pith. sign in

arxiv: 2604.08638 · v1 · submitted 2026-04-09 · 🧮 math.LO · math.AC

Ordered henselian valued fields: definability and Borel sets

Lothar Sebastian Krapp , Floris Vermeulen This is my paper

Pith reviewed 2026-05-10 17:16 UTC · model grok-4.3

classification 🧮 math.LO math.AC
keywords ordered henselian valued fieldsquantifier eliminationBorel setsDenef-Pas languageresplendencyNIP fieldsorder topologydefinable sets
0
0 comments X

The pith

Resplendent ordered henselian valued fields admit relative field quantifier elimination in the expanded Denef-Pas language, making their definable sets Borel in the order topology.

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

The paper shows that resplendency allows ordered henselian valued fields to have relative quantifier elimination for the field sort in a language that adds linear orders to the Denef-Pas language. This result is then used with a dimensionality reduction theorem to prove that definable sets are Borel sets under the order topology. A reader might care because this advances the understanding of definable sets in valued fields and connects to broader questions about NIP fields in model theory.

Core claim

Due to their resplendency, ordered henselian valued fields admit relative field quantifier elimination in the Denef--Pas language expanded by linear orders in the field and residue field sort. From this, together with a dimensionality reduction theorem, it follows that any set definable over an ordered henselian valued field is a Borel set with respect to the order topology. The work is placed in the context of Shelah's classification conjecture for NIP fields and related topics.

What carries the argument

The resplendency property enabling relative field quantifier elimination in the order-expanded Denef-Pas language, combined with dimensionality reduction to establish the Borel property for definable sets.

If this is right

  • Definable sets over these fields are Borel sets in the order topology.
  • The results support investigations into definable henselian valuations.
  • This contributes to Shelah's classification conjecture of NIP fields.
  • The work relates to the Fundamental Theorem of Statistical Learning.

Where Pith is reading between the lines

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

  • These techniques might be applied to other classes of fields to determine their definable sets' topological properties.
  • Checking the Borel property in concrete examples like the reals with valuation could test the result.
  • The connection to NIP fields suggests potential for classifying more structures in model theory.
  • Extensions could involve other topologies beyond the order topology.

Load-bearing premise

The ordered henselian valued fields are resplendent, and the dimensionality reduction theorem can be applied to derive the Borel property from the quantifier elimination.

What would settle it

A counterexample would be an ordered henselian valued field where a definable set in the expanded language is not Borel in the order topology, or where the relative quantifier elimination fails for a resplendent field.

read the original abstract

We firstly show that due to their resplendency ordered henselian valued fields admit relative field quantifier elimination in the Denef--Pas language expanded by linear orders in the field and residue field sort. Secondly, we deduce from a dimensionality reduction theorem that any set definable over an ordered henselian valued field is a Borel set with respect to the order topology. Our results are contextualised within Shelah's classification conjecture of NIP fields and its connections to the study of definable henselian valuations and the Fundamental Theorem of Statistical Learning.

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.

Referee Report

2 major / 2 minor

Summary. The manuscript proves that resplendent ordered henselian valued fields admit relative quantifier elimination for the field sort in the Denef-Pas language expanded by linear orders on the field and residue field. It then uses a dimensionality reduction theorem to show that definable sets are Borel sets in the order topology. The work is framed within Shelah's classification conjecture for NIP fields and related topics in definable valuations and statistical learning.

Significance. If the proofs are correct, this contributes significantly to the model theory of valued fields by establishing strong elimination and topological regularity results for definable sets. The Borel property is a strong conclusion that could aid in understanding NIP properties in these structures. Explicit credit is due for connecting to the Fundamental Theorem of Statistical Learning, though the direct impact depends on further development.

