From Boolean Valued Analysis to Quantum Set Theory: Mathematical Worldview of Gaisi Takeuti
Gaisi Takeuti introduced Boolean valued analysis around 1974 to provide systematic applications of the Boolean valued models of set theory to analysis. Later, his methods were further developed by his followers, leading to solving several open problems in analysis and algebra. Using the methods of B...
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MDPI AG
2021-02-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/9/4/397 |
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| author | Masanao Ozawa |
| author_facet | Masanao Ozawa |
| author_sort | Masanao Ozawa |
| collection | DOAJ |
| description | Gaisi Takeuti introduced Boolean valued analysis around 1974 to provide systematic applications of the Boolean valued models of set theory to analysis. Later, his methods were further developed by his followers, leading to solving several open problems in analysis and algebra. Using the methods of Boolean valued analysis, he further stepped forward to construct set theory that is based on quantum logic, as the first step to construct "quantum mathematics", a mathematics based on quantum logic. While it is known that the distributive law does not apply to quantum logic, and the equality axiom turns out not to hold in quantum set theory, he showed that the real numbers in quantum set theory are in one-to-one correspondence with the self-adjoint operators on a Hilbert space, or equivalently the physical quantities of the corresponding quantum system. As quantum logic is intrinsic and empirical, the results of the quantum set theory can be experimentally verified by quantum mechanics. In this paper, we analyze Takeuti’s mathematical world view underlying his program from two perspectives: set theoretical foundations of modern mathematics and extending the notion of sets to multi-valued logic. We outlook the present status of his program, and envisage the further development of the program, by which we would be able to take a huge step forward toward unraveling the mysteries of quantum mechanics that have persisted for many years. |
| first_indexed | 2024-03-09T00:47:41Z |
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| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-09T00:47:41Z |
| publishDate | 2021-02-01 |
| publisher | MDPI AG |
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| series | Mathematics |
| spelling | doaj.art-71c7e2a892014436a476e6d943df5ba72023-12-11T17:25:41ZengMDPI AGMathematics2227-73902021-02-019439710.3390/math9040397From Boolean Valued Analysis to Quantum Set Theory: Mathematical Worldview of Gaisi TakeutiMasanao Ozawa0College of Engineering, Chubu University, 1200 Matsumoto-cho, Kasugai 487-8501, JapanGaisi Takeuti introduced Boolean valued analysis around 1974 to provide systematic applications of the Boolean valued models of set theory to analysis. Later, his methods were further developed by his followers, leading to solving several open problems in analysis and algebra. Using the methods of Boolean valued analysis, he further stepped forward to construct set theory that is based on quantum logic, as the first step to construct "quantum mathematics", a mathematics based on quantum logic. While it is known that the distributive law does not apply to quantum logic, and the equality axiom turns out not to hold in quantum set theory, he showed that the real numbers in quantum set theory are in one-to-one correspondence with the self-adjoint operators on a Hilbert space, or equivalently the physical quantities of the corresponding quantum system. As quantum logic is intrinsic and empirical, the results of the quantum set theory can be experimentally verified by quantum mechanics. In this paper, we analyze Takeuti’s mathematical world view underlying his program from two perspectives: set theoretical foundations of modern mathematics and extending the notion of sets to multi-valued logic. We outlook the present status of his program, and envisage the further development of the program, by which we would be able to take a huge step forward toward unraveling the mysteries of quantum mechanics that have persisted for many years.https://www.mdpi.com/2227-7390/9/4/397TakeutiBoolean algebrasset theoryBoolean valued modelsforcingcontinuum hypothesis |
| spellingShingle | Masanao Ozawa From Boolean Valued Analysis to Quantum Set Theory: Mathematical Worldview of Gaisi Takeuti Mathematics Takeuti Boolean algebras set theory Boolean valued models forcing continuum hypothesis |
| title | From Boolean Valued Analysis to Quantum Set Theory: Mathematical Worldview of Gaisi Takeuti |
| title_full | From Boolean Valued Analysis to Quantum Set Theory: Mathematical Worldview of Gaisi Takeuti |
| title_fullStr | From Boolean Valued Analysis to Quantum Set Theory: Mathematical Worldview of Gaisi Takeuti |
| title_full_unstemmed | From Boolean Valued Analysis to Quantum Set Theory: Mathematical Worldview of Gaisi Takeuti |
| title_short | From Boolean Valued Analysis to Quantum Set Theory: Mathematical Worldview of Gaisi Takeuti |
| title_sort | from boolean valued analysis to quantum set theory mathematical worldview of gaisi takeuti |
| topic | Takeuti Boolean algebras set theory Boolean valued models forcing continuum hypothesis |
| url | https://www.mdpi.com/2227-7390/9/4/397 |
| work_keys_str_mv | AT masanaoozawa frombooleanvaluedanalysistoquantumsettheorymathematicalworldviewofgaisitakeuti |