Authors: Feng Xu
Comments: 6 pages, first published in 2006 in Hadronic Journal, volume 29, page 227
The set of all the subsets of a set is its power set, and the cardinality of the power set is always larger than the set and its subsets. Based on the definition and the inequality in cardinality, a set cannot include its power set as element, and a power set cannot include itself as element. "Russell's set" is a putative set of all the sets that don't include themselves as element. It can be shown, however, that "Russell's set" can never take in all such sets. This is because its own power set, which (like any power set) is a set that doesn't include itself (thus qualifies as an element for "Russell's set"), cannot (although should) be taken in due to the cardinality inequality. Thus "Russell's set" can never be formed. Without it, Russell's paradox, which forced the modification of Cantor's intuitive set theory into a more restricted axiomatic theory, can never be formulated. The reported approach to resolve Russell's paradox is fundamentally different from the conventional approaches. It may restore the self-consistency of Cantor's original set theory, make the Axiom of Regularity unnecessary, and expand the coverage of set to assemblies that include themselves as element.
Category: Set Theory and Logic