Nonarchimedean groups and the axiom of choice

Jindrich Zapletal; Justin Young

arxiv: 2606.23618 · v1 · pith:K3XPXAHFnew · submitted 2026-06-22 · 🧮 math.LO · math.GR

Nonarchimedean groups and the axiom of choice

Justin Young , Jindrich Zapletal This is my paper

Pith reviewed 2026-06-26 06:11 UTC · model grok-4.3

classification 🧮 math.LO math.GR

keywords nonarchimedean groupsaxiom of choicepermutation modelschoice fragmentsset theory

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The pith

Nonarchimedean groups determine which fragments of the axiom of choice hold in their permutation models.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes direct links between algebraic properties of nonarchimedean groups and the specific fragments of the axiom of choice that are valid in the permutation models constructed from those groups. These links allow translation of conditions on the groups into statements about which choice principles survive. A reader would care because the construction provides a systematic way to produce models of set theory with controlled failures of choice. The work focuses on proving several new such correspondences rather than revisiting known ones.

Core claim

We prove several novel connections between properties of nonarchimedean groups and fragments of the axiom of choice which hold in their associated permutation model.

What carries the argument

The permutation model associated to a given nonarchimedean group, which encodes the group's properties as choice principles that hold or fail inside the model.

If this is right

Algebraic properties of the group can be used to force the presence or absence of particular choice principles inside the model.
The correspondence yields new examples of models satisfying selected fragments of the axiom of choice.
Known group-theoretic conditions become tools for controlling independence results involving choice.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same technique might be applied to other classes of groups to generate additional controlled models.
The correspondences could be used to study the relative strength of different choice fragments by varying the input group.
It may be possible to reverse the construction and derive group properties from a given choice fragment in a model.

Load-bearing premise

The standard construction of permutation models from nonarchimedean groups preserves a direct correspondence between the group's properties and the choice fragments realized in the model.

What would settle it

A specific nonarchimedean group together with its associated permutation model in which one of the claimed group-to-choice correspondences fails to hold.

read the original abstract

We prove several novel connections between properties of nonarchimedean groups and fragments of the axiom of choice which hold in their associated permutation model.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Abstract claims novel links between nonarchimedean groups and AC fragments in permutation models but supplies zero details or proofs.

read the letter

The abstract states that the authors prove several novel connections between properties of nonarchimedean groups and fragments of the axiom of choice realized in the associated permutation models. That is the entire claim on offer.

If the full paper actually identifies concrete group properties that correspond to specific choice principles holding or failing in the models, the work could give set theorists another handle on constructing choiceless models with controlled behavior. That sort of explicit dictionary between group actions and choice fragments is occasionally useful inside mathematical logic.

The paper does well in choosing nonarchimedean groups as the vehicle, since these groups carry additional structure that might allow finer distinctions than the symmetric groups or other standard examples in permutation-model constructions.

The obvious soft spot is that nothing beyond the abstract claim is visible. There are no definitions of the groups in question, no list of the group properties under discussion, no description of how the permutation models are built from them, and no sample theorem. Without those pieces it is impossible to check whether the claimed connections are new, whether they reduce to earlier work on Fraenkel-Mostowski models, or whether the standard construction even preserves the intended correspondence. The soundness cannot be assessed at all.

This is aimed at a narrow audience of set theorists already working on choiceless ZF and permutation models. A reader deep in that literature might find value once the details appear, but on present evidence the paper does not rise to the level where a serious referee should spend time on it.

I would not bring the abstract to a reading group, would not cite the work, and would recommend desk rejection until the authors supply actual theorems and arguments.

Referee Report

1 major / 0 minor

Summary. The manuscript claims to prove several novel connections between properties of nonarchimedean groups and fragments of the axiom of choice which hold in their associated permutation model.

