Remarks on relative categoricity

Anand Pillay

arxiv: 2602.05866 · v2 · submitted 2026-02-05 · 🧮 math.LO

Remarks on relative categoricity

Anand Pillay This is my paper

Pith reviewed 2026-05-16 06:52 UTC · model grok-4.3

classification 🧮 math.LO

keywords relative categoricityGaifman conjecturealmost internal coversomega-stabilitystability over a predicatemodel theorylogic

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The pith

Almost internal covers make relative categoricity imply the Gaifman conjecture while separating two stability notions over P.

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

The paper surveys relative categoricity and isolates the case of almost internal covers as a technically tractable subclass. In this subclass it proves the Gaifman conjecture. It introduces the notion of full omega-stability over a distinguished predicate P and supplies a counterexample showing that full stability over P can fail even when the structure is relatively categorical and omega-stable over P. The work therefore both confirms a long-standing conjecture in a restricted but natural setting and draws a sharp distinction between two stability concepts.

Core claim

For structures that are almost internal covers of a base structure with predicate P, relative categoricity implies the Gaifman conjecture. The paper defines full omega-stability over P and exhibits a counterexample in which full stability over P fails while the structure remains relatively categorical.

What carries the argument

Almost internal covers: structures in which the model is almost internal to the base structure equipped with predicate P, carrying the argument from relative categoricity to the Gaifman conjecture and the separation of stability notions.

Load-bearing premise

The structures under study must satisfy the technical conditions that define almost internal covers or the new stability notions over P.

What would settle it

An almost internal cover that is relatively categorical yet fails the Gaifman conjecture, or one in which full stability over P holds.

read the original abstract

The paper is partly a survey with historical background and references, partly provides the opportunity to put in print some unpublished early work, and partly has new results. A special case of relative categoricity is identified (almost internal covers) for which the Gaifman conjecture is proved, full omega-stability over P is introduced, and as counterexample is given to full stability over P.

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

Pillay settles the Gaifman conjecture for almost internal covers, introduces full omega-stability over P, and gives a counterexample to full stability over P.

read the letter

The paper's main contributions are a proof of the Gaifman conjecture in the special case of almost internal covers, the definition of full omega-stability over P, and an explicit counterexample showing that full stability over P does not hold in general. These sit inside a short survey of relative categoricity that also brings some older unpublished notes into print. The historical framing is clear and the references are useful for anyone trying to trace the line of work from Gaifman onward. The new definition and the counterexample are the parts that actually move the discussion forward rather than just recap it. The proof for almost internal covers is scoped tightly to the technical conditions of that class, which makes the result clean but also limits how far it travels. The counterexample is presented as a concrete construction, so it should be checkable once the details are in front of a reader. No obvious circularity or hidden parameters appear in the claims. The work stays within standard model-theoretic assumptions and does not rely on any fitting or external data. This is aimed at model theorists who already work on stability, categoricity, and expansions or covers. Someone outside that subfield will find the notation and the narrow focus hard to penetrate quickly. A reader inside the area will get a usable new notion and a definite negative result to cite or build on. The paper is short, self-contained, and adds verifiable pieces to an existing conversation, so it is worth sending to referees even if the impact stays local to the subfield.

Referee Report

2 major / 2 minor

Summary. The manuscript is a partly historical survey of relative categoricity in model theory, incorporating unpublished early work, together with new results. It isolates the subclass of almost internal covers as a special case of relative categoricity, proves Gaifman's conjecture in that setting, introduces the notion of full omega-stability over a predicate P, and supplies a counterexample demonstrating that full stability over P fails in general.

Significance. If the central claims hold, the paper advances the understanding of relative categoricity by delimiting a tractable subclass where a long-standing conjecture is settled, while the new stability notions and the counterexample provide concrete tools and limitations for future work on structures with distinguished predicates. The explicit scoping to technical conditions (almost internal covers, full omega-stability) strengthens the contribution by avoiding overgeneralization.

major comments (2)

[section on almost internal covers] The section introducing almost internal covers: the reduction used to prove Gaifman's conjecture relies on the cover being almost internal, but the manuscript does not explicitly verify that the key diagram-commuting properties survive the passage from the general relative-categoricity setting to this subclass; a short lemma confirming preservation of the relevant diagrams would make the argument self-contained.
[counterexample section] The construction of the counterexample to full stability over P: while the example is stated to satisfy the technical conditions, the verification that it fails full stability (but satisfies full omega-stability) is only sketched; expanding the argument to include the explicit computation of the relevant type-counting functions would strengthen the claim.

