"Discussions on Two Major Problems in Computation and Contradiction"
Weihan Huang · International Journal of Software & Hardware Research in Engineering · 2022
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.