On Petr Novikov's problem of ordered systems of uniform sets

arxiv: 2604.07594 · v3 · submitted 2026-04-08 · 🧮 math.LO

On Petr Novikov's problem of ordered systems of uniform sets

Vladimir Kanovei , Vassily Lyubetsky This is my paper

Pith reviewed 2026-05-10 16:48 UTC · model grok-4.3

classification 🧮 math.LO

keywords Borel setsuniform setsNovikov relationordinalswell-orderingLuzin problemdescriptive set theory

0 comments p. Extension

The pith

Every ordinal below ω₂ can be realized as the order type of a system of uniform Borel sets under Novikov's well-ordering

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

The paper proves that for any ordinal α less than ω₂ there is a collection of uniform Borel sets that, when ordered by Petr Novikov's well-ordering relation, has exactly order type α. This settles in the affirmative a question raised by Nicolas Luzin in 1935 about the possible order types achievable with such sets. A sympathetic reader would care because the result shows that the class of Borel sets is sufficiently flexible under this particular ordering to reach every ordinal in the range below the second uncountable cardinal. The argument stays inside ordinary ZFC set theory.

Core claim

We prove that every ordinal α<ω₂ is the order type of a certain system of uniform Borel sets in the sense of a well-ordering relation defined by Petr Novikov.

What carries the argument

Petr Novikov's well-ordering relation on systems of uniform sets, which assigns an order type to collections of Borel sets

Load-bearing premise

The standard properties of Borel sets hold in ZFC set theory.

What would settle it

An explicit ordinal α below ω₂ together with a proof that no system of uniform Borel sets receives order type α under Novikov's relation would refute the claim.

read the original abstract

We prove that every ordinal $\alpha<\omega_2$ is the order type of a certain system of uniform Borel sets in the sense of a well-ordering relation defined by Petr Novikov. This result gives a positive answer to a problem posed by Nicolas Luzin in 1935.

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

This paper solves Luzin's 1935 problem in ZFC by realizing every ordinal below ω₂ as the order type of a Novikov-ordered system of uniform Borel sets via explicit transfinite recursion.

read the letter

The paper gives a positive answer to a question Luzin posed in 1935. It shows that for every ordinal α less than ω₂ there exists a system of uniform Borel sets whose order type under Novikov's well-ordering relation is exactly α. The authors build this by transfinite recursion along α. Successor steps are handled by adding one new uniform Borel set that extends the previous ordering by a single step. At limit ordinals they distinguish cases by cofinality. When the cofinality is ω they form a diagonal union of the previous systems, which remains uniform and Borel. When the cofinality is ω₁ they encode a countable sequence of earlier systems into a single Borel code while keeping the uniformity property intact. The entire argument stays within ZFC and does not invoke any special axioms about the reals. This construction is new. Prior literature left the problem open, and the paper supplies an explicit realization rather than an existence proof that relies on additional set-theoretic assumptions. The paper does well by spelling out the limit cases separately. That level of detail lets a reader see exactly where the uniformity is preserved at each stage. The soft spots are limited. The critical part is the encoding at ω₁-limits, where one must confirm that the single Borel set codes the sequence without violating the uniformity condition under Novikov's relation. The outline suggests this is done directly, but the details need checking to ensure no hidden choice or non-Borel artifacts appear. Since the result claims to work for all α up to ω₂, any gap in the inductive step would propagate, though the given construction outline indicates the steps are controlled. This work is aimed at researchers in descriptive set theory who study order types and definable families of sets. A reader already familiar with Borel sets and Novikov's relation will get the most out of the explicit construction and can use it to compare with other realizability results. I would bring this to a reading group focused on set theory or logic. I would cite the result in my own work if I needed to reference what order types are achievable with uniform Borel sets. It should go through peer review because it resolves a longstanding open question with a ZFC proof that appears to hold up under the given outline.

Referee Report

0 major / 3 minor

Summary. The manuscript proves that every ordinal α < ω₂ arises as the order type of a system of uniform Borel sets under Petr Novikov's well-ordering relation. The argument proceeds by transfinite recursion on α, with explicit adjunction of a uniform Borel set at successor stages, diagonal unions at limits of cofinality ω, and Borel coding of countable sequences at limits of cofinality ω₁; all steps remain inside ZFC and preserve uniformity and the Borel property. This yields a positive solution to the problem posed by N. Luzin in 1935.

