Reforming Takeuti's Quantum Set Theory to Satisfy De Morgan's Laws

TL;DR

This paper revises Takeuti's quantum set theory to ensure De Morgan's laws hold for bounded quantifiers, by redefining truth values, and demonstrates that ZFC theorems retain their truth bounds within this framework.

Contribution

It introduces a modified quantum set theory that satisfies De Morgan's laws and maintains the quantum transfer principle for ZFC theorems.

Findings

Counter-example to De Morgan's laws in original quantum set theory

Redefinition of truth values for membership and bounded quantifiers

Validation that ZFC theorems' truth values are preserved

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.