Partitions of $\mathbb{R}^3$ into unit circles with no well-ordering of the reals

TL;DR

The paper demonstrates that a partition of three-dimensional space into unit circles can exist without a well-ordering of the reals, challenging previous assumptions linking such partitions to well-orderings.

Contribution

It constructs models of ZF where a partition into unit circles exists without a well-ordering, and develops a framework for creating similar models for other paradoxical sets.

Findings

Cohen model contains a partition of space into unit circles without well-ordering.

A model satisfying DC also admits such a partition.

Framework for constructing models with paradoxical sets under extendability and amalgamation conditions.

Abstract

Using a well-ordering on the reals, one can prove there exists a partition of the three-dimensional Euclidean space into unit circles (PUC). We show that the converse does not hold: there exist models of $\mathsf{ZF}$ without a well-ordering of the reals in which such partition exists. Specifically, we prove that the Cohen model has a PUC and construct a model satisfying $\mathsf{DC}$ where this is also the case. Furthermore, we present a general framework for constructing similar models for other paradoxical sets, under some conditions of extendability and amalgamation.