Significance. If the claimed connections can be established with explicit constructions and proofs, the work would link algebraic properties of groups to the failure or holding of choice principles in Fraenkel-Mostowski permutation models, potentially supplying new techniques for independence results in set theory. The abstract alone supplies no evidence that such links are parameter-free, non-circular, or novel relative to existing literature on group actions and symmetric models.

major comments (1)

No definitions of the nonarchimedean groups, no description of the associated permutation models, and no statements of the claimed theorems or proofs are supplied anywhere in the manuscript. The central claim therefore cannot be evaluated for soundness or for whether the standard permutation-model construction preserves the asserted correspondence between group properties and choice fragments.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their report. The primary concern raised is that the manuscript lacks necessary definitions, model descriptions, and theorem statements, preventing evaluation of the claims. We address this directly below.

read point-by-point responses

Referee: No definitions of the nonarchimedean groups, no description of the associated permutation models, and no statements of the claimed theorems or proofs are supplied anywhere in the manuscript. The central claim therefore cannot be evaluated for soundness or for whether the standard permutation-model construction preserves the asserted correspondence between group properties and choice fragments.

Authors: The referee is correct that the submitted manuscript does not contain explicit definitions of the nonarchimedean groups, descriptions of the associated permutation models, or statements and proofs of the claimed theorems. This omission makes it impossible to assess the results as presented. In the revised manuscript we will supply: (i) precise definitions of the relevant nonarchimedean groups and their actions, (ii) a self-contained description of the Fraenkel-Mostowski permutation models constructed from those groups, and (iii) full statements of the theorems together with their proofs. These additions will make the asserted correspondences between group properties and choice fragments in the models directly verifiable. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The provided abstract states only that novel connections are proved between nonarchimedean group properties and AC fragments in associated permutation models. No equations, parameter fits, self-citations, ansatzes, or uniqueness theorems are quoted or referenced in the given text. No load-bearing step reduces by construction to an input, fitted value, or prior self-citation. The claim is therefore independent of the inputs and receives the default non-circularity finding.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no information on free parameters, background axioms, or new entities introduced.

pith-pipeline@v0.9.1-grok · 5529 in / 964 out tokens · 25774 ms · 2026-06-26T06:11:25.301092+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

13 extracted references · 1 canonical work pages · 1 internal anchor

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The class and dynamics of $\alpha$-balanced Polish groups

Shaun Allison and Aristotelis Panagiotopoulos. The class and dynamics of α-balanced Polish groups. 2026.arXiv:2406.06082

work page internal anchor Pith review Pith/arXiv arXiv 2026

[2] [2]

Injectivity, projectivity, and the axiom of choice.Transac- tions of the American Mathematical Society, 255:31–59, 1979

Andreas Blass. Injectivity, projectivity, and the axiom of choice.Transac- tions of the American Mathematical Society, 255:31–59, 1979

1979

[3] [3]

Peter J. Cameron. The random graph has the strong small index property. Discrete Math., 291(1-3):41–43, 2005

2005

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Cambridge University Press, Cambridge, 1993

Wilfrid Hodges.Model theory, volume 42 ofEncyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 1993

1993

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Rubin.Consequences of the Axiom of Choice

Paul Howard and Jean E. Rubin.Consequences of the Axiom of Choice. Math. Surveys and Monographs. Amer. Math. Society, Providence, 1998

1998

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Models of ZFA in which every lin- early ordered set can be well ordered.Arch

Paul Howard and Eleftherios Tachtsis. Models of ZFA in which every lin- early ordered set can be well ordered.Arch. Math. Logic, 62(7-8):1131–1157, 2023

2023

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Springer Verlag, New York, 2002

Thomas Jech.Set Theory. Springer Verlag, New York, 2002

2002

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Julia F. Knight. A completeL ω1ω-sentence characterizingℵ 1.The Journal of Symbolic Logic, 42(1):59–62, 1977

1977

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The automorphism groups of regular trees.J

R¨ ognvaldur M¨ oller. The automorphism groups of regular trees.J. London Math. Soc., 43:236–252, 1991

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The strong small index property for free homogeneous structures

Gianluca Paolini and Saharon Shelah. The strong small index property for free homogeneous structures. InResearch Trends in Contemporary Logic. to appear

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Global and local boundedness of Polish groups.Indi- ana University Mathematics Journal, 62:1621–1678, 2013

Christian Rosendal. Global and local boundedness of Polish groups.Indi- ana University Mathematics Journal, 62:1621–1678, 2013

2013

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J. K. Truss. Infinite permutation groups. II. Subgroups of small index.J. Algebra, 120(2):494–515, 1989

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Dynamical ideals and the axiom of choice.Canadian Journal of Mathematics, 2026

Jindˇ rich Zapletal. Dynamical ideals and the axiom of choice.Canadian Journal of Mathematics, 2026. 28

2026