# Bounded inductive dichotomy: separation of open and clopen determinacies with finite alternatives in constructive contexts

Bounded inductive dichotomy: separation of open and clopen determinacies with finite alternatives... In his previous work, the author has introduced the axiom schema of inductive dichotomy, a weak variant of the axiom schema of inductive definition, and used this schema for elementary (Δ01\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\varDelta ^1_0$$\end{document}) positive operators to separate open and clopen determinacies for those games in which two players make choices from infinitely many alternatives in various circumstances. Among the studies on variants of inductive definitions for bounded (Δ00\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\varDelta ^0_0$$\end{document}) positive operators, the present article investigates inductive dichotomy for these operators, and applies it to constructive investigations of variants of determinacy statements for those games in which the players make choices from only finitely many alternatives. As a result, three formulations of open determinacy, that are all classically equivalent with each other, are equivalent to three different semi-classical principles, namely Markov’s Principle, Lesser Limited Principle of Omniscience and Limited Principle of Omniscience, over a suitable constructive base theory that proves clopen determinacy. Open and clopen determinacies for these games are thus separated. Some basic results on variants of inductive definitions for Δ00\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\varDelta ^0_0$$\end{document} positive operators will also be given. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Archive for Mathematical Logic Springer Journals

# Bounded inductive dichotomy: separation of open and clopen determinacies with finite alternatives in constructive contexts

, Volume OnlineFirst – Oct 16, 2021
37 pages

/lp/springer-journals/bounded-inductive-dichotomy-separation-of-open-and-clopen-FRx731fJ3N
Publisher
Springer Journals
Copyright © The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2021
ISSN
0933-5846
eISSN
1432-0665
DOI
10.1007/s00153-021-00795-2
Publisher site
See Article on Publisher Site

### Abstract

In his previous work, the author has introduced the axiom schema of inductive dichotomy, a weak variant of the axiom schema of inductive definition, and used this schema for elementary (Δ01\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\varDelta ^1_0$$\end{document}) positive operators to separate open and clopen determinacies for those games in which two players make choices from infinitely many alternatives in various circumstances. Among the studies on variants of inductive definitions for bounded (Δ00\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\varDelta ^0_0$$\end{document}) positive operators, the present article investigates inductive dichotomy for these operators, and applies it to constructive investigations of variants of determinacy statements for those games in which the players make choices from only finitely many alternatives. As a result, three formulations of open determinacy, that are all classically equivalent with each other, are equivalent to three different semi-classical principles, namely Markov’s Principle, Lesser Limited Principle of Omniscience and Limited Principle of Omniscience, over a suitable constructive base theory that proves clopen determinacy. Open and clopen determinacies for these games are thus separated. Some basic results on variants of inductive definitions for Δ00\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\varDelta ^0_0$$\end{document} positive operators will also be given.

### Journal

Archive for Mathematical LogicSpringer Journals

Published: Oct 16, 2021

Keywords: Inductive definition; Bounded positive operator; Constructive reverse mathematics; Semi-classical principles; Markov’s principle; Lesser limited principle of omniscience; primary 03D70; 03E65; 03F35; 03F55; secondary 03D30; 03F50