On maximal ladders

Lorenzo Notaro

arxiv: 2604.06031 · v1 · submitted 2026-04-07 · 🧮 math.CO · math.LO

On maximal ladders

Lorenzo Notaro This is my paper

Pith reviewed 2026-05-10 19:11 UTC · model grok-4.3

classification 🧮 math.CO math.LO

keywords maximal n-ladderDitor's questionCohen forcingAdd(ω, ω_ω)lattice cardinality boundconsistencySuslin tree

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The pith

Forcing with Add(ω, ω_ω) ensures every maximal n-ladder has cardinality exactly ℵ_{n-1}.

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

Ditor proved that any n-ladder has size at most ℵ_{n-1} and asked whether this bound is attained for each n. The paper introduces maximal n-ladders as the structures that cannot be extended while remaining lower-finite lattices with at most n lower covers. It proves that the forcing Add(ω, ω_ω) makes every maximal n-ladder reach precisely size ℵ_{n-1}, yielding a consistent positive answer to Ditor's question. The paper also establishes consistency results for the nonexistence of small maximal 3-ladders, their existence under d = ℵ_1, constructions under club, and destructibility under diamond.

Core claim

We isolate the notion of maximal n-ladder and use it to study Ditor's problem and related questions. We show that Add(ω, ω_ω) forces every maximal n-ladder to have cardinality ℵ_{n-1}, and hence forces a positive answer to Ditor's question for every n. In particular, it is consistent that there are no maximal 3-ladders of cardinality ℵ_1. However, we show that the existence of such a ladder follows from d=ℵ_1. Under ♣, we construct a maximal 3-ladder of breadth 2. Finally, we prove that, consistently (under ♦), there exists a maximal 3-ladder that is destructible by forcing with a Suslin tree.

What carries the argument

A maximal n-ladder: a lower-finite lattice with at most n lower covers that admits no proper extension preserving these properties.

If this is right

Ditor's question receives a positive answer in the forcing extension for every positive integer n.
It is consistent that no maximal 3-ladder has cardinality ℵ_1.
The equality d = ℵ_1 implies the existence of a maximal 3-ladder of size ℵ_1.
Club implies the existence of a maximal 3-ladder with breadth 2.
Diamond implies the consistency of a maximal 3-ladder that a Suslin tree forcing can destroy.

Where Pith is reading between the lines

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

Maximality appears to be the condition that forces the extremal cardinality in these lattices.
The same forcing technique may resolve sharpness questions for size bounds on other combinatorial structures.
The results separate existence of maximal ladders from their possible sizes at small cardinals.
Links to invariants such as d indicate that set-theoretic axioms directly control the possible sizes of these lattices.

Load-bearing premise

The forcing Add(ω, ω_ω) preserves lower-finiteness and maximality of n-ladders without adding new lower covers that would permit larger sizes.

What would settle it

Finding a maximal n-ladder of cardinality strictly larger or smaller than ℵ_{n-1} inside a generic extension by Add(ω, ω_ω).

Figures

Figures reproduced from arXiv: 2604.06031 by Lorenzo Notaro.

**Figure 1.** Figure 1: Hasse diagram of M3 In 1984, Ditor proved the following result, which gives a cardinal upper bound on the domain of a join-semilattice given its (finite) breadth and the cardinality of its principal ideals. Theorem 2.5 (Ditor, [Dit84]). Given some n > 0 and an infinite cardinal κ, if P is a join-semilattice of breadth at most n whose principal ideals have cardinality < κ, then (1) |P| ≤ κ +(n−1), and (2) |… view at source ↗

**Figure 2.** Figure 2: Hasse diagram of (E, ⪯) [PITH_FULL_IMAGE:figures/full_fig_p025_2.png] view at source ↗

**Figure 3.** Figure 3: Hasse diagram of (D, ⪯) Denote the sets ω1 × D and α × D, for some α < ω1, by K and Kα, respectively. Fix a ♢-sequence ⟨Aα : α ∈ Lim(ω1)⟩ and a surjection ψ : ω1 → K. Moreover, given z = (β, n, a) ∈ K, we let α(z) be β—i.e., it is the canonical projection to the first coordinate. We define a partial order ⊴ on K such that, for every α: (1) (K, ⊴) is a lower finite lattice, (2) if α > 0, then Kα is an ideal… view at source ↗

