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

TL;DR

This work shows that a partition of $\mathbb{R}^3$ into unit circles (PUC) can exist in models of $\mathsf{ZF}$ without a well-ordering of the reals, by developing a general forcing framework that yields $L(\mathbb{R}, \mathcal{P})$ models with $\mathsf{DC}$ and without $\mathsf{WO}(\mathbb{R})$. It proves a flexible setup: start from a ground model $V$, add reals with a homogeneous forcing $\mathbf{Q}$, force a paradoxical partition $\mathcal{P}$ with a real-absolute, $\sigma$-closed forcing $\mathbf{P}$ that satisfies extendability and amalgamation and is $\mathbf{Q}$-balanced, and obtain $L(\mathbb{R}, \mathcal{P})^{V[g,h]}$ with the desired properties. The paper applies this to PUCs, proving a PUC exists in the Cohen model, thereby separating the existence of a PUC from Countable Choice. It also develops a Cohen-model extension of the PUC forcing and closes with an appendix showing the framework yields existing results for Hamel bases and Mazurkiewicz sets, highlighting the method’s versatility for generating paradoxical sets under minimal choice assumptions.

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.

Figures (6)

• Figure 1: The condition $p\in \mathbf{P}$ is $\mathbf{Q}$-balanced. The compatibility of $p_1$ and $p_2$ is witnessed by $p^\ast$.
• Figure 2: For each point $t$ that we want to avoid, there are two options for $o_\beta$ that we have to discard.
• Figure 3: $t$ is one of the (at most two) intersection points of the only unit circle $\tilde{C}$ given by $(o, \pi)$ and the plane $\pi_\beta$, and $o_\beta$ is then one of the (at most two) points in $\pi_\beta$ such that $d(o_\beta,x_\beta)=d(o_\beta,t)=1$.
• Figure 4: Two circles from different models intersecting in two points $t$ and $u$.
• Figure 5: There are circles $C$ and $C_1$ with parameters in $V[x,y]$ such that $C\cap C_2=\{u\}$, $C_1 \cap C_2=\{t\}$, and $u\neq t$.

Theorems & Definitions (49)

• Theorem A: see Theorem \ref{['th: partition no wo']}
• Theorem B: see Theorem \ref{['th: PUC - no wo']}
• Theorem C: see Theorem \ref{['th: PUC in Cohen model']}
• Definition 2.1
• Lemma 2.2
• Theorem 2.3: Grigorieff1975
• Theorem 2.4: The Solovay basis result Grigorieff1975
• Lemma 2.5
• Lemma 2.6
• Definition 2.7