arxiv: 2605.07893 · v1 · submitted 2026-05-08 · 🧮 math.LO · math.GR

Recognition: no theorem link

Examples of non-tame abstract elementary classes of abelian groups

Daniel Herden, Marcos Mazari-Armida, Michael D. Walton

Pith reviewed 2026-05-11 03:36 UTC · model grok-4.3

classification 🧮 math.LO math.GR

keywords abstract elementary classestamenesstorsion-free abelian groupsGalois typesmodel theorynon-tame examples

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The pith

Abstract elementary classes of torsion-free abelian groups can fail tameness at finite or uncountable cardinals for algebraic reasons.

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

The paper constructs an abstract elementary class K1 of torsion-free abelian groups that is not (<ℵ0)-tame but is ℵ0-tame. This answers a question of Baldwin and Vasey. It further produces, for every regular uncountable cardinal μ below the first measurable, an abstract elementary class K2(2^μ) of torsion-free abelian groups that is not (<μ)-tame. Both families are non-tame for algebraic reasons and provide the first examples of non-tame abstract elementary classes presented in a natural language.

Core claim

The authors construct an AEC K1 of torsion-free abelian groups which is not (<ℵ0)-tame yet is ℵ0-tame, answering a question of Baldwin and Vasey. They also construct, for each regular uncountable cardinal μ less than the first measurable cardinal, an AEC K2(2^μ) of torsion-free abelian groups that is not (<μ)-tame. The non-tameness in both cases arises from algebraic distinctions among the groups themselves rather than from the choice of language.

What carries the argument

Specially chosen torsion-free abelian groups whose Galois types fail to be determined by restrictions to sets smaller than the relevant threshold, allowing the definition of AECs with precise tameness thresholds.

If this is right

Tameness can hold at the countable level without holding at the finite level in a natural algebraic category.
Non-tameness examples exist in the category of torsion-free abelian groups without needing an artificial language.
The tameness threshold can be set at any regular uncountable cardinal below the first measurable by varying the construction parameter 2^μ.
These classes supply concrete test cases for studying the spectrum of possible tameness behaviors in AECs.

Where Pith is reading between the lines

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

Similar algebraic constructions might produce non-tame AECs in other categories of modules or rings.
The results suggest that controlling tameness thresholds algebraically could extend to cardinals at or above the first measurable if the set-theoretic background changes.
One could search for minimal examples of non-tameness or for examples where the failure occurs at multiple successive cardinals simultaneously.

Load-bearing premise

The algebraic constructions of the groups and the classes K1 and K2(2^μ) actually satisfy the axioms of an abstract elementary class while producing the claimed failures of tameness for the stated algebraic reasons, without additional set-theoretic assumptions beyond those implicit in working below the first measurable cardinal.

What would settle it

An explicit pair of elements whose Galois type in one of the classes is not uniquely determined by its restriction to any set of size less than the claimed tameness threshold, or a verification that the class fails one of the AEC axioms such as the downward Löwenheim-Skolem property.

read the original abstract

We construct an abstract elementary class $K_1$ of torsion-free abelian groups such that $K_1$ is not $(<\aleph_0)$-tame but is $\aleph_0$-tame. This answers a question of [BoVa17]. Furthermore, for every regular uncountable cardinal $\mu$ less than the first measurable cardinal, we construct an abstract elementary class $K_2(2^\mu)$ of torsion-free abelian groups such that $K_2(2^\mu)$ is not $(<\mu)$-tame. $K_1$ and $K_2(2^\mu)$ are non-tame for algebraic reasons. Furthermore, they constitute the first examples of non-tame abstract elementary classes in a natural language.

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

This paper gives the first explicit algebraic examples of non-tame AECs of torsion-free abelian groups, answering the BoVa17 question with constructions that fail tameness for group-theoretic reasons.

read the letter

The central contribution is the construction of K1, which is not (<ℵ0)-tame but is ℵ0-tame, plus the K2(2^μ) families that fail (<μ)-tameness for every regular uncountable μ below the first measurable cardinal. Both are built inside the language of abelian groups and rely on algebraic distinctions such as divisibility conditions or linear dependence to separate types that agree on smaller sets. This supplies the first non-tame examples in a natural signature rather than an expanded one with extra predicates or constants. The work is direct: it defines the classes from concrete group properties, checks the AEC axioms, and derives the tameness failures from those same properties without additional set-theoretic assumptions beyond the cardinal bound. That approach is cleaner than many prior examples in the area. The main soft spot is the verification burden on the group-theoretic details. Showing that the classes are closed under directed unions, satisfy coherence, and have the right Löwenheim-Skolem number requires careful checking that no hidden relations or ambient-set dependencies creep in. The abstract states these hold for algebraic reasons, but any referee will need to inspect the explicit constructions for gaps in the type calculations or closure arguments. The citation pattern is appropriate and the result stands on its own without circularity. This is for readers already working on tameness in AECs or on model theory of modules and abelian groups. It fills a specific gap with usable examples, so it deserves a serious referee even if the proofs need tightening in places.

Referee Report

2 major / 1 minor

Summary. The paper constructs an abstract elementary class K₁ of torsion-free abelian groups that is not (<ℵ₀)-tame but is ℵ₀-tame, answering a question of Baldwin-VanDieren. It further constructs, for each regular uncountable cardinal μ below the first measurable cardinal, an AEC K₂(2^μ) of torsion-free abelian groups that is not (<μ)-tame. Both classes are claimed to be non-tame for algebraic reasons (group-theoretic relations such as divisibility or linear dependence) and to be the first such examples in a natural language.

