Totalities of Infinite Sets

Abstract

Four constructions result from a desire to create enhancements to Cantor's infinite real set cardinality. Each continues to keep Cantor's cardinality formulation in place while providing new comparisons of arbitrary infinite sets. To distinguish their features from a single determinant of a bijection between sets, three of the constructions are characterized by the term "totality" in place of cardinality. We use outer measure in one construction to attain a comparison between subsets of an arbitrary metric space $X$ vis-a-vis subsets of the power set $\mathbb{P}(X)$.