An effective analysis of the denjoy rank

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Abstract

We analyze the descriptive complexity of several ∏1 1-ranks from classical analysis which are associated to Denjoy integration. We show that VBG, VBG*, ACG, and ACG* are ∏1 1-complete, answering a question of Walsh in case of ACG*. Furthermore, we identify the precise descriptive complexity of the set of functions obtainable with at most α steps of the transfinite process of Denjoy totalization: if ∣·∣ j is the ∏1 1-rank naturally associated to VBG, VBG*, or ACG*, and if (equation presented)-complete. These finer results are an application of the author's previous work on the limsup rank on well-founded trees. Finally, (equation presented) is Denjoy integrableºare ∏1 1-complete, answering more questions of Walsh.

Original languageEnglish (US)
Pages (from-to)245-263
Number of pages19
JournalNotre Dame Journal of Formal Logic
Volume61
Issue number2
DOIs
StatePublished - May 2020

All Science Journal Classification (ASJC) codes

  • Logic

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