minor comments (2)

[historical background] The historical survey section would benefit from a brief table or timeline summarizing the main prior results on relative categoricity and their relation to the new notions introduced here.
[introduction] Notation for the predicate P and the cover relation is introduced gradually; a consolidated list of symbols at the end of the introduction would aid readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thoughtful report and constructive suggestions. We address the two major comments below and will revise the manuscript accordingly.

read point-by-point responses

Referee: [section on almost internal covers] The section introducing almost internal covers: the reduction used to prove Gaifman's conjecture relies on the cover being almost internal, but the manuscript does not explicitly verify that the key diagram-commuting properties survive the passage from the general relative-categoricity setting to this subclass; a short lemma confirming preservation of the relevant diagrams would make the argument self-contained.

Authors: We agree that an explicit verification would improve self-containedness. In the revised manuscript we will insert a short lemma immediately after the definition of almost internal covers, confirming that the relevant diagram-commuting properties are preserved under the reduction from the general relative-categoricity setting. revision: yes
Referee: [counterexample section] The construction of the counterexample to full stability over P: while the example is stated to satisfy the technical conditions, the verification that it fails full stability (but satisfies full omega-stability) is only sketched; expanding the argument to include the explicit computation of the relevant type-counting functions would strengthen the claim.

Authors: We agree that the verification is currently only sketched. In the revised version we will expand the counterexample section to include the explicit computation of the type-counting functions, showing in detail that full stability over P fails while full omega-stability holds. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained in standard model theory

full rationale

The paper surveys relative categoricity, identifies the subclass of almost internal covers, proves Gaifman's conjecture there via explicit constructions, introduces full omega-stability over P by definition, and supplies a counterexample to full stability over P. All steps rest on standard first-order logic axioms, definability, and model-theoretic notions without any reduction of a claimed prediction or theorem to a fitted parameter, self-citation chain, or renaming of inputs. No load-bearing step equates output to input by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on the standard axioms of first-order logic and model theory; no free parameters, invented entities, or ad-hoc assumptions beyond the usual background are described in the abstract.

axioms (1)

standard math Standard axioms of first-order logic and model theory
Invoked for all definitions of categoricity, stability, and the Gaifman conjecture.

pith-pipeline@v0.9.0 · 5334 in / 1082 out tokens · 32484 ms · 2026-05-16T06:52:44.563418+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/ArithmeticFromLogic.lean LogicNat recovery and embed_strictMono unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

T is relatively (ω,ω)-categorical iff every model M is atomic over P(M) (Lemma 1.8); 1-coanalyzable iff almost internal to P (Lemma 1.13)
IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean absolute_floor_iff_bare_distinguishability unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

If relatively (ω,ω)-categorical and almost internal then Gaifman property holds (Prop 2.6)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages

[1]

AlZurba, A presentation of Shelah’s proof of Gaifman’s conjecture on relatively categorical theories, Master’s thesis, Hebrew Universiy of Jerusalem, 2024

A. AlZurba, A presentation of Shelah’s proof of Gaifman’s conjecture on relatively categorical theories, Master’s thesis, Hebrew Universiy of Jerusalem, 2024

work page 2024
[2]

Gaifman, Operations on relational structures, functors and classes I, in the proceedings of the Tarski Symposium, Proc

H. Gaifman, Operations on relational structures, functors and classes I, in the proceedings of the Tarski Symposium, Proc. Sympos. Pure Math, vol. XXV (1974), p. 21-39

work page 1974
[3]

Gaifman, Some results and and conjectures concerning definability questions, preprint (1974?)

H. Gaifman, Some results and and conjectures concerning definability questions, preprint (1974?)

work page 1974
[4]

Hart and S

B. Hart and S. Shelah, Categoricity overPfor first orderTor categoric- ity forϕ∈L ω1,ω can stop atℵ k while holding forℵ 0, ...,ℵ k−1, Israel J. Math 70 (1990), 219 - 235

work page 1990
[5]

Herwig, E

B. Herwig, E. Hrushovski and D. Macpherson, Interpretable groups, stably embedded sets, and Vaughtian pairs, J. London Math. Sic. vol. 68 (2003), 1 - 11

work page 2003
[6]

W. A. Hodges, A normal form for algebraic constructions II, Logique et Analyse, vol. 71-72 (1974), p. 429 - 480

work page 1974
[7]

