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

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

Vladimir Kanovei , Vassily Lyubetsky This is my paper

Pith reviewed 2026-05-14 21:34 UTC · model grok-4.3

classification 🧮 math.LO MSC 03E1503E35
keywords set theoryprojective equivalence relationslocally countable graphsdescriptive set theoryconsistency resultsforcingROD functions
0
0 comments X p. Extension

The pith

There is a model of set theory with a countable Π¹₂ equivalence relation on the reals not generated by countably many projective functions.

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

The paper constructs a model of set theory containing a countable equivalence relation of Π¹₂ complexity on a subset of the real line. This relation cannot be obtained from any countable collection of projective functions. Its irreflexive part is a locally countable Π¹₂ graph with the same non-generation property. A sympathetic reader would care because the construction separates the projective definability of the relation itself from the definability of functions that could generate it, showing that countable generation is not automatic at this complexity level.

Core claim

The authors define a model of set theory in which there exists a countable Π¹₂ 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 Π¹₂ graph not generated in the same way.

What carries the argument

The model of set theory containing the constructed countable Π¹₂ equivalence relation that resists generation by any countable family of projective functions.

If this is right

  • Equivalence relations of Π¹₂ complexity need not reduce to countable unions of projective function graphs.
  • Locally countable graphs at the second projective level exist without being generated by countably many projective functions.
  • The generation question for projective objects receives a negative consistency answer at this level of complexity.

Where Pith is reading between the lines

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

  • Similar model constructions might separate generation properties for equivalence relations at other projective levels.
  • The result connects to broader questions about the minimal number of generators needed for definable equivalence relations in the reals.
  • Further work could test whether the non-generation property persists when additional set-theoretic assumptions like large cardinals are added.

Load-bearing premise

The required model can be built consistently from ZFC using forcing or inner-model techniques that keep the projective complexity intact while enforcing the non-generation property.

What would settle it

A proof in ZFC that every countable Π¹₂ equivalence relation on the reals is generated by some countable family of projective functions would refute the existence of the described model.

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.

Referee Report

2 major / 1 minor

Summary. The paper answers a question of Rettich and Serafin by constructing a model of set theory containing a countable Π¹₂ equivalence relation E on a subset of the reals that is not generated by any countable family of projective (or even ROD) functions; the irreflexive part of E is therefore a locally countable Π¹₂ graph with the same non-generation property.

Significance. If the forcing construction succeeds, the result separates projective complexity from generation properties for locally countable graphs and equivalence relations, showing that Π¹₂ objects need not be generated by projective functions. This has implications for descriptive set theory questions about the projective hierarchy and the extent to which forcing can control definability while enforcing combinatorial features such as local countability.

major comments (2)
  1. [Model Construction] The central claim requires the poset to simultaneously (i) preserve the Π¹₂ definition of E (no new reals witness non-equivalence) and (ii) ensure that no name for a projective or ROD function generates E. The abstract and model-construction section provide no explicit verification that the chosen iteration or product forcing satisfies both absoluteness and non-generation simultaneously; a concrete lemma addressing the universal quantifiers in the Π¹₂ formula is needed.
  2. [Non-generation argument] The non-generation argument must rule out both ground-model and extension projective functions. The sketch that any projective name is forced into the ground model or its orbits fail to cover the E-classes is load-bearing but lacks a reference to the specific fusion or properness properties used to control projective names; without this, the separation between Π¹₂ complexity and projective generation remains incomplete.
minor comments (1)
  1. [Introduction] The introduction should clarify the precise definition of 'ROD functions' and how it relates to the projective hierarchy before the main construction.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive report. The comments highlight places where the presentation of the forcing construction and non-generation argument can be strengthened with explicit lemmas. We address each point below and will revise the manuscript to incorporate the requested clarifications.

read point-by-point responses

  1. Referee: [Model Construction] The central claim requires the poset to simultaneously (i) preserve the Π¹₂ definition of E (no new reals witness non-equivalence) and (ii) ensure that no name for a projective or ROD function generates E. The abstract and model-construction section provide no explicit verification that the chosen iteration or product forcing satisfies both absoluteness and non-generation simultaneously; a concrete lemma addressing the universal quantifiers in the Π¹₂ formula is needed.

    Authors: We agree that an explicit lemma would improve clarity. The product forcing is constructed so that it is proper and adds no new reals that could witness failure of the Π¹₂ equivalence (by the usual absoluteness for Σ¹₂ statements under proper forcing), while the iteration is arranged so that any name for a projective function is forced to lie in an earlier stage and thus cannot generate the full E-classes. In the revision we will insert a dedicated lemma immediately after the definition of the poset that directly verifies preservation of the universal quantifiers in the Π¹₂ formula for E. revision: yes

  2. Referee: [Non-generation argument] The non-generation argument must rule out both ground-model and extension projective functions. The sketch that any projective name is forced into the ground model or its orbits fail to cover the E-classes is load-bearing but lacks a reference to the specific fusion or properness properties used to control projective names; without this, the separation between Π¹₂ complexity and projective generation remains incomplete.

    Authors: The argument proceeds by showing that any name for a projective (or ROD) function is forced by a condition whose fusion decides the values on a dense set of E-classes, using the properness of the iterands to ensure the name is already interpreted in the ground model. We will expand the non-generation section to cite the specific fusion lemma for the countable-support iteration and the properness preservation theorem that together force any such name into an earlier stage, thereby preventing it from covering the new E-classes added later. revision: yes

Circularity Check

0 steps flagged

No circularity: forcing construction is independent of target relation

full rationale

The paper establishes consistency of a countable Π¹₂ equivalence relation E on a subset of ℝ that is not generated by any countable family of projective or ROD functions. This is achieved by an explicit forcing construction (or inner-model technique) whose poset properties are chosen to preserve the Π¹₂ complexity of E while ensuring that no name for a projective function can generate the E-classes. The derivation chain consists of standard absoluteness and genericity arguments that do not define E in terms of itself, do not rename a fitted parameter as a prediction, and do not rely on a load-bearing self-citation whose content is the target result. The construction is externally falsifiable via the forcing axioms and does not reduce to any of the enumerated circular patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The construction rests on the standard axioms of ZFC together with forcing techniques to produce the desired model; no free parameters are fitted to data and no new entities are postulated beyond the equivalence relation realized inside the model.

axioms (1)
  • standard math ZFC set theory
    Base theory in which the model is constructed and the projective hierarchy is interpreted.

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Reference graph

Works this paper leans on

8 extracted references · 8 canonical work pages · 1 internal anchor

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