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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].

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
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