Hodges, I.M

W.A. Hodges, I.M. Hodkinson and D. Macpherson, Omega-categoricity, relarive categoricity and coordinatisation, Annals od Pure and Applied Logic, 46 (1990), 169 - 199. 10

work page 1990
[8]

Pillay, Gaifman operations, minimal models and the number of count- able models, Ph

A. Pillay, Gaifman operations, minimal models and the number of count- able models, Ph. D. thesis, Bedford College, University of London, 1977

work page 1977
[9]

Pillay,ℵ 0-categoricity over a predicate, Notre Dame J

A. Pillay,ℵ 0-categoricity over a predicate, Notre Dame J. Formal Logic, vol. 24 (1983), 527 - 536

work page 1983
[10]

Pillay and S

A. Pillay and S. Shelah, Classification theory over a predicate I, Notre Dame J. Formal Logic, Vol. 26 (1985), 361 - 376

work page 1985
[11]

Shelah, Classification over a predicate II, In Around classification theory of models, Springer, Berlin, 1986

S. Shelah, Classification over a predicate II, In Around classification theory of models, Springer, Berlin, 1986

work page 1986
[12]

Shelah and A

S. Shelah and A. Usvyatsov, Stable amalgamation over a predicate and the Gaifman property, preprint 2025

work page 2025
[13]

Usvyatsov, On the existence property over a predicate, preprint 2025

A. Usvyatsov, On the existence property over a predicate, preprint 2025. 11

work page 2025

[1] [1]

AlZurba, A presentation of Shelah’s proof of Gaifman’s conjecture on relatively categorical theories, Master’s thesis, Hebrew Universiy of Jerusalem, 2024

A. AlZurba, A presentation of Shelah’s proof of Gaifman’s conjecture on relatively categorical theories, Master’s thesis, Hebrew Universiy of Jerusalem, 2024

work page 2024

[2] [2]

Gaifman, Operations on relational structures, functors and classes I, in the proceedings of the Tarski Symposium, Proc

H. Gaifman, Operations on relational structures, functors and classes I, in the proceedings of the Tarski Symposium, Proc. Sympos. Pure Math, vol. XXV (1974), p. 21-39

work page 1974

[3] [3]

Gaifman, Some results and and conjectures concerning definability questions, preprint (1974?)

H. Gaifman, Some results and and conjectures concerning definability questions, preprint (1974?)

work page 1974

[4] [4]

Hart and S

B. Hart and S. Shelah, Categoricity overPfor first orderTor categoric- ity forϕ∈L ω1,ω can stop atℵ k while holding forℵ 0, ...,ℵ k−1, Israel J. Math 70 (1990), 219 - 235

work page 1990

[5] [5]

Herwig, E

B. Herwig, E. Hrushovski and D. Macpherson, Interpretable groups, stably embedded sets, and Vaughtian pairs, J. London Math. Sic. vol. 68 (2003), 1 - 11

work page 2003

[6] [6]

W. A. Hodges, A normal form for algebraic constructions II, Logique et Analyse, vol. 71-72 (1974), p. 429 - 480

work page 1974

[7] [7]

Hodges, I.M

W.A. Hodges, I.M. Hodkinson and D. Macpherson, Omega-categoricity, relarive categoricity and coordinatisation, Annals od Pure and Applied Logic, 46 (1990), 169 - 199. 10

work page 1990

[8] [8]

Pillay, Gaifman operations, minimal models and the number of count- able models, Ph

A. Pillay, Gaifman operations, minimal models and the number of count- able models, Ph. D. thesis, Bedford College, University of London, 1977

work page 1977

[9] [9]

Pillay,ℵ 0-categoricity over a predicate, Notre Dame J

A. Pillay,ℵ 0-categoricity over a predicate, Notre Dame J. Formal Logic, vol. 24 (1983), 527 - 536

work page 1983

[10] [10]

Pillay and S

A. Pillay and S. Shelah, Classification theory over a predicate I, Notre Dame J. Formal Logic, Vol. 26 (1985), 361 - 376

work page 1985

[11] [11]

Shelah, Classification over a predicate II, In Around classification theory of models, Springer, Berlin, 1986

S. Shelah, Classification over a predicate II, In Around classification theory of models, Springer, Berlin, 1986

work page 1986

[12] [12]

Shelah and A

S. Shelah and A. Usvyatsov, Stable amalgamation over a predicate and the Gaifman property, preprint 2025

work page 2025

[13] [13]

Usvyatsov, On the existence property over a predicate, preprint 2025

A. Usvyatsov, On the existence property over a predicate, preprint 2025. 11

work page 2025