Todorčević, Stevo Hierarchies of Ackermann, Baire and von Neumann. (English) Zbl 1003.03041 Proceedings of the 10th congress of Yugoslav mathematicians, Belgrade, Yugoslavia, January 21-24, 2001. Belgrade: University of Belgrade, Faculty of Mathematics. 11-20 (2001). The paper is a survey (without proofs) on some results and problems related to one of the hierarchies from the title. Chapter 1 contains results on estimating the rate of growth of certain functions arising naturally from some famous theorems (van der Waerden on arithmetic progression, Mittag-Leffler theorem), by functions related to the Ackermann hierarchy of fast-growing functions. In Chapter 2 the impact of a large-cardinal axiom (there exists a nontrivial elementary embedding of some \(V_\alpha\) into itself) to Artin’s braid group and some related problems are discussed. Chapter 3 contains some of the author’s recent results on whether a measurable real function has a continuous restriction to some \(\zeta\)-set (\(\zeta:N \to N\)).For the entire collection see [Zbl 0981.00012]. Reviewer: Predrag Tanović (Beograd) MSC: 03D55 Hierarchies of computability and definability 03-02 Research exposition (monographs, survey articles) pertaining to mathematical logic and foundations 03D20 Recursive functions and relations, subrecursive hierarchies 11B25 Arithmetic progressions 03E55 Large cardinals 20F36 Braid groups; Artin groups 28A20 Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence 03F30 First-order arithmetic and fragments Keywords:Davenport-Schinzel sequences; primitive recursive arithmetic; Ackermann function; growth rates; survey PDFBibTeX XMLCite \textit{S. Todorčević}, in: Proceedings of the 10th congress of Yugoslavian mathematicians, Belgrade, Yugoslavia, January 21--24, 2001. Belgrade: University of Belgrade, Faculty of Mathematics. 11--20 (2001; Zbl 1003.03041)