major comments (2)
  1. [Deduction of Borel property] The deduction that definable sets are Borel relies on applying a dimensionality reduction theorem to the QE result in the expanded language. It is unclear whether the hypotheses of that theorem (such as dimension counting for definable sets) are preserved when adding linear orders and considering the order topology rather than the valuation topology; this needs explicit confirmation as it is central to the second claim.
  2. [Quantifier elimination result] The claim of relative field QE due to resplendency is central; the manuscript should specify which resplendency properties are invoked and how they interact with the added order predicates to achieve elimination, particularly in the residue field sort.
minor comments (2)
  1. [Abstract] The abstract refers to 'a dimensionality reduction theorem' without citation or reference; the full paper should include the specific theorem or paper referenced.
  2. [Introduction] Ensure that the Denef-Pas language is briefly recalled or referenced for accessibility, as expansions are central.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the constructive major comments. We address each point below and will revise the text accordingly to improve clarity.

read point-by-point responses

  1. Referee: [Deduction of Borel property] The deduction that definable sets are Borel relies on applying a dimensionality reduction theorem to the QE result in the expanded language. It is unclear whether the hypotheses of that theorem (such as dimension counting for definable sets) are preserved when adding linear orders and considering the order topology rather than the valuation topology; this needs explicit confirmation as it is central to the second claim.

    Authors: We agree that an explicit confirmation is needed. The relative quantifier elimination in the Denef-Pas language expanded by the linear orders reduces definable sets in the field sort to Boolean combinations of conditions on the residue field (with its order) and the value group. The dimensionality reduction theorem applies because the added orders are linear and compatible with the field structure; they do not increase the VC-dimension or violate the finite-dimensionality hypotheses used in the theorem. Moreover, in ordered henselian fields the order topology is generated by intervals that are definable in the expanded language, and the Borel property transfers from the residue field topology. We will add a short clarifying paragraph immediately after the statement of the Borel result, citing the relevant dimension bounds and confirming their invariance under the order expansion. revision: yes

  2. Referee: [Quantifier elimination result] The claim of relative field QE due to resplendency is central; the manuscript should specify which resplendency properties are invoked and how they interact with the added order predicates to achieve elimination, particularly in the residue field sort.

    Authors: We accept the request for greater precision. The resplendency properties used are: (i) every type over a small parameter set is realized in an elementary extension, and (ii) the structure remains henselian after such extensions. These allow us to eliminate field quantifiers by realizing residue-field types that satisfy the order conditions. The order predicates on the residue field are preserved under resplendency because the residue field of a resplendent ordered henselian field is itself resplendent as an ordered field; this ensures that the QE reduction to the residue sort works uniformly. We will expand the proof of the QE theorem (currently in Section 3) with a dedicated subsection that lists the exact resplendency axioms invoked and sketches their interaction with the order predicates, including a short argument for the residue-field case. revision: yes

Circularity Check

0 steps flagged

No circularity: QE proved from resplendency; Borel via independent dimensionality reduction theorem

full rationale

The paper proves relative field quantifier elimination directly from the resplendency assumption in the expanded Denef-Pas language with orders. It then invokes a separate dimensionality reduction theorem to obtain the Borel conclusion for definable sets in the order topology. No step reduces by construction to its own inputs, no self-citation is load-bearing for the central claims, and the derivation chain does not rename or smuggle prior results as new predictions. The applicability of the reduction theorem to the ordered expansion is a question of correctness rather than circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Based on the abstract alone, the central claims rest on the resplendency property of the fields and the applicability of an unspecified dimensionality reduction theorem; no free parameters, invented entities, or additional axioms are stated.

axioms (2)
  • domain assumption Ordered henselian valued fields are resplendent
    Invoked to obtain relative field quantifier elimination in the expanded Denef-Pas language.
  • domain assumption A dimensionality reduction theorem holds for definable sets in this setting
    Used to deduce that definable sets are Borel in the order topology.

pith-pipeline@v0.9.0 · 5379 in / 1322 out tokens · 54972 ms · 2026-05-10T17:16:44.286365+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

24 extracted references · 24 canonical work pages

  1. [1]

    N.\ L.\ Alling , Foundations of Analysis over Surreal Number Fields , North-Holland Math.\ Stud.\ 141, Notas Mat.\ 117 (North-Holland, Amsterdam, 1987)

    work page 1987

  2. [2]

