Applications of Multivalued Logic to Set Theory and Calculus
Ravi Ahmad Salim, Ravi Ahmad Salim · Journal of Hunan University Natural Sciences · 2023
The objectives of this paper are to make set theory a step farther from catastrophic contradictions such as the Russell paradox, which blow away some collections of objects such as the set of all sets, which intuitively should exist since their members exist, and make use of multivalued logic in developing non-standard models in set theory and calculus to make work easier. The set theoretic contradictions are derived through two-valued logic, which ends with sentences of the form “P and not P,” which always have the value “false.” By introducing other truth values, such sentences can end up with a non-false value if both “P” and “not P” are not false. Non-standard objects are obtained through collections of ordinary objects indicating that none of the elements are meant, whereas what is more desirable is indicated. For example, a positive number smaller than all positive real numbers can be indicated by a set of positive real numbers with values ever approaching but never truly attaining “true” as they approach zero. In this way, infinite sets can be indicated by multivalued collections of finite sets, and both infinite and infinitesimal numbers are reintroduced to calculus, improving the scope of integration and differentiation. This paper also presents a new way to accommodate truth values for set theory where the objects can be far more numerous than real numbers between zero and one representing values between “false” and “true.” Keywords: multivalued logic, the Zermelo-Fraenkel set theory, equipollence classes, truth values, multivalued derivatives and integrals, sums of arbitrary series. https://doi.org/10.55463/issn.1674-2974.50.12.5