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arxiv: 1907.08323 · v1 · pith:4FYEOLOKnew · submitted 2019-07-19 · 🧮 math.GN

Universal sets for ideals

Aleksander Cie\'slak , Marcin Michalski This is my paper

Pith reviewed 2026-05-24 19:15 UTC · model grok-4.3

classification 🧮 math.GN
keywords universal setsidealsBorel complexitynull idealmeager idealFubini productsLaver ideal
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The pith

There exist universal sets of minimal Borel complexity for the null ideal on 2^ω and the meager ideal on Polish spaces.

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

The paper establishes the existence of universal sets at the lowest possible Borel complexity for several important ideals on Polish spaces. These include the ideal of null sets and the ideal of meager sets. The existence of these sets helps in proving properties of the real line that hold in generic extensions by the corresponding forcings. The authors provide constructions for additional ideals including E, K_sigma, the Laver ideal, and show that Sigma^0_3 level suffices for the Fubini products of null and meager.

Core claim

Universal sets of minimal Borel complexity exist for the null subsets of 2^ω, meager subsets of any Polish space, the σ-ideal E generated by closed null subsets of 2^ω, K_σ subsets of ω^ω, the Laver ideal, and there are Σ^0_3 universal sets for N⊗M and M⊗N. The existence is helpful in establishing facts about the real line in generic extensions.

What carries the argument

Universal set for an ideal: a set in a product space whose sections generate all members of the ideal at the specified Borel complexity level.

If this is right

  • The existence simplifies establishing facts about the real line in generic extensions.
  • Universal sets can be constructed for the E ideal, K_sigma ideal, and Laver ideal.
  • Sigma^0_3 universal sets exist for the Fubini products N⊗M and M⊗N.

Where Pith is reading between the lines

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

  • These universal sets could be used to investigate other sigma-ideals associated with forcing notions.
  • Minimal complexity might allow for better bounds in cardinal invariant calculations involving these ideals.
  • Similar constructions may apply to other Polish spaces or different product ideals.

Load-bearing premise

The definitions of universal sets and the Borel hierarchy on Polish spaces permit explicit constructions of minimal complexity in ZFC alone.

What would settle it

Demonstrating that no universal set of the claimed minimal complexity exists for the null ideal would falsify the main result.

read the original abstract

In this paper we consider a notion of universal sets for ideals. We show that there exist universal sets of minimal Borel complexity for classic ideals like null subsets of $2^\omega$ and meager subsets of any Polish space, and demonstrate that the existence of such sets is helpful in establishing some facts about the real line in generic extensions. We also construct universal sets for $\mathcal{E}$ - the $\sigma$-ideal generated by closed null subsets of $2^\omega$, and for some ideals connected with forcing notions: $\mathcal{K}_\sigma$ subsets of $\omega^{\omega}$ and the Laver ideal. We also consider Fubini products of ideals and show that there are $\Sigma^0_3$ universal sets for $\mathcal{N}\otimes\mathcal{M}$ and $\mathcal{M}\otimes\mathcal{N}$.

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

0 major / 2 minor

Summary. The paper defines universal sets for ideals on Polish spaces and constructs them at the lowest possible Borel complexity for the null ideal on 2^ω, the meager ideal on any Polish space, the σ-ideal E generated by closed null sets, the K_σ ideal on ω^ω, the Laver ideal, and Σ^0_3 universal sets for the Fubini products N⊗M and M⊗N. It shows these constructions are useful for proving facts about the real line in generic extensions, all within ZFC using standard coding and diagonalization.

Significance. The explicit ZFC constructions of minimal-complexity universal sets for these classic and forcing-related ideals strengthen the toolkit in descriptive set theory. They enable cleaner arguments about properties of the reals in generic extensions without extra assumptions, and the parameter-free nature of the constructions (via standard Borel coding) is a strength.

minor comments (2)
  1. The abstract states results for N⊗M and M⊗N at Σ^0_3 but does not indicate whether the same complexity is achieved for other Fubini products; a brief remark in the introduction on why these two are treated would improve readability.
  2. Notation for the ideals (e.g., script N, script M, script E) is introduced without an early consolidated list; adding a short table or paragraph in §1 listing all ideals considered would aid navigation.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, the assessment of significance, and the recommendation to accept the manuscript. We are pleased that the explicit ZFC constructions and their applications were viewed as strengthening the toolkit in descriptive set theory.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper consists of explicit ZFC constructions of universal sets at claimed minimal Borel complexities for ideals such as the null ideal on 2^ω, the meager ideal, E, K_σ, the Laver ideal, and Fubini products N⊗M and M⊗N. These are carried out via standard coding, diagonalization, and descriptive-set-theoretic techniques without any derivations that reduce by construction to fitted parameters, self-definitions, or load-bearing self-citations. All results are framed as existence proofs and direct constructions that remain self-contained within ZFC and do not invoke uniqueness theorems or ansatzes from the authors' prior work in a circular manner. This is the normal case of an independent mathematical construction paper.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract provides no information on free parameters, invented entities or non-standard axioms; relies on standard ZFC and descriptive set theory notions.

