Partitions of groups and matroids into independent subsets

TL;DR

This paper investigates whether the real line minus zero can be covered by countably many independent subsets over rationals, using matroid theory to show the answer depends on the Continuum Hypothesis.

Contribution

It introduces a matroid-based approach to connect set coverings with the Continuum Hypothesis, providing a novel perspective on independence in real line subsets.

Findings

Under CH, the real line minus zero can be covered by countably many independent subsets.

Without CH, such a covering is impossible.

The result links set theory hypotheses to combinatorial properties of the real line.

Abstract

Can the real line with removed zero be covered by countably many linearly (algebraically) independent subsets over the field of rationals? We use a matroid approach to show that an answer is "Yes" under the Continuum Hypothesis, and "No" under its negation.