Abstract
In 1981, Takeuti introduced set theory based on quantum logic by constructing a model analogous to Boolean-valued models for Boolean logic. He defined the quantum logical truth value for every sentence of set theory. He showed that equality axioms do not hold, while axioms of ZFC set theory hold if appropriately modified with the notion of commutators. Here, we consider the problem in Takeuti’s quantum set theory that De Morgan’s laws do not hold for bounded quantifiers. We construct a counter-example to De Morgan’s laws for bounded quantifiers in Takeuti’s quantum set theory. We redefine the truth value for the membership relation and bounded existential quantification to ensure that De Morgan’s laws hold. Then, we show that the truth value of every theorem of ZFC set theory is lower bounded by the commutator of constants therein as quantum transfer principle.
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Acknowledgements
The author acknowledges the support of the JSPS KAKENHI, No. 26247016, No. 17K19970, and the support of the IRI-NU collaboration. The author thanks the referee for calling his attention to the well-definedness of restrictions of quantum sets.
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Ozawa, M. (2021). Reforming Takeuti’s Quantum Set Theory to Satisfy de Morgan’s Laws. In: Arai, T., Kikuchi, M., Kuroda, S., Okada, M., Yorioka, T. (eds) Advances in Mathematical Logic. SAML 2018. Springer Proceedings in Mathematics & Statistics, vol 369. Springer, Singapore. https://doi.org/10.1007/978-981-16-4173-2_7
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DOI: https://doi.org/10.1007/978-981-16-4173-2_7
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