We analyze the descriptive complexity of several -ranks from classical analysis which are associated to Denjoy integration. We show that , , , and are -complete, answering a question of Walsh in case of . Furthermore, we identify the precise descriptive complexity of the set of functions obtainable with at most steps of the transfinite process of Denjoy totalization: if is the -rank naturally associated to , , or , and if , then is -complete. These finer results are an application of the author’s previous work on the limsup rank on well-founded trees. Finally, and are -complete, answering more questions of Walsh.
"An Effective Analysis of the Denjoy Rank." Notre Dame J. Formal Logic 61 (2) 245 - 263, May 2020. https://doi.org/10.1215/00294527-2020-0006