Maximal sets without Choice

TL;DR

This paper demonstrates the consistency, relative to ZF, of the existence of various special sets of reals such as Hamel bases and Vitali sets without requiring a well-ordering of the reals, thus challenging traditional assumptions in set theory.

Contribution

It establishes new consistency results showing the existence of special sets of reals and maximal independent sets under weaker assumptions than the Axiom of Choice, extending previous work significantly.

Findings

Existence of special sets of reals without well-ordering of 

Consistency of maximal independent sets in projective hypergraphs

Results hold under  with or without -choice for reals

Abstract

We show that it is consistent relative to ZF, that there is no well-ordering of $\mathbb{R}$ while a wide class of special sets of reals such as Hamel bases, transcendence bases, Vitali sets or Bernstein sets exists. To be more precise, we can assume that every projective hypergraph on $\mathbb{R}$ has a maximal independent set, among a few other things. For example, we get transversals for all projective equivalence relations. Moreover, this is possible while either $\mathsf{DC}_{\omega_1}$ holds, or countable choice for reals fails. Assuming the consistency of an inaccessible cardinal, "projective" can even be replaced with "$L(\mathbb{R})$". This vastly strengthens earlier consistency results in the literature.