    Anscombe and F

    S. Anscombe and F. Jahnke , `Characterizing NIP henselian fields', J.\ Lond.\ Math.\ Soc., II.\ Ser. 109 (2024), Article ID e12868, doi:10.1112/jlms.12868 https://doi.org/10.1112/jlms.12868

    work page doi:10.1112/jlms.12868 2024

  3. [3]

    Aschenbrenner , A

    M. Aschenbrenner , A. Chernikov , A. Gehret and M. Ziegler , `Distality in valued fields and related structures', Trans. Amer. Math. Soc. 375 (2022), 4641--4710, doi:10.1090/tran/8661 https://doi.org/10.1090/tran/8661

    work page doi:10.1090/tran/8661 2022

  4. [4]

    Cluckers , I

    R. Cluckers , I. Halupczok and S. Rideau , `Hensel minimality I', Forum Math.\ Pi 10 (2022) e11

    work page 2022

  5. [5]

    Cluckers , I

    R. Cluckers , I. Halupczok , S. Rideau and F. Vermeulen , `Hensel minimality II: Mixed characteristic and a diophantine application', Forum Math.\ Sigma 11 (2023) e89

    work page 2023

  6. [6]

    Delon and R

    F. Delon and R. Farré , `Some model theory for almost real closed fields', J.\ Symb.\ Log.\ 61 (1996) 1121--1152, doi:10.2307/2275808 https://doi.org/10.2307/2275808

    work page doi:10.2307/2275808 1996

  7. [7]

    Dittmann , F

    P. Dittmann , F. Jahnke , L. S. Krapp and S. Kuhlmann , `Definable valuations on ordered fields', Model Theory 2 (2023) 101--120, doi:10.2140/mt.2023.2.101 https://doi.org/10.2140/mt.2023.2.101

    work page doi:10.2140/mt.2023.2.101 2023

  8. [8]

    van den Dries , `Lectures on the Model Theory of Valued Fields', Model Theory in Algebra, Analysis and Arithmetic , Lect.\ Notes Math.\ 2111 (eds D

    L. van den Dries , `Lectures on the Model Theory of Valued Fields', Model Theory in Algebra, Analysis and Arithmetic , Lect.\ Notes Math.\ 2111 (eds D. Macpherson and C. Toffalori; Springer, Heidelberg, 2014) 55--157, doi:10.1007/978-3-642-54936-6\_4 https://doi.org/10.1007/978-3-642-54936-6_4

    work page doi:10.1007/978-3-642-54936-6 2014

  9. [9]

    A. J. Engler and A. Prestel , Valued Fields , Springer Monogr. Math. (Springer, Berlin, 2005), doi:10.1007/3-540-30035-X https://doi.org/10.1007/3-540-30035-X

    work page doi:10.1007/3-540-30035-x 2005

  10. [10]

    Fehm and F

    A. Fehm and F. Jahnke , `Recent progress on definability of Henselian valuations', Ordered Algebraic Structures and Related Topics , Contemp. Math. 697 (Amer.\ Math.\ Soc., Providence, RI, 2017), 135--143, doi:10.1090/conm/697/14049 https://doi.org/10.1090/conm/697/14049

    work page doi:10.1090/conm/697/14049 2017

  11. [11]

    Hils , `Model theory of valued fields', Lectures in Model Theory , Münster Lect.\ Math.\ (eds F

    M. Hils , `Model theory of valued fields', Lectures in Model Theory , Münster Lect.\ Math.\ (eds F. Jahnke, D. Palacín and K. Tent; European Mathematical Society, Zürich, 2018) 151--180, doi:10.4171/184-1/6 https://doi.org/10.4171/184-1/6

    work page doi:10.4171/184-1/6 2018

  12. [12]

    Jahnke , `Henselian expansions of NIP fields', J.\ Math.\ Log

    F. Jahnke , `Henselian expansions of NIP fields', J.\ Math.\ Log. 24 (2024), Article ID 2350006, doi:10.1142/S021906132350006X https://doi.org/10.1142/S021906132350006X

    work page doi:10.1142/s021906132350006x 2024

  13. [13]