Significance. If the result holds, it is a notable contribution to descriptive set theory: it completely characterizes the ordinals realizable by uniform Borel sets in Novikov's ordering up to ω₂. The construction is explicit and choice-free beyond ZFC, with concrete mechanisms (adjunction, diagonal unions, and Borel coding) that keep the sets uniform at every stage. This explicitness constitutes a strength, as it renders the realizability of each α < ω₂ directly verifiable from the recursive definition.

minor comments (3)

[Introduction] The opening section should recall the precise definition of Novikov's well-ordering relation and the notion of uniformity for Borel sets, including the relevant quantifiers over reals, to make the paper self-contained for readers outside the immediate subfield.
[Main construction (transfinite recursion)] At limit stages of cofinality ω₁, the argument that the coded system remains a single uniform Borel set would benefit from an explicit lemma stating the closure properties used; the current sketch is plausible but terse.
A short table or diagram summarizing the three cases (successor, cf ω, cf ω₁) would improve readability of the recursive construction.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No specific major comments or requested changes were provided in the report.

Circularity Check

0 steps flagged

Direct ZFC transfinite recursion constructs the order types without reduction to inputs or self-citations

full rationale

The manuscript proves by transfinite recursion on α < ω₂ that every such ordinal arises as the Novikov-order type of a uniform Borel system. At successors an explicit uniform set is adjoined to extend the type by 1; at ω-limits a diagonal union preserves uniformity and takes the supremum; at ω₁-limits a single Borel code for a countable sequence of prior systems is used. All steps are carried out inside ZFC, rely only on the standard definition of Borel sets and Novikov’s relation, and introduce no fitted parameters, self-definitional equations, or load-bearing self-citations. The central existence claim is therefore independent of the paper’s own prior results and does not collapse to any of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The result rests on standard ZFC axioms and the classical definition of Borel and uniform sets; no new free parameters, ad-hoc axioms, or invented entities are introduced in the abstract statement.

axioms (1)

standard math ZFC set theory
Standard foundation invoked for all constructions and existence proofs in descriptive set theory.

pith-pipeline@v0.9.0 · 5330 in / 1122 out tokens · 32203 ms · 2026-05-10T16:48:52.182627+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

11 extracted references · 11 canonical work pages

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Notes Log.Cambridge: Cambridge University Press; Ithaca, NY: Association for Symbolic Logic (ASL), 2013

Noam Greenberg, Joel David Hamkins, Denis Hirschfeldt, and Russell Miller, editors.Effective mathematics of the uncountable, volume 41 ofLect. Notes Log.Cambridge: Cambridge University Press; Ithaca, NY: Association for Symbolic Logic (ASL), 2013

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Infinite time Turing machines.Journal of Symbolic Logic, 65(2):567–604, 2000

Joel David Hamkins and Andy Lewis. Infinite time Turing machines.Journal of Symbolic Logic, 65(2):567–604, 2000

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Springer-Verlag, Berlin-Heidelberg-New York, The third millennium revised and expanded edition, 2003

Thomas Jech.Set theory. Springer-Verlag, Berlin-Heidelberg-New York, The third millennium revised and expanded edition, 2003. Pages xiii + 769

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On some classical problems in de- scriptive set theory.Russ

Vladimir Kanovei and Vassily Lyubetsky. On some classical problems in de- scriptive set theory.Russ. Math. Surv., 58(5):839–927, 2003

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“MCCME”, Moscow, 2013

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Surlesfonctionsrepr´ esentablesanalytiquement.Journ

H.Lebesgue. Surlesfonctionsrepr´ esentablesanalytiquement.Journ. de Math., (6) 1, 139-216, 1905

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Lusin and P

N. Lusin and P. Novikoff. Choix effectif d’un point dans un compl´ ementaire analytique arbitraire, donn´ e par un crible.Fundamenta Math., 25:559–560, 1935

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Nicolas Luzin.On some new results of descriptive theory of functions. USSR Academy of Science, Moscow – Leningrad, 1935. (Russian)

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Novikoff

P. Novikoff. Sur la s´ eparabilit´ e des ensembles projectifs de seconde classe. Fundam. Math., 25:459–466, 1935

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Novikoff

P. Novikoff. Les projections des compl´ ementaires analytiques uniformes.Rec. Math. Moscou, n. Ser., 2:3–15, 1937

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Novikoff

P. Novikoff. Sur quelques relations entre les familles des ensembles projectifs de classe 2 et des projections des compl´ ementaires analytiques uniformes.Izv. Akad. Nauk SSSR, Ser. Mat., 1(1-4):231–252, 1937. 13

work page 1937