Totalities of Infinite Sets

TL;DR

The paper introduces four new constructions related to Cantor's infinite set cardinality, emphasizing 'totality' as an alternative comparison method for infinite sets, including one using outer measure for metric space subsets.

Contribution

It presents novel set comparison frameworks that extend Cantor's cardinality, notably replacing it with 'totality' and incorporating outer measure techniques.

Findings

Four new constructions related to infinite set comparison.

Three constructions use 'totality' instead of cardinality.

One construction employs outer measure for metric space subsets.

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 $\mathbb{P}(X)$.