    Johnson , `The classification of dp-minimal and dp-small fields', J.\ Eur.\ Math.\ Soc.\ (JEMS) 25 (2023) 467--513, doi:10.4171/JEMS/1187 https://doi.org/10.4171/JEMS/1187

    W. Johnson , `The classification of dp-minimal and dp-small fields', J.\ Eur.\ Math.\ Soc.\ (JEMS) 25 (2023) 467--513, doi:10.4171/JEMS/1187 https://doi.org/10.4171/JEMS/1187

    work page doi:10.4171/jems/1187 2023

  14. [14]

    Kaiser , `First order tameness of measures', Ann.\ Pure Appl.\ Logic 163 (2012) 1903--1927

    T. Kaiser , `First order tameness of measures', Ann.\ Pure Appl.\ Logic 163 (2012) 1903--1927

    work page 2012

  15. [15]

    Karpinski and A

    M. Karpinski and A. Macintyre , `Approximating Volumes and Integrals in o-Minimal and p-Minimal Theories', Connections between Model Theory and Algebraic and Analytic Geometry , Quad.\ Math. 6 (2000) 49–177

    work page 2000

  16. [16]

    L. S. Krapp , S. Kuhlmann and G. Lehéricy , `Strongly NIP almost real closed fields', Math.\ Log.\ Quart. 67 (2021) 321--328, doi:10.1002/malq.202000060 https://doi.org/10.1002/malq.202000060

    work page doi:10.1002/malq.202000060 2021

  17. [17]

    L.\ S.\ Krapp , `Ordered Algebraic Structures: Hahn Fields, Convex Valuations, and Exponentials', Habilitation Thesis, Universität Konstanz, 2023, urn:nbn:de:bsz:352-2-1ljrlhqax4j190 http://nbn-resolving.de/urn:nbn:de:bsz:352-2-1ljrlhqax4j190

    work page 2023

  18. [18]

    L. S. Krapp and L. Wirth , `Measurability in the Fundamental Theorem of Statistical Learning', Preprint, 2025, arXiv:2410.10243v2 https://arxiv.org/abs/2410.10243v2

    work page arXiv 2025

  19. [19]

    L. S. Krapp , M. Vermeil and L. Wirth , `On Tameness, Measurability and the Independence Property', Preprint, 2025, arXiv:2506.08733v2 https://arxiv.org/abs/2506.08733v2

    work page arXiv 2025

  20. [20]

    K. H. Nguyen , M. Stout and F. Vermeulen , `Almost real closed fields with real analytic structure', J.\ Math.\ Log.\ 2550026, doi:10.1142/S0219061325500266 https://doi.org/10.1142/S0219061325500266

    work page doi:10.1142/s0219061325500266

  21. [21]

    Pas `Uniform p -adic cell decomposition and local zeta functions', J.\ Reine Angew.\ Math

    J. Pas `Uniform p -adic cell decomposition and local zeta functions', J.\ Reine Angew.\ Math. 399 (1989), 137--172, doi:10.1515/crll.1989.399.137 https://doi.org/10.1515/crll.1989.399.137

    work page doi:10.1515/crll.1989.399.137 1989

  22. [22]

    S. Rideau , `Some properties of analytic difference valued fields', J.\ Inst.\ Math.\ Jussieu 16 (2017) 447--499, doi:10.1017/S1474748015000183 https://doi.org/10.1017/S1474748015000183

    work page doi:10.1017/s1474748015000183 2017

  23. [23]

    Shelah , `Strongly dependent theories', Israel J.\ Math

    S. Shelah , `Strongly dependent theories', Israel J.\ Math. 204 (2014) 1--83, doi:10.1007/s11856-014-1111-2 https://doi.org/10.1007/s11856-014-1111-2

    work page doi:10.1007/s11856-014-1111-2 2014

  24. [24]

    P. Simon , A Guide to NIP Theories , Lect.\ Notes Log.\ 44 (Association for Symbolic Logic, Cambridge University Press, Cambridge, 2015), doi:10.1017/CBO9781107415133 https://doi.org/10.1017/CBO9781107415133

    work page doi:10.1017/cbo9781107415133 2015