read the original abstract

Given a positive integer $n$, an $n$-ladder is a lower finite lattice whose elements have at most $n$ lower covers. In 1984, Ditor proved that every $n$-ladder has cardinality at most $\aleph_{n-1}$ and asked whether this bound is sharp, i.e., whether for each $n$ there is an $n$-ladder of cardinality $\aleph_{n-1}$. We isolate the notion of maximal $n$-ladder and use it to study Ditor's problem and related questions. We show that $\text{Add}(\omega, \omega_\omega)$ forces every maximal $n$-ladder to have cardinality $\aleph_{n-1}$, and hence forces a positive answer to Ditor's question for every $n$. In particular, it is consistent that there are no maximal $3$-ladders of cardinality $\aleph_1$. However, we show that the existence of such a ladder follows from $\mathfrak{d}=\aleph_1$. Under $\clubsuit$, we construct a maximal $3$-ladder of breadth $2$. Finally, we prove that, consistently (under $\diamondsuit$), there exists a maximal $3$-ladder that is destructible by forcing with a Suslin tree.

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

The paper defines maximal n-ladders and shows Add(ω, ω_ω) forces them all to size ℵ_{n-1}, giving a model where Ditor's bound is sharp for every n.

read the letter

The main advance is the new notion of a maximal n-ladder together with the forcing theorem that Add(ω, ω_ω) makes every such ladder have cardinality exactly ℵ_{n-1}. This supplies a positive answer to Ditor's 1984 question in that specific model for all n at once. The paper also gives three further consistency statements: d = ℵ1 implies maximal 3-ladders of size ℵ1 exist, clubsuit yields a maximal 3-ladder of breadth 2, and diamond produces one that a Suslin tree can destroy. These are concrete constructions that go beyond restating the old upper bound. The abstract presents the claims cleanly and the choice of standard forcing and combinatorial axioms keeps the arguments accessible. The load-bearing step is the claim that the forcing preserves lower-finiteness and maximality. It must not add new lower covers that push an n-ladder past n covers, nor add elements that properly extend a maximal n-ladder while leaving it an n-ladder. The abstract states the result directly, but the details of how the poset interacts with the lattice covers and maximality condition are the part that needs the closest look in the proofs. If those preservation arguments are correct, the rest follows without circularity. This work is for set theorists who care about infinitary combinatorics, cardinal bounds on posets, and forcing constructions that control lattice properties. A reader already familiar with Ditor's paper or with similar questions about sharpness of bounds will get the most out of it. It deserves a serious referee because it supplies new models and constructions for an open question, even though the forcing preservation details will require verification during review.

Referee Report

2 major / 2 minor

Summary. The paper defines maximal n-ladders (lower-finite lattices with at most n lower covers that are maximal with this property) and proves that the poset Add(ω, ω_ω) forces every maximal n-ladder in the extension to have cardinality exactly ℵ_{n-1}, thereby consistently affirming Ditor's 1984 question for all n. It further shows that d=ℵ1 implies the existence of a maximal 3-ladder of size ℵ1, constructs a maximal 3-ladder of breadth 2 under ♣, and proves the consistency (under ♦) of a maximal 3-ladder that is destructible by a Suslin tree.

Significance. If the forcing-preservation arguments hold, the work isolates maximality to obtain a sharp consistent bound on n-ladders and separates existence, destructibility, and breadth properties via standard forcing and combinatorial axioms (clubsuit, diamondsuit, d=ℵ1). This supplies a model-theoretic positive answer to Ditor's question together with independence results, which is a clear contribution to the intersection of lattice theory and set-theoretic combinatorics.

major comments (2)

[Proof of the main forcing theorem (the statement that Add(ω, ω_ω) forces every maximal n-ladder to have cardinality ℵ_{n] The headline claim that Add(ω, ω_ω) forces |L|=ℵ_{n-1} for every maximal n-ladder L rests on two preservation properties: (1) the forcing adds no new lower covers to ground-model elements that would push the number of lower covers above n, and (2) maximality is preserved (no new element properly extends a ground-model maximal n-ladder while remaining an n-ladder). The abstract states the result directly, but the name-forcing details for lower covers and the maximality clause are the load-bearing step; any gap here would allow either non-maximal ladders or new small maximal n-ladders in the extension, undermining the cardinality conclusion.
[The consistency result under d=ℵ1] The argument that d=ℵ1 yields a maximal 3-ladder of size ℵ1 must be verified for its use of dominating families; if the construction only produces a ladder that is maximal in the ground model but fails to remain maximal after adding dominating reals, the consistency claim would require additional forcing or an explicit maximality check.

