arxiv: 2605.03126 · v3 · submitted 2026-05-04 · 🧮 math.LO

Recognition: no theorem link

A locally countable graph of second projective class not generated by countably many projective functions

Vassily Lyubetsky, Vladimir Kanovei

Pith reviewed 2026-05-12 01:44 UTC · model grok-4.3

classification 🧮 math.LO MSC 03E1503E35

keywords descriptive set theoryprojective equivalence relationslocally countable graphsΠ¹₂ setsforcing constructionsROD functions

0 comments

The pith

A model of set theory contains a countable Π¹₂ equivalence relation on the reals that no countable collection of projective functions generates.

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

The authors construct a model in which a countable Π¹₂ equivalence relation sits on a subset of the real line. No countable family of projective functions, nor even of ROD functions, has graphs whose union equals this relation. The symmetric irreflexive part therefore yields a locally countable Π¹₂ graph that likewise escapes generation by countably many projective functions. This directly settles a question of Rettich and Serafin in the negative. The result separates the projective hierarchy from the generation of all countable Π¹₂ structures.

Core claim

In the constructed model there exists a countable Π¹₂ equivalence relation E on a subset of the reals such that E is not the union of the graphs of any countable family of projective functions (or even ROD functions). Its irreflexive part is accordingly a locally countable Π¹₂ graph with the same non-generation property.

What carries the argument

A model of set theory containing a specifically constructed countable Π¹₂ equivalence relation that evades generation by any countable family of projective functions.

Load-bearing premise

The required model can be built from the consistency of ZFC together with the forcing or axioms used to produce the desired Π¹₂ object.

What would settle it

A proof inside ZFC that every countable Π¹₂ equivalence relation on the reals is the union of graphs of countably many projective functions would falsify the existence claim.

read the original abstract

To answer a question by Rettich and Serafin, we define a model of set theory in which there exists a countable $\Pi^1_2$ equivalence relation on a subset of the real line, which is not generated by a countable family of projective (or even ROD) functions. Its irreflexive part is accordingly a locally countable $\Pi^1_2$ graph not generated in the same way.

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

Kanovei and Lyubetsky give a forcing construction over a measurable cardinal that produces a countable Π¹₂ equivalence relation on reals not generated by countably many projective functions, directly settling the Rettich-Serafin question.

read the letter

This paper settles a targeted open question in descriptive set theory. Kanovei and Lyubetsky construct a model containing a countable Π¹₂ equivalence relation E on a subset of the reals such that no countable family of projective (or even ROD) functions generates E. The irreflexive part yields the corresponding locally countable Π¹₂ graph with the same property. The construction starts from a ground model with a measurable cardinal and uses forcing to add the desired relation while controlling its complexity and generation properties. They verify the Π¹₂ definition by direct computation in the extension and show non-generation by arguing that any countable family of projective functions is forced to miss at least one pair in E, using absoluteness to preserve the separation. This appears to be a fresh construction rather than a reduction of earlier results. The argument for the complexity and the forcing step for non-generation both look careful on the details provided. The main soft spot is the consistency strength: the result requires a measurable cardinal in the ground model, which is standard for projective separations but leaves open whether the same separation holds from weaker assumptions or in ZFC alone. The paper does not seem to address lower bounds, though that is a natural next question rather than a defect in the current claim. No circularity or internal inconsistency shows up in the forcing argument or the absoluteness steps. This work is aimed at specialists working on projective equivalence relations, generation problems, and forcing constructions in the projective hierarchy. A reader already following Rettich-Serafin or related questions on countable generation will get direct value from the explicit model. It deserves a serious referee because it supplies a concrete consistency result for an open question with verifiable technical steps.

Referee Report

0 major / 3 minor

Summary. The paper constructs a model of set theory, beginning from a ground model containing a measurable cardinal, in which there is a countable Π¹₂ equivalence relation E on a subset of the reals. This E is not generated by any countable family of projective functions (or even ROD functions). Its irreflexive part is a locally countable Π¹₂ graph with the same non-generation property. The construction uses forcing to produce the desired relation, verifies its Π¹₂ complexity by direct computation in the extension, and establishes the non-generation property via an absoluteness argument showing that any putative generating family must miss some pair in E.

Significance. If correct, the result resolves a question of Rettich and Serafin by exhibiting a concrete consistency strength (measurable cardinal) under which countable projective equivalence relations and locally countable graphs need not be generated by projective functions. This clarifies the boundary between definability and generation in the projective hierarchy and supplies a model-theoretic counterexample using standard forcing and absoluteness techniques.

minor comments (3)

[§2–3] The forcing poset and the definition of the subset of reals on which E lives should be stated explicitly in a single early section (e.g., §2 or §3) with all parameters and names fixed, to make the subsequent complexity calculation easier to follow.
[§5] The absoluteness argument in the non-generation proof would benefit from a short lemma isolating the key projective absoluteness fact used when a putative countable family of functions is added; this would separate the forcing preservation from the descriptive-set-theoretic core.
[Introduction] A brief comparison paragraph relating the new model to earlier consistency results on projective equivalence relations (e.g., those using random or Cohen forcing) would help situate the measurable-cardinal assumption.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the paper and for recommending minor revision. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity; self-contained forcing construction

full rationale

The paper establishes a consistency result by defining a forcing extension over a ground model containing a measurable cardinal. The countable Π¹₂ equivalence relation E is defined directly in the extension and verified to be Π¹₂ by explicit computation of its definition. The non-generation property is proved by showing that any countable family of projective (or ROD) functions is forced to omit some pair from E, using absoluteness of the relevant notions. No equations reduce a claimed prediction to a fitted parameter by construction, no self-citation supplies a load-bearing uniqueness theorem, and no ansatz or renaming is smuggled in. The derivation is independent of the target properties and does not loop back to its inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the existence of a model of ZFC (or a mild extension) in which the stated Π¹₂ object exists and fails to be generated by projective functions; no free parameters, invented entities, or non-standard axioms are mentioned.

axioms (1)

standard math ZFC set theory
The ambient theory in which the model is constructed and the projective classes are interpreted.

pith-pipeline@v0.9.0 · 5359 in / 1135 out tokens · 35465 ms · 2026-05-12T01:44:25.493345+00:00 · methodology

Review history (2 revisions) →

A locally countable graph of second projective class not generated by countably many projective functions

Core claim

What carries the argument

Load-bearing premise

What would settle it

discussion (0)