, that is: =, <, and an arbitrary number of arithmetical functions such as $x+y, x . y, x^{y}$ etc.

I shall show that no finite number of proper axioms, involving only these primitive terms, suffices to prove all the particular cases of the scheme of induction. Thus, Peano's arithmetic is not finitely axiomatizable if, only, the traditional primitive notions are allowed in the axioms.

From the methodological point of view, it may be interesting to note that the non-classical models of arithmetic (the existence of which was first proved by Skolem in 1934) are the chief tools used in my proof.

This paper is self-contained and all auxiliary theorems are explicitly stated and proved. The author believes that some of them may also prove useful in further investigations of related problems.

Following Skolem, we may interpret the notion of "a property" which occurs in the principle of mathematical induction for the positive integers as a predicate of one variable which can be expressed in the lower functional calculus in terms of the primitive relations and functors (some or all of the notions of equality, order, sum, product, etc.). Writing down the principle of induction successively for all these predicates, we obtain a sequence of axioms. If we add the usual rules of arithmetic and order, there results an infinite set of axioms for the system of positive integers. The present author establishes the important result that an equivalent finite set of axioms for the positive integers does not exist in the lower functional calculus, even if new functors are added as primitive notions. An important part in the proof is played by the non-normal models of axiomatic systems for positive integers whose existence was first established by Skolem. Given any finite system of axioms in the lower functional calculus, the author constructs a predicate of one variable which is inductive although it is not satisfied by all the elements of a certain non-normal model for the positive integers, as mentioned. The existence of such a predicate proves the main result.

In this review I will try to present how Czesław Ryll-Nardzewski contributed directly or indirectly to the development of ergodic theory. I consider the theorems and works directly related to issues from ergodic theory as direct contributions, while the indirect contribution is the application of his more general theorems to ergodic theory (by other authors). I will also mention results that, although they do not use Ryll-Nardzewski's theorems, were partly inspired by his question and, as a result, led to groundbreaking results in ergodic theory and beyond. It is not clear to what extent Ryll-Nardzewski's question was the real stimulus to undertake this research, since it could have developed quite independently. Nevertheless, the connection undoubtedly exists and is worth discussing.

Czesław Ryll-Nardzewski's contribution to the development of ergodic theory was extremely important. His most famous achievements in this area include the demonstration of ergodicity (with respect to the so-called Gaussian measure) of the transformation transforming a number from the interval [0,1] to the fractional part of its reciprocal, known as the Gaussian transformation, which has a direct impact on the expansion of a number into a continued fraction. The application of ergodic theory methods to the study of such expansions was innovative. This result is cited in almost every book on the subject of continued fractions.

Czesław Ryll-Nardzewski published over a hundred articles with a thematic diversity unheard of in today's times. CRN (in the Wrocław environment this abbreviation was quite commonly used as a simplification for the Professor's long name) wrote about issues from analysis, theory of stochastic processes, functional analysis, dynamic systems to general topology, set theory and logic. A significant part of CRN's scientific achievements concerns measure theory and this is the subject of this article. The following selection of works is not fully representative even in such a limited scope. I would like to mention the Professor's earlier works from the period of his first stay in Wrocław, in the early fifties, which were the result of cooperation with Edward Marczewski.

Czesław Ryll-Nardzewski (popularly known as CRN) was one of the most original and versatile mathematicians in post-war Poland. His elegant ideas, proofs, and discoveries in many areas of mathematics - including model theory, measure theory, probability theory, topology, and functional analysis - are widely considered legendary. In this article, we review his contributions to probability theory, starting with point processes and de Finetti sequences, and moving on to random functional processes and ergodic theory, and discuss the influence of his work on other mathematicians.

One of the most frequently cited results of CRN (according to the Mathematical Reviews) is the so-called Kuratowski-Ryll-Nardzewski's theorem on selectors proven. Although this theorem is not, strictly speaking, a probabilistic result, it has found diverse applications in statistics, stochastic geometry, stochastic games, stochastic control, and also in economics, deterministic and stochastic dynamic optimisation, and other areas. This theorem is fundamental in the theory of multivalued functions, random fixed point theorems for a multivalued contraction mappings, and the theory of martingales in Banach spaces. It is basic in stochastic geometry as it allows to define, for example, the expectation of a random set. An unexpected application was found in the proof of an extension of the Choquet-Bishop-de Leeuw theorem due to Gerry Edgar. Diestel and Uhl, describing his result, write "the proof is a beautiful mixture of martingale methods, ... a selection theorem of Kuratowski and Ryll-Nardzewski."

All of his works are characterised by elegance, which makes reading them a feast for the mathematician's soul. This is best described by the opinion of the commission that awarded him the title of full professor in 1964:

Ryll-Nardzewski's works are not a product of routine, even in the best sense of the word, the routine of the creative process in a restricted special area. Each of them is original through and through, new in terms of the result, new in terms of the method, new in comparison with the achievements of other scholars and the author himself. In Ryll-Nardzewski's work, one can rarely find proofs burdened with long calculations or laborious technique, almost always the reasoning is light and shiny, based on theorems, which the author's insight and erudition allow him to find in various fields of mathematics, sometimes seemingly distant from the topic of the work, and effectively apply to the solution of the problem under investigation. Banach defined a good mathematician as one who sees connections between ideas, between theories. It would be extremely beneficial to apply this definition to Professor Ryll-Nardzewski.

... he showed civic courage, maintaining honesty and a dignified attitude. Among other things, he presided over the underground master's examination of Rafał Dutkiewicz.

The above list of awards could be at least a few items longer, if it were not for the widely known aversion of Professor Ryll-Nardzewski both to any kind of celebration of his person, as well as to advertising the results of his research. In particular, he rarely participated in conferences, and even more rarely reported his own results. He did not care about the rank of the journals in which he published his works, publishing them mainly in Polish journals. Probably all this contributed to the fact that the Professor's contribution to the development of mathematics is poorly realised by all mathematicians, including Polish ones, and therefore underestimated.

It was said that his genius was that when someone explained something completely new to him, Professor Ryll-Nardzewski immediately understood it better than the explainer.