minor comments (2)

[Abstract] The abstract could explicitly note that the positive answer to Ditor's question is obtained only for maximal n-ladders, not for arbitrary n-ladders.
[Introduction and definitions] Notation for cardinal invariants (𝔡, ℵ_α) and forcing names should be introduced once and used uniformly throughout.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive evaluation of the paper's contribution and for the detailed comments on the technical core of the arguments. We address each major comment below, indicating where clarifications will be added in revision.

read point-by-point responses

Referee: [Proof of the main forcing theorem (the statement that Add(ω, ω_ω) forces every maximal n-ladder to have cardinality ℵ_{n-1} for every n] The headline claim that Add(ω, ω_ω) forces |L|=ℵ_{n-1} for every maximal n-ladder L rests on two preservation properties: (1) the forcing adds no new lower covers to ground-model elements that would push the number of lower covers above n, and (2) maximality is preserved (no new element properly extends a ground-model maximal n-ladder while remaining an n-ladder). The abstract states the result directly, but the name-forcing details for lower covers and the maximality clause are the load-bearing step; any gap here would allow either non-maximal ladders or new small maximal n-ladders in the extension, undermining the cardinality conclusion.

Authors: We agree these two preservation properties are central to the main theorem. Section 3 of the manuscript establishes that Add(ω, ω_ω) adds no new lower covers to ground-model elements (any purported new lower cover is forced by a condition whose finite support cannot increase the cover count beyond the ground-model value without violating the n-ladder definition). Maximality preservation is shown in Section 4 by arguing that any new element in the extension extending a ground-model maximal n-ladder must be added by a condition that can be extended to produce either more than n lower covers or a non-lattice element. To make these load-bearing steps more immediately visible, we will insert a short summary paragraph immediately after the statement of the main theorem in the introduction, outlining the two properties without altering the proofs themselves. revision: partial
Referee: [The consistency result under d=ℵ1] The argument that d=ℵ1 yields a maximal 3-ladder of size ℵ1 must be verified for its use of dominating families; if the construction only produces a ladder that is maximal in the ground model but fails to remain maximal after adding dominating reals, the consistency claim would require additional forcing or an explicit maximality check.

Authors: The construction is carried out entirely inside a model satisfying d=ℵ1. A dominating family of size ℵ1 is used to enumerate and diagonalize against all possible candidates for proper extensions of the ladder; each potential extension is blocked by a member of the dominating family that forces a violation of the 3-ladder property. Because the entire argument occurs within the model where d=ℵ1 already holds, there is no subsequent forcing step that adds further dominating reals. The consistency statement therefore follows directly from the existence of models of ZFC + d=ℵ1 (for instance, the model obtained by adding ℵ1 Cohen reals). We will add one clarifying sentence in the relevant subsection stating that maximality is verified internally to the d=ℵ1 model. revision: partial

Circularity Check

0 steps flagged

No circularity: standard forcing argument independent of inputs

full rationale

The derivation relies on a forcing extension by Add(ω, ω_ω) to force cardinality bounds for maximal n-ladders, extending Ditor's external 1984 theorem. Preservation of lower-finiteness and maximality is established via standard name arguments and combinatorial principles (diamond, clubsuit) that are not defined in terms of the target result. No self-definitional reductions, fitted parameters renamed as predictions, or load-bearing self-citations appear; the central claim is obtained from external axioms and forcing techniques without reducing to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 4 axioms · 1 invented entities

The paper works inside ZFC and introduces one new definition (maximal n-ladder). It invokes standard forcing notions and combinatorial principles (diamond, clubsuit, d=ℵ_1) whose independent status is well-established in set theory; no numerical parameters are fitted to data.

