A model with everything except for a well-ordering of the reals

TL;DR

The paper constructs a model of set theory with specific subsets like Luzin and Sierpiński sets and a Burstin basis, but without a well-ordering of the continuum, highlighting independence results in set theory.

Contribution

It demonstrates the consistency of having certain special sets and bases without a well-ordering of the continuum in ZF + DC.

Findings

Existence of Luzin and Sierpiński sets in the model

Presence of a Burstin basis without a well-ordering

Model satisfies ZF + DC but lacks a well-ordering of the continuum

Abstract

We construct a model of $\mathsf{ZF} + \mathsf{DC}$ containing a Luzin set, a Sierpi\'{n}ski set, as well as a Burstin basis but in which there is no a well ordering of the continuum.