# Definable Hamel bases and $AC_\omega(R)$

## TL;DR

This paper constructs a model of ZF set theory where a definable Hamel basis exists but the countable axiom of choice for reals ($AC_(R)$) fails, highlighting the independence of these concepts.

## Contribution

It demonstrates the existence of a ZF model with a definable Hamel basis where $AC_(R)$ does not hold, showing their independence.

## Key findings

- Existence of a $_3$ definable Hamel basis in a ZF model.
- Failure of $AC_(R)$ in the same model.
- Illustration of the independence between definable bases and choice axioms.

## Abstract

There is a model of ZF with a $\Delta^1_3$ definable Hamel basis in which $AC_\omega(R)$ fails.