axioms (1)
  • standard math ZFC set theory
    Standard foundation assumed for all proofs in descriptive set theory and forcing.

pith-pipeline@v0.9.0 · 5656 in / 1161 out tokens · 28575 ms · 2026-05-24T19:15:27.230153+00:00 · methodology

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Reference graph

Works this paper leans on

17 extracted references · 17 canonical work pages

  1. [1]

    Balcerzak, Sz

    M. Balcerzak, Sz. Głąb, Measure-category properties of Borel plane sets and Borel functions of two variables , Acta Math. Hungar., vol. 126, no. 3 (2010), 241-252

    work page 2010

  2. [2]

    Bartoszyński, H

    T. Bartoszyński, H. Judah, Set Theory: On the structure of the real line, A K Peters, 1995

    work page 1995

  3. [3]

    Borodulin-Nadzieja, Sz

    P. Borodulin-Nadzieja, Sz. Głąb, Ideals with bases of unbounded Borel complexity, Mathematical Logic Quarterly, vol. 57, no. 6 (2011), 582- 590. Universal sets for ideals 11

    work page 2011

  4. [4]

    Brendle, Y

    J. Brendle, Y. Khomskii, W. Wohofsky, Cofinalities of Marczewski-like ideals, Colloquium Mathematicum, vol. 150 (2017), 269-279

    work page 2017

  5. [5]

    Cichoń, J

    J. Cichoń, J. Pawlikowski, On ideals of subsets of the plane and on Cohen reals , The Journal of Symbolic Logic, vol. 51, no. 3 (1986), 561-569

    work page 1986

  6. [6]

    D. H. Fremlin, The partially ordered sets of measure theory and Tukey’s ordering , Note Mat., vol. 11 (1991), 177-214

    work page 1991

  7. [7]

    Jech, Set theory, millenium edition , Springer Monographs in Math- ematics, Springer-Verlag, 2003

    T. Jech, Set theory, millenium edition , Springer Monographs in Math- ematics, Springer-Verlag, 2003

    work page 2003

  8. [8]

    Judah, A

    H. Judah, A. Miller, S. Shelah, Sacks forcing, Laver forcing, and Mar- tin ’s axiom , Arch. Math. Logic, vol. 31 (1992), 145-161

    work page 1992

  9. [9]

    A. S. Kechris, Classical descriptive set theory , Graduate Texts in Mathematics 156, Springer-Verlag New York, Inc., 1995

    work page 1995

  10. [10]

    Krzeszowiec, Uniquely universal sets in R × ω and [0, 1] × ω, Topol- ogy and its Applications, vol

    A. Krzeszowiec, Uniquely universal sets in R × ω and [0, 1] × ω, Topol- ogy and its Applications, vol. 182 (2015), 132-134

    work page 2015

  11. [11]

    Kunen, Random and Cohen reals, Handbook of set-theoretic topol- ogy, Elsevier Science Publishers B.V., 1984

    K. Kunen, Random and Cohen reals, Handbook of set-theoretic topol- ogy, Elsevier Science Publishers B.V., 1984

    work page 1984

  12. [12]

    A. W. Miller, Uniquely Universal sets , Topology and its Applications, vol. 159 (2012), 3033-3041

    work page 2012

  13. [13]

    Pawlikowski, I

    J. Pawlikowski, I. Recław, Parametrized Cichoń’s diagram and small sets, Fundamenta Mathematicae, vol. 147 (1995), 135-155

    work page 1995

  14. [14]

    Rałowski, Sz

    R. Rałowski, Sz. Żeberski, Generalized Luzin sets , Houston Journal of Mathematics, vol. 39, no. 3 (2013), 983-993

    work page 2013

  15. [15]

    Recław, P

    I. Recław, P. Zakrzewski, Fubini properties of ideals , Real Analysis Exchange, vol. 25, no. 2 (1999), 565-578

    work page 1999

  16. [16]

    S. M. Srivastava, A course on Borel sets , Graduate Texts in Mathe- matics 180, Springer-Verlag New York, Inc., 1998

    work page 1998

  17. [17]

    Zapletal, Descriptive set theory and definable forcing , Memoirs of the American Mathematical Society, vol

    J. Zapletal, Descriptive set theory and definable forcing , Memoirs of the American Mathematical Society, vol. 167, no. 793 (2004)

    work page 2004