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arxiv: 2605.19044 · v1 · pith:WP55WBPZnew · submitted 2026-05-18 · 🧮 math.LO

On antichain numbers and the splitting ideal

Aleksander Cie\'slak This is my paper

Pith reviewed 2026-05-20 07:51 UTC · model grok-4.3

classification 🧮 math.LO
keywords antichain numbersplitting idealBorel idealscardinal invariantsF sigma idealsanalytic P idealsKatětov orderconsistency
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The pith

For many Borel ideals J on the naturals, min{b, cov^+_h(J)} is a lower bound on the antichain number a(J).

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

The paper investigates the splitting ideal on omega together with the antichain numbers a(J) of the quotient algebras P(omega)/J for various Borel ideals. It proves the inequality min{b, cov^+_h(J)} ≤ a(J) for all F_sigma ideals, all analytic P-ideals, and many additional examples. Separate arguments establish b ≤ a(J) for the convergent ideal and the Boring ideal, while forcing constructions yield models in which a(J) < b for Van der Waerden's ideal and the linear growth ideal. These relations locate the antichain numbers among the classical cardinal invariants and clarify the Katětov position of the splitting ideal.

Core claim

The central claim is that min{b, cov^+_h(J)} ≤ a(J) holds for a wide class of ideals including all F_σ-ideals and analytic P-ideals, that b ≤ a(J) for the convergent and Boring ideals, and that a(J) < b is consistent for Van der Waerden's ideal and the linear growth ideal; the splitting ideal itself is located in the Katětov order and its cardinal invariants are computed explicitly.

What carries the argument

The antichain number a(J), the smallest cardinality of a maximal antichain in the quotient Boolean algebra P(ω)/J.

If this is right

  • The splitting ideal occupies a definite place among definable ideals in the Katětov order.
  • The antichain number a(J) is forced to lie above min{b, cov^+_h(J)} for every F_sigma ideal.
  • Consistency of a(J) < b is obtained for two concrete ideals by standard forcing techniques.
  • These bounds connect a(J) to the bounding number b for convergent and Boring ideals.

Where Pith is reading between the lines

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

  • The same style of argument may extend the inequality min{b, cov^+_h(J)} ≤ a(J) to still larger classes of ideals not treated in the paper.
  • The Katětov-order calculations for the splitting ideal could be used to compare it with other definable ideals whose antichain numbers remain uncomputed.
  • The consistency results suggest that further separations between a(J) and other invariants may be achievable by iterating the same forcing notions.

Load-bearing premise

The combinatorial arguments assume that the standard definitions of the splitting ideal, the listed Borel ideals, and the cardinal invariants b and cov^+_h(J) behave as expected in ZFC and in the forcing extensions used.

What would settle it

An explicit Borel ideal J together with a ZFC proof that min{b, cov^+_h(J)} > a(J), or a model in which a(J) ≥ b for Van der Waerden's ideal, would refute the stated inequalities or consistency claim.

read the original abstract

In this article, we study combinatorial properties of a certain ideal on $\omega$, called the \emph{Splitting ideal}. We calculate its cardinal invariants and its position in the Kat\v{e}tov order among other definable ideals. We also study the antichain numbers $\mathfrak{a}(\mathcal{J})$ of algebras $\mathcal{P}(\omega)/\mathcal{J}$ for various Borel ideals. We show that $\textrm{min}\{\mathfrak{b},\textrm{cov}^{+}_{h}(\mathcal{J})\}\leq\mathfrak{a}(\mathcal{J})$ holds for a wide class of ideals, including all $F_{\sigma}$-ideals, all analytic $P$-ideals and many other examples. We also show that $\mathfrak{b}\leq\mathfrak{a}(\mathcal{J})$ holds for \emph{convergent ideal} and for \emph{Boring ideal}. Finally, we will show the consistency of $\mathfrak{a}(\mathcal{J})<\mathfrak{b}$ for the \emph{Van der Waerden's ideal} and the linear growth ideal

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

1 major / 3 minor

Summary. The paper studies the splitting ideal on ω, computing its cardinal invariants and its position in the Katětov order relative to other definable ideals. It establishes that min{b, cov⁺_h(J)} ≤ a(J) for a wide class of Borel ideals J (including all F_σ-ideals and analytic P-ideals), shows b ≤ a(J) for the convergent ideal and the Boring ideal, and proves the consistency of a(J) < b for Van der Waerden's ideal and the linear growth ideal.

Significance. If the results hold, the work contributes concrete calculations for the splitting ideal and new inequalities and consistency results for antichain numbers a(J) of quotient algebras P(ω)/J. These provide lower bounds in terms of b and cov⁺_h(J) for many standard ideals and exhibit examples where a(J) can be strictly smaller than b, which is relevant to the broader study of cardinal invariants and forcing over ideals on ω.

major comments (1)
  1. [§4] §4 (Consistency results): The forcing construction establishing a(J) < b for Van der Waerden's ideal is only sketched; it is unclear whether the iteration preserves the relevant properties of the ideal or the value of b without additional verification that the generic extension does not collapse the inequality.
minor comments (3)
  1. [§2] The definition of the splitting ideal in §2 should include an explicit reference to its generating sets or a comparison with the standard splitting number s to clarify its relation to known invariants.
  2. [Introduction] Notation for cov⁺_h(J) is introduced without a prior definition or citation; add a short paragraph recalling the standard definition from the literature on covering numbers for ideals.
  3. [§3] Figure 1 (Katětov diagram) has overlapping arrows that reduce readability; consider separating the F_σ and analytic P-ideal classes for clarity.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and the helpful comment on our manuscript. We address the major comment below and will incorporate the suggested clarifications in the revised version.

read point-by-point responses

  1. Referee: [§4] §4 (Consistency results): The forcing construction establishing a(J) < b for Van der Waerden's ideal is only sketched; it is unclear whether the iteration preserves the relevant properties of the ideal or the value of b without additional verification that the generic extension does not collapse the inequality.

    Authors: We agree that the forcing argument in §4 was presented concisely and that a more explicit verification of the preservation properties is warranted. In the revision we will expand the construction to include a detailed argument that the iterated forcing is proper, that it preserves the relevant combinatorial properties of the Van der Waerden ideal (in particular, that the ideal remains Borel and that the antichain number remains small), and that b is not collapsed in the extension. We will also verify that the generic extension satisfies a(J) < b by showing that the forcing does not add dominating reals or destroy the relevant unbounded families. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected in derivation chain

full rationale

The paper derives inequalities such as min{b, cov^+_h(J)} ≤ a(J) for F_σ-ideals, analytic P-ideals and other classes, plus b ≤ a(J) for convergent and Boring ideals, and consistencies a(J) < b for specific ideals, via standard ZFC combinatorial arguments and forcing constructions. These rest on the usual definitions of the splitting ideal, Katětov order, and cardinal invariants without any self-definitional reductions, fitted inputs renamed as predictions, or load-bearing self-citations that collapse the claims to their own inputs. The manuscript supplies explicit verifications aligned with external set-theoretic techniques, rendering the derivation self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, invented entities, or non-standard axioms are mentioned in the abstract; all claims appear to rest on ZFC plus standard definitions of Borel ideals and cardinal invariants.

pith-pipeline@v0.9.0 · 5703 in / 1183 out tokens · 34266 ms · 2026-05-20T07:51:12.101227+00:00 · methodology

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