Hausdorff Measure: Lost in Translation

TL;DR

This paper explores a novel approach to defining Hausdorff measure using only countable coverings and discusses how different interpretations of 'countable set' can lead to contradictions in mathematical theory.

Contribution

It introduces a new definition of Hausdorff measure allowing empty coverings and analyzes the impact of varying interpretations of 'countable set' on measure theory.

Findings

Hausdorff measure can be defined with only countable coverings and empty elements

Different interpretations of 'countable set' can cause contradictions in measure theory

The paper clarifies the implications of these interpretations for mathematical consistency

Abstract

In the present article we describe how one can define Hausdorff measure allowing empty elements in coverings, and using infinite countable coverings only. In addition, we discuss how the use of different nonequivalent interpretations of the notion "countable set", that is typical for classical and modern mathematics, may lead to contradictions.