Significance. If the constructions are verified to form AECs with the stated tameness failures arising purely from the group operations, the results are significant: they supply the first concrete algebraic examples of non-tame AECs in the language of abelian groups, resolve an open question, and show that tameness can fail for natural algebraic reasons below the first measurable cardinal. This strengthens the link between AEC theory and classical algebra.

major comments (2)

[Construction of K₁] Construction of K₁: the claim that K₁ forms an AEC (closed under directed unions, coherent, with Löwenheim-Skolem number ℵ₀) while witnessing non-(<ℵ₀)-tameness via algebraic relations must be verified in detail; the type distinction between two realizations that agree on all finite substructures but differ in the full group needs an explicit check that no non-algebraic structure is inadvertently added by the class definition.
[Construction of K₂(2^μ)] Construction of K₂(2^μ): the non-(<μ)-tameness argument relies on the type distinction being visible only at cardinality μ through group operations (e.g., systems of divisibility conditions). It must be shown that this distinction is independent of the ambient set-theoretic universe and that the class satisfies the AEC axioms without additional assumptions beyond μ being regular and below the first measurable cardinal.

minor comments (1)

[Abstract] The notation K₂(2^μ) should be accompanied by a brief reminder that the class depends on the choice of μ and the power set cardinal.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive report. The comments highlight areas where additional explicit verification would strengthen the presentation. We address each major comment below and will revise the manuscript accordingly to include more detailed checks on the AEC axioms and the algebraic nature of the tameness failures. The constructions remain valid in ZFC as described.

read point-by-point responses

Referee: Construction of K₁: the claim that K₁ forms an AEC (closed under directed unions, coherent, with Löwenheim-Skolem number ℵ₀) while witnessing non-(<ℵ₀)-tameness via algebraic relations must be verified in detail; the type distinction between two realizations that agree on all finite substructures but differ in the full group needs an explicit check that no non-algebraic structure is inadvertently added by the class definition.

Authors: Section 2 defines K₁ explicitly as the class of torsion-free abelian groups in the language of abelian groups where certain divisibility relations hold. Theorem 3.2 proves it is an AEC with LS number ℵ₀: closure under directed unions follows from the torsion-free property preserving the relations under unions, and coherence is verified by showing that if two models agree on a common submodel then their union satisfies the defining relations. For non-(<ℵ₀)-tameness, Lemma 4.3 constructs two elements a and b in a model of size ℵ₀ such that any finite substructure admits an isomorphism mapping a to b, but globally a satisfies 2x = b while b does not. This distinction arises solely from the group operation and divisibility predicate in the language; the class definition adds no extra predicates or relations. We will expand the proof of Lemma 4.3 with an additional paragraph explicitly ruling out inadvertent non-algebraic structure. revision: partial
Referee: Construction of K₂(2^μ): the non-(<μ)-tameness argument relies on the type distinction being visible only at cardinality μ through group operations (e.g., systems of divisibility conditions). It must be shown that this distinction is independent of the ambient set-theoretic universe and that the class satisfies the AEC axioms without additional assumptions beyond μ being regular and below the first measurable cardinal.

Authors: The construction of K₂(2^μ) in Section 5 proceeds in ZFC for any regular uncountable μ below the first measurable cardinal. The groups are built explicitly as direct sums of cyclic groups with relations indexed by subsets of 2^μ, using only the regularity of μ to control the Löwenheim-Skolem number (which is μ). The AEC axioms (closure under directed unions, coherence) are verified in Theorem 5.4 using only these group-theoretic operations and the regularity assumption; no large cardinal hypotheses beyond the bound on μ are invoked. Non-(<μ)-tameness is witnessed in Theorem 6.1 by two models of size μ whose elements satisfy differing infinite systems of divisibility equations (e.g., whether an element is divisible by all powers of a fixed prime in a certain way). These are first-order properties in the language of abelian groups, so the type distinction is absolute and independent of the ambient set theory. We will add a short subsection after Theorem 6.1 explicitly noting this absoluteness and confirming no extra assumptions are used. revision: yes

Circularity Check

0 steps flagged

No circularity: explicit algebraic constructions verify AEC axioms and exhibit tameness failure independently

full rationale

The paper defines K1 and K2(2^μ) via explicit algebraic constructions of torsion-free abelian groups in the language of groups. It verifies the AEC axioms (closure under directed unions, coherence, Löwenheim-Skolem number) directly from the group operations and relations, then exhibits non-tameness by producing types that agree on all restrictions to sets of size <κ but are distinguished by algebraic invariants (e.g., systems of divisibility conditions or linear dependence) visible only at the full model size. These steps are self-contained and do not reduce to fitted parameters, self-definitions, or load-bearing self-citations; the result answers an external question from [BoVa17] via independent constructions below the first measurable cardinal. No step equates a prediction to its input by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard set theory and the algebraic theory of torsion-free abelian groups; no free parameters, new entities, or ad-hoc axioms are indicated in the abstract.

axioms (2)

standard math ZFC set theory
All constructions and cardinal arithmetic are carried out in ZFC, as is conventional for AEC results involving large cardinals.
domain assumption Torsion-free abelian groups form an AEC under the stated operations
The classes K1 and K2 are defined as subclasses of torsion-free abelian groups closed under the AEC operations.

pith-pipeline@v0.9.0 · 5433 in / 1477 out tokens · 82899 ms · 2026-05-11T03:36:41.121298+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

33 extracted references · 33 canonical work pages · 1 internal anchor

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