Iterated failures of choice (1911.09285v3)

Abstract: We combine several folklore observations to provide a working framework for iterating constructions which contradict the axiom of choice. We use this to define a model in which any kind of structural failure must fail with a proper class of counterexamples. For example, the rational numbers have a proper class of non-isomorphic algebraic closures, every partial order embeds into the cardinals of the model, every set is the image of a Dedekind-finite set, every weak choice axiom of the form $\mathsf{AC}_X^Y$ fails with a proper class of counterexamples, every field has a vector space with two linearly independent vectors but without endomorphisms that are not scalar multiplication, etc.

• The paper develops a framework using iterated symmetric extensions to construct models where the Axiom of Choice fails extensively and irredeemably across various mathematical structures.
• Key findings demonstrate how these non-choice models enable constructions of structures like vector spaces with trivial endomorphisms and fields with non-isomorphic algebraic closures.
• The research explores non-trivial cardinal structures, Dedekind-finiteness, and group structures under non-choice assumptions, impacting foundational understanding across mathematics.

Iterated Failures of Choice: An Exploration of Framework for Non-Choice Constructions

The paper "Iterated Failures of Choice" by Asaf Karagila presents an advanced framework to explore the limitations imposed by the absence of the Axiom of Choice (AC) in set theory. The research establishes a method to iteratively construct models where the AC fails extensively, providing a structured way to investigate the implications of non-choice scenarios on a broad spectrum of mathematical structures.

Several key insights are cemented in this work, primarily accomplished through the framework of iterated symmetric extensions. These extensions, built upon symmetric forcing, allow for the crafting of models where the AC not only fails but does so with comprehensive ramifications across different mathematical entities. The paper focuses on demonstrating the intensity of these failures, going beyond traditional models to explore their extensive iterations, which are notably robust against AC restoration efforts due to the need for ordinal addition.

Key Contributions and Findings

1. Framework for Iterations: Karagila systematically combines folklore observations and formalises a framework that enables iterations of symmetric extensions. This framework answers, "how bad can it get?" by striving for models where the AC is irredeemably lost, thereby necessitating the introduction of ordinals to restore it.
2. Structural Failures in Mathematical Entities: The paper tackles issues like the construction of vector spaces over fields devoid of non-trivial endomorphisms except for scalar multiplications, and fields with non-isomorphic algebraic closures. These constructions often employ the idea of symmetric copies, which are manipulated to inhibit certain choice-dependent properties.
3. Non-trivial Cardinal Structures: An exploration into embedding partial orders into cardinals is presented, significantly expanding previous results by showing that any given partial order can manifest in the cardinal structure across the model. This exploration questions prior limits on the comparability of cardinals and fundamentally contradicts the necessity of the AC for cardinality ordering.
4. Dedekind-Finiteness and Countable Unions: An impressive construct emerges where every set can be seen as an image of a Dedekind-finite set. Similarly, the iterations include sets that are countable unions of countable sets having power sets that map onto large portions of the universe, a condition steep with implications on collection principles like Kinna–Wagner Principles.
5. Cohomology and Group Structures: Through nonabelian cohomology, the paper establishes scenarios where the typical application of choice is obstructed. Non-trivialities within structures like G-torsors over inhomogeneous sets are thereby constructed, leveraging symmetric extensions.

Implications and Future Directions

Practically, the insights offer profound implications for how set theory is interpreted in contexts without choice. The research lays the groundwork for further exploration into dependent choice principles and large cardinal axioms under non-choice assumptions. It poses questions on their interaction with iterated extensions and potential preservation insights that could lead to deeper understanding.

Theoretically, the results implicate foundational understanding across mathematics, particularly in demonstrating that many classical assertions can remain unaffected by choice, while some, like algebraic structural implications, demand re-evaluation. This connects to broader domains such as ring theory, topology, and possibly analytic contexts, especially in understanding how these constructs could extend to affect classical analysis or category theory.

The framework proposed by Karagila spotlights a paradigm where choice-based methodologies are scrutinised. Future work might leverage this to develop additional set theoretic constructs or further elucidation of large cardinal implications, as well as connections to algebraic axioms and their intrinsic properties outside well-ordered universes.

Overall, "Iterated Failures of Choice" stands as a seminal foray into non-choice set theory, presenting a comprehensive framework that intertwines previously isolated notions into a coherent system, pushing beyond classical boundaries of choice-related constructs.