Discussions on Two Major Problems in Computation and Contradiction

Abstract

Russell's paradox shows that the set of all sets does not exist, which gives an example of set theoretical paradox. To resolve a version of this paradox, Russell proposed a theory of types, and then the type hierarchy to infinity appeared in the resolution of the paradox. Victor gave a solution to remove the infinite type hierarchy by  proposing his Duality Principle, which not only can solve Russell's paradox, but also can solve the Liar's paradox. In this paper, I contend the completeness of meanings in Duality Principle. Firstly, in solving the Russell's paradox, I assert that the complete meaning of a set should contain Miscellaneous Sets such as A={a,{b}}, in addition to Pure Sets {a,b,c} and Set of Sets {{a},{{b},c}}. And this will require solution of Russel's paradox by Duality principle change its disjunction condition to have 3 conditions. i.e A set is either a Pure Set, or a Set of Sets, or a Miscellaneous Set. Then his original solution can not follow these new conditions of the Duality Principle. Secondly, in resolving the Liar's paradox by Victor's Duality Principle, I argue that the completeness of meaning of a statement, in addition to {a statement, a statement about statements}, there should be also { a statement about (statement about statements) }. It means that the original meaning of a statement is incomplete, there are more meanings of a statement such that it is a complete set of meanings. However, this will push us to Russell's infinite type hierarchy which Victor tried to avoid. Therefore, I myself would like to give the meaning of a statement in Liar's paradox as a Boolean function, which in turn renders the Dialetheism.