arxiv: 2605.13683 · v1 · submitted 2026-05-13 · 🧮 math.LO

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O-minimal open core is not an elementary property

Alexi Block Gorman, Esther Elbaz Saban

Authors on Pith no claims yet

Pith reviewed 2026-05-14 18:04 UTC · model grok-4.3

classification 🧮 math.LO

keywords o-minimalityopen coreelementary propertymodel theoryordered structuresdefinable setsexpansions

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The pith

Having an o-minimal open core is not an elementary property.

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

Structures with definable topologies have an open core made from their definable open sets. The paper proves that o-minimality of this open core is not an elementary property, meaning it is not preserved under elementary equivalence. By building a particular expansion of the ordered rationals, the authors show a structure whose open core is o-minimal, yet some elementary extensions have open cores that are not o-minimal. This directly addresses open questions about the nature of open cores in model theory.

Core claim

Given a structure M with a definable topology, its open core is the structure on the same universe with language consisting of all definable open sets. The authors prove that having an o-minimal open core is not an elementary property by constructing an expansion of (Q, <) with o-minimal open core such that some elementary superstructures do not have this property.

What carries the argument

A specific expansion of the ordered rationals (Q, <) in which the open core is o-minimal, but this o-minimality does not hold in all elementary extensions.

If this is right

Elementary extensions of structures with o-minimal open cores may lose this property.
The open core's o-minimality depends on the choice of model, not solely on the theory.
Similar non-elementary behaviors may exist for other geometric properties defined via definable sets.

Where Pith is reading between the lines

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

This implies that when studying o-minimal structures, one must consider specific models rather than just their theories.
Future work could explore whether other core properties, like cell decomposition, are elementary or not.
The construction technique might be adapted to show non-elementarity in related ordered structures.

Load-bearing premise

It is possible to expand (Q, <) in a way that makes its open core o-minimal while ensuring that first-order properties allow for elementary extensions with non-o-minimal open cores.

What would settle it

An explicit computation or proof showing that the open core in every elementary extension of the constructed expansion remains o-minimal would falsify the central claim.

read the original abstract

Given a structure $\mathcal{M}$ with a definable topology, its open core is a structure defined on the same universe whose language consists of all open sets of all arities definable in $\mathcal{M}$. In response to questions raised by Dolich, Miller, and Steinhorn in their early work on open core, we prove that having an o-minimal open core is not an elementary property. In particular, we construct an expansion of the structure $(\mathbb{Q},<)$ that has an o-minimal open core, but some of its elementary superstructures do not.

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

The paper gives an explicit counterexample showing that o-minimality of the open core is not preserved under elementary extensions.

read the letter

This paper settles the open question from Dolich, Miller, and Steinhorn by constructing an expansion of (Q, <) whose open core is o-minimal, yet some elementary superstructure has a non-o-minimal open core. The main new thing is the concrete counterexample itself rather than a general non-existence argument. That makes the negative result usable for anyone who wants to see exactly where the property breaks. The construction follows standard model-theoretic lines for expansions and definable open sets, and the abstract indicates the first-order properties are preserved in the base structure while failing in the extension. That approach is appropriate for the claim. The soft spot is the usual one for construction-based papers: the details of how the open sets are defined and why o-minimality holds initially but not later need full verification. From the abstract alone it looks plausible, but a referee would want to check that no definability slips occur when moving to the superstructure. The citation pattern is clean and points back to the original questions without unnecessary self-reference. This is for model theorists working on o-minimality, expansions, and definable topologies. A reader already familiar with open cores will get immediate value from the counterexample and can test whether similar constructions work in other ordered structures. It shows clear engagement with the literature and the problem as stated. I would send it to peer review because it resolves a specific open question with an explicit, checkable example that fits the standards of the subfield.

Referee Report

0 major / 1 minor

Summary. The paper proves that having an o-minimal open core is not an elementary property. It constructs an expansion of (ℚ, <) whose open core (the structure whose language consists of all definable open sets) is o-minimal, while some elementary superstructures have non-o-minimal open cores, thereby answering a question of Dolich, Miller, and Steinhorn.