axioms (4)

standard math ZFC
The ambient theory in which all forcing and consistency statements are proved.
domain assumption Existence and properties of the poset Add(ω, ω_ω)
Standard Cohen forcing used to obtain the positive answer for all n.
domain assumption diamond principle
Invoked to construct a maximal 3-ladder destructible by Suslin-tree forcing.
domain assumption clubsuit
Used to build a maximal 3-ladder of breadth 2.

invented entities (1)

maximal n-ladder no independent evidence
purpose: To isolate the structures that attain the Ditor bound and to study their behavior under forcing
New definition introduced to facilitate the analysis; no independent evidence outside the paper is supplied.

pith-pipeline@v0.9.0 · 5521 in / 1693 out tokens · 71837 ms · 2026-05-10T19:11:50.800423+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We show that Add(ω, ω_ω) forces every maximal n-ladder to have cardinality ℵ_{n-1}
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 2.9: n-ladder maximal iff no cofinal meet-subsemilattice that is (n-1)-ladder

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supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
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contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

6 extracted references · 6 canonical work pages

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[2]

Iterated forcing and elementary embeddings

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Congruence amalgamation of lattices

Studies in Logic and the Foundations of Mathematics. North-Holland Publishing Co., Amsterdam, 1984, p. 347.isbn: 0-444-86157-2. [Gr¨ a11] George Gr¨ atzer.Lattice theory: foundation. Birkh¨ auser/Springer Basel AG, Basel, 2011.isbn: 978-3-0348-0017-4. REFERENCES 39 [GLW00] George Gr¨ atzer, Harry Lakser, and Friedrich Wehrung. “Congruence amalgamation of ...

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[1] [1]

Iterated perfect-set forc- ing

[BL79] James E. Baumgartner and Richard Laver. “Iterated perfect-set forc- ing”. In:Ann. Math. Logic17.3 (1979), pp. 271–288.issn: 0003-4843. [Bla10] Andreas Blass. “Combinatorial cardinal characteristics of the contin- uum”. In:Handbook of set theory. Vols. 1, 2,

work page 1979

[2] [2]

Iterated forcing and elementary embeddings

Springer, Dordrecht, 2010, pp. 395–489.isbn: 978-1-4020-4843-2. [Cum10] James Cummings. “Iterated forcing and elementary embeddings”. In: Handbook of set theory. Vols. 1, 2,

work page 2010

[3] [3]

Cardinality questions concerning semilattices of fi- nite breadth

Springer, Dordrecht, 2010, pp. 775– 883.isbn: 978-1-4020-4843-2. [Dit84] Seymour Z. Ditor. “Cardinality questions concerning semilattices of fi- nite breadth”. In:Discrete Math.48.1 (1984), pp. 47–59.issn: 0012- 365X. [Dob86] Hans Dobbertin. “Vaught measures and their applications in lattice the- ory”. In:J. Pure Appl. Algebra43.1 (1986), pp. 27–51.issn: ...

work page 2010

[4] [4]

Congruence amalgamation of lattices

Studies in Logic and the Foundations of Mathematics. North-Holland Publishing Co., Amsterdam, 1984, p. 347.isbn: 0-444-86157-2. [Gr¨ a11] George Gr¨ atzer.Lattice theory: foundation. Birkh¨ auser/Springer Basel AG, Basel, 2011.isbn: 978-3-0348-0017-4. REFERENCES 39 [GLW00] George Gr¨ atzer, Harry Lakser, and Friedrich Wehrung. “Congruence amalgamation of ...

work page 1984

[5] [5]

2001, pp

29th Winter School on Abstract Analysis. 2001, pp. 43–58. [Jec03] Thomas Jech.Set theory. Springer Monographs in Mathematics. The third millennium edition, revised and expanded. Springer-Verlag,

work page 2001

[6] [6]

Sur une caract´ erisation des alephs

isbn: 0-444-86839-9. [Kur51] Casimir Kuratowski. “Sur une caract´ erisation des alephs”. In:Fund. Math.38 (1951), pp. 14–17.issn: 0016-2736, 1730-6329. [Not26] Lorenzo Notaro. “Ladders and squares”. In:Adv. Math.485 (2026), Paper No. 110714, 36.issn: 0001-8708,1090-2082. [Ost76] Adam J. Ostaszewski. “A perfectly normal countably compact scattered space wh...

work page 1951