Partitions of $\mathbb{R}^3$ into unit circles with no well-ordering of the reals (2501.03131v1)

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.

• The paper demonstrates the existence of a partition of R³ into unit circles in models of set theory lacking a well-ordering of the reals, such as the Cohen model.
• The author uses forcing with the finite support product of Cohen forcing to construct these partitions in models satisfying ZF + DC + ¬WO(R).
• This research explores how complex geometric structures exist without certain forms of the Axiom of Choice and contributes to understanding paradoxical sets.

In the paper titled "Partitions of $R^3$ into Unit Circles with No Well-Ordering of the Reals" by Azul Fatalini, the author explores the intriguing relationship between paradoxical sets in Euclidean spaces and the Axiom of Choice (AC). Specifically, the paper investigates whether it is possible to partition the three-dimensional real space, $R^3$, into unit circles without assuming a well-ordering of the reals, a question bound by the constraints of set theory and topology.

The paper is set within models of set theory derived from $\mathsf{ZF}$ (Zermelo-Fraenkel set theory without AC), with particular attention to the Principle of Dependent Choices ($\mathsf{DC}$) and different forms of the Axiom of Choice, such as countable choice ($\mathsf{AC}_\omega$). The central result shows the existence of a partition of $R^3$ into unit circles in the Cohen model and some models satisfying $\mathsf{ZF} + \mathsf{DC} + \neg \mathsf{WO}(R)$, where $\mathsf{WO}(R)$ denotes a well-order of the reals.

Key Contributions and Methodology:

1. The Cohen Model Application: Demonstrating the existence of a Partition of Unit Circles (PUC) within the Cohen model, which is known to violate standard forms of the Axiom of Choice, such as countable choice, is a significant accomplishment. The partition in this case implies that while $\mathsf{AC}_\omega$ fails, it's still feasible to construct highly non-trivial geometric entities in such a model.
2. Forcing Technique with Special Partitions: The author employs a powerful set-theoretical method known as forcing with the finite support product of Cohen forcing. This technique is adapted to construct models with particular partitions by ensuring that these partitions extend beyond any partial conditions — demonstrating that partitions can be amalgamated for algebraic closure.
3. General Framework for Paradoxical Sets: The establishment of a more general framework to construct models where other types of paradoxical sets can exist without certain forms of AC is explored. This notably involves leveraging properties like extendability and amalgamation within the forcing extensions, both crucial for establishing non-trivial partitions.
4. Preliminary Results and Extensions: By exploring the consequences of not having a well-ordered real line, but instead asserting particular choice principles like $\mathsf{DC}$, the article provides a nuanced take on canonicity and independence within set-theoretical extensions. This includes an analysis of how other geometrical paradoxical sets, such as Mazurkiewicz sets, might fit into similar or more general frameworks.
5. Analytical and Co-analytical Extensions: The paper verifies properties of PUCs such as their non-Borel nature unless analytically defined and extends results from models like $V=L$ (the constructible universe) to yield coanalytic PUCs, reflecting advances in descriptive set theory's interaction with classical set-theoretical paradoxes.

In conclusion, Fatalini's work presents an elegant, rigorous exploration of how very complex geometrical constructs can be unearthed in models of set theory that purposefully distance themselves from certain forms of choice. This research notably contributes to the understanding of paradoxical sets, advancing thought on how AC influences such objects' existence, properties, and the fine interplay of set theory, logic, and topology.