Significance. This provides a concrete negative answer showing that o-minimality of the open core is not preserved under elementary extensions. The explicit construction of the expansion of (ℚ, <) supplies a falsifiable counterexample using standard model-theoretic notions, which strengthens the result and clarifies the scope of elementary properties for structures with definable topologies.

minor comments (1)

§2 (or the section defining the specific expansion): the precise language of the expansion and the verification that the open core remains o-minimal in the base structure but fails in the extension should be cross-referenced to the relevant lemmas to aid readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the paper and the recommendation for minor revision. The result provides an explicit counterexample in an expansion of (Q, <) showing that o-minimality of the open core is not preserved under elementary extensions, directly answering the question of Dolich, Miller, and Steinhorn.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper establishes its main result via an explicit construction of an expansion of (Q,<) whose open core is o-minimal, together with verification that certain elementary extensions fail to preserve this property. This is a standard model-theoretic counterexample argument that relies on the definitions of open core, definable sets, and elementary extensions without any self-definitional loops, fitted parameters renamed as predictions, or load-bearing self-citations. No equations or steps reduce the claimed non-elementarity to the inputs by construction; the derivation remains independent of the result itself.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The proof relies on standard model theory without introducing new free parameters or invented entities; axioms are background definitions of open core and o-minimality.

axioms (3)

standard math Standard axioms of first-order logic and elementary extensions
Invoked for the definition of elementary superstructure and preservation of first-order sentences.
domain assumption Definition of open core as the structure of all definable open sets
Core concept taken from prior literature on open cores by Dolich et al.
domain assumption Definition of o-minimality for ordered structures
Standard notion that every definable set is a finite union of intervals.

pith-pipeline@v0.9.0 · 5385 in / 1300 out tokens · 39924 ms · 2026-05-14T18:04:46.125770+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

9 extracted references · 9 canonical work pages

[1]

Alexi Block Gorman, Philipp Hieronymi, and Elliot Kaplan, Pairs of Theories Satisfying a Mordell-Lang Condition. Fund. Math., (2) 251:131-160, (2020)

work page 2020
[2]

Gareth Boxall and Philipp Hieronymi, Expansions which introduce no new open sets.J. Symb. Log., 77(1):111-121, (2012)

work page 2012
[3]

Alfred Dolich, Chris Miller, and Charles Steinhorn, Structures having o-minimal open core.Trans. Am. Math. Soc., 362(3):1371–1411, (2010)

work page 2010
[4]

Pure Appl

Alfred Dolich, Chris Miller, and Charles Steinhorn, Expansions of o-minimal structures by dense independent sets.Ann. Pure Appl. Logic, 167(8):684–706, (2016)

work page 2016
[5]

Pure Appl

Philipp Hieronymi, Travis Nell, and Erik Walsberg, Wild theories with o-minimal open core.Ann. Pure Appl. Logic, (2) 169:146–163, (2018)

work page 2018
[6]

Deacon Linkhorn, Monadic Second Order Logic and Linear Orders.The University of Manchester (United Kingdom) ProQuest Dissertations & Theses, 2021.29051746, (2021)

work page arXiv 2021
[7]

Math.,162(3):193–208 (1999)

Chris Miller and Patrick Speissegger, Expansions of the real line by open sets: o-minimality and open cores.Fund. Math.,162(3):193–208 (1999)

work page 1999
[8]

Marcus Tressl, On The Strength of Some Topological Lattices.Ordered Algebraic Structures and Related Topics, 607: 325–347, (2017)

work page 2017
[9]

Pure Appl

Yilong Zhang, Green points in the reals.Ann. Pure Appl. Logic, (3) 177: 103666 (2026). Universiteit van Amsterdam, Institute for Logic, Language and Computation, Science Park 900, 1098 XH Amsterdam, Netherlands Email address:a.t.blockgorman@uva.nl

work page 2026