On the Compatibility Between the Minimalist Foundation and Constructive Set Theory
The Minimalist Foundation MF $$\mathbf {MF}$$ was ideated by M.E. Maietti and G. Sambin and then completed as a formal system by M.E. Maietti in order to provide a foundation for constructive mathematics compatible with the main classical and intuitionistic, predicative and impredicative, foundation...
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| Published in | Revolutions and Revelations in Computability Vol. 13359; pp. 172 - 185 |
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| Main Authors | , |
| Format | Book Chapter |
| Language | English |
| Published |
Switzerland
Springer International Publishing AG
2022
Springer International Publishing |
| Series | Lecture Notes in Computer Science |
| Online Access | Get full text |
| ISBN | 3031087399 9783031087394 |
| ISSN | 0302-9743 1611-3349 |
| DOI | 10.1007/978-3-031-08740-0_15 |
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| Summary: | The Minimalist Foundation MF $$\mathbf {MF}$$ was ideated by M.E. Maietti and G. Sambin and then completed as a formal system by M.E. Maietti in order to provide a foundation for constructive mathematics compatible with the main classical and intuitionistic, predicative and impredicative, foundational theories. Here we show that MF $$\mathbf {MF}$$ is in fact compatible with Aczel’s constructive set theory CZF $$\mathbf {CZF}$$ . We prove this by extending the extensional level of MF $$\mathbf {MF}$$ with rules obtaining a system which turns out to be equivalent to CZF $$\mathbf {CZF}$$ . |
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| Bibliography: | Original Abstract: The Minimalist Foundation MF\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {MF}$$\end{document} was ideated by M.E. Maietti and G. Sambin and then completed as a formal system by M.E. Maietti in order to provide a foundation for constructive mathematics compatible with the main classical and intuitionistic, predicative and impredicative, foundational theories. Here we show that MF\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {MF}$$\end{document} is in fact compatible with Aczel’s constructive set theory CZF\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {CZF}$$\end{document}. We prove this by extending the extensional level of MF\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {MF}$$\end{document} with rules obtaining a system which turns out to be equivalent to CZF\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {CZF}$$\end{document}. |
| ISBN: | 3031087399 9783031087394 |
| ISSN: | 0302-9743 1611-3349 |
| DOI: | 10.1007/978-3-031-08740-0_15 |