# Transcendence bases, well-orderings of the reals and the axiom of choice

## TL;DR

This paper demonstrates the relative consistency of the existence of a transcendence basis for the reals without a well-ordering of the reals, addressing a question in set theory and the axiom of choice.

## Contribution

It proves the consistency of having a transcendence basis for the reals while lacking a well-ordering, resolving a question posed by Larson and Zapletal.

## Key findings

- Consistency of $ZF+DC$ with a transcendence basis for reals
- Non-existence of a well-ordering of the reals in the same model
- Addresses a longstanding open question in set theory

## Abstract

We prove that $ZF+DC+"$there exists a transcendence basis for the reals$"+"$there is no well-ordering of the reals$"$ is consistent relative to $ZFC$. This answers a question of Larson and Zapletal.