Ranks on the Baire class ξ functions

Elekes, Márton and Kiss, Viktor and Vidnyánszky, Zoltán (2016) Ranks on the Baire class ξ functions. TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, 368 (11). pp. 8111-8143. ISSN 0002-9947

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Abstract

In 1990 Kechris and Louveau developed the theory of three very natural ranks on the Baire class 1 functions. A rank is a function assigning countable ordinals to certain objects, typically measuring their complexity. We extend this theory to the case of Baire class ξ functions and generalize most of the results from the Baire class 1 case. We also show that their assumption of the compactness of the underlying space can be eliminated. As an application, we solve a problem concerning the so-called solvability cardinals of systems of difference equations, arising from the theory of geometric decompositions. We also show that certain other very natural generalizations of the ranks of Kechris and Louveau surprisingly turn out to be bounded in ω1. Finally, we prove a general result showing that all ranks satisfying some natural properties coincide for bounded functions. © 2016 American Mathematical Society.

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