Towers and clubs

Pierre Matet

REVIEW 3 major objections 5 minor 49 references

Towers and clubs

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Club guessing produces long ascending or descending towers in every ideal extending the nonstationary ideal over a successor cardinal.

desk verdict Matet's tower theorems are a useful generalization but Theorem 3.9 has a genuine cardinality gap that needs fixing before the centerpiece holds. read the letter →

arxiv 1908.05336 v1 pith:IF7OCWWY submitted 2019-08-14 math.LO

Pierre Matet This is my paper · ORCID

classification math.LO MSC 03E05

keywords clubprinciplesaturatedidealtowerdiamondguessingnonstationarydepthofthefunctionspacesettheory

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper tries to show that weak club-guessing principles force every $\kappa$-complete ideal extending the nonstationary ideal on a regular uncountable cardinal $\kappa$ to be non-saturated, and to witness this non-saturation by explicit towers of positive sets. The central theorem states that when $\kappa=\nu^+$ and the cofinal club-guessing principle $\clubsuit^{\mathrm{cof},-}_{\kappa}[J]$ holds, every regular $\tau<\mathrm{Depth}(\kappa^\kappa)$ admits either an ascending $(J,I_\kappa)$-tower or a descending $(J,J)$-tower of length $\tau$. Tower existence is a direct certificate that the ideal cannot be $\tau$-saturated, and the result covers ideals that are not normal. The same mechanism is extended to guessing generalized clubs and to full stationary reflection, and several known diamond and club results are reproved in passing.

What carries the argument

The carrier of the argument is the cofinal club-guessing principle $\clubsuit^{\mathrm{cof},-}_{\kappa}[J]$, which supplies small sets $B^i_\delta\subseteq\delta$ for $i<\delta$ such that for every $W\in[\kappa]^\kappa$, the collection of $\delta$ where some $B^i_\delta$ has cofinal intersection with $W$ is $J^+$. Lemma 3.8 converts these into $\nu$-sized pieces $A^i_\delta$ of $\kappa\times\kappa$ that capture $f\upharpoonright Z$ for cofinal $Z\subseteq\delta$. Towers are sequences $\langle A_\alpha:\alpha<\tau\rangle$ of $J^+$ sets whose successive differences lie in a prescribed family $Y$; an ascending $(J,I_\kappa)$-tower means each new part is almost disjoint from the union of earlier parts, and a descending $(J,J)$-tower means the leftover part stays $J^+$. The proof tracks the guessed sets $S_\alpha$ and the crossing sets $S_{\alpha\beta}$, and uses $\mathrm{Depth}(\kappa^\kappa)$, the least length of an unavoidable increasing chain in $(\kappa^\kappa,<^{*})$, as the ceiling for $\tau$.

What would settle it

Build a model with $\kappa=\nu^+$, a $\kappa$-complete ideal $J$ extending the nonstationary ideal, $\clubsuit^{\mathrm{cof},-}_{\kappa}[J]$, and a regular $\tau<\mathrm{Depth}(\kappa^\kappa)$ for which $J$ is $\tau$-saturated; then neither tower of length $\tau$ can exist and Theorem 3.9 fails. Concretely, inspect the proof's choice of $T$ and $\theta$: if every $J^+$-set carries guesses of size at least $\tau$, the recursion $g:\theta^+\to\tau$ cannot be defined, so exhibiting such an ideal with $\mathrm{Depth}(\kappa^\kappa)>\tau$ would pinpoint the gap.

Watch

Extended reading notes

Core claim

The paper's central claim is a uniform transfer: cofinal club guessing converts any strictly $<^{*}$-increasing chain of functions in $\kappa^\kappa$ of length $\tau$ into either an ascending $(J,I_\kappa)$-tower or a descending $(J,J)$-tower of length $\tau$, for any $\kappa$-complete ideal $J$ extending the nonstationary ideal. Theorem 3.9 states this for $\kappa=\nu^+$ under $\clubsuit^{\mathrm{cof},-}_{\kappa}[J]$ and every regular $\tau<\mathrm{Depth}(\kappa^\kappa)$. The proof first turns club guesses into sets $A^i_\delta\subseteq\kappa\times\kappa$ that capture restrictions of functions to cofinal subsets of $\delta$, then defines $J^+$ sets $S_\alpha$ where a function is guessed and $S_{\alpha\beta}$ where $\kappa$-truncations of two functions cross. Assuming neither tower exists, a diagonal recursion on those sets produces a contradiction. The paper derives corollaries for ideals extending $NS_\kappa|E^\kappa_\theta$ under cardinal arithmetic assumptions, for ideals with generalized-club guessing, and for stationary reflection.

Load-bearing premise

The proof of Theorem 3.9 secretly assumes that the guessed sets have size smaller than the tower length $\tau$; the stated hypotheses only bound them by $\nu$, and no argument is given that this smaller bound holds.

Editorial extensions

If this is right

Under $\kappa=\nu^+$ and $\clubsuit^{\mathrm{cof},-}_{\kappa}[J]$, every regular $\tau<\mathrm{Depth}(\kappa^\kappa)$ is witnessed by a tower, so $J$ is not $I_\kappa$-$\tau$-saturated (by Observation 2.29).
For ideals extending $NS_\kappa|E^\kappa_\theta$ with $\mathrm{cf}(\nu)\neq\theta$, Fact 3.12 supplies the club guessing and Proposition 3.14 yields the same tower dichotomy without extra hypotheses.
The non-normal case is covered, so ideals that are not subnormal still admit long towers whenever the relevant club principle holds.
Guessing generalized clubs (Theorem 4.7) and full stationary reflection (Theorem 5.12) each produce the same ascending-or-descending tower conclusion.
The slow-train propositions show that $\clubsuit^{\mathrm{cof},-}_{\kappa}[J]$ together with $2^{<\kappa}=\kappa$ upgrades to diamond $\lozenge_\kappa[J]$, recovering and extending an earlier diamond theorem.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the missing assumption that the guessed sets have size below $\tau$ can be proved from the stated hypotheses, then club guessing and $\mathrm{Depth}(\kappa^\kappa)$ alone control non-saturation; if not, the theorem likely needs an explicit size bound on the guesses.
The same construction might yield ascending $(J,J)$-towers or descending $(J,I_\kappa)$-towers by symmetry, which would answer the paper's Questions 6.1 and 6.2 affirmatively under club guessing.
The bound $\tau<\mathrm{Depth}(\kappa^\kappa)$ may be improvable to $\tau<\mathrm{Depth}(C_\kappa)$, since Fact 2.47 shows the two depths differ by at most one; testing this would show whether the tower length is really tied to the function space.
A natural test is whether $\clubsuit^{\mathrm{cof},-}_{\kappa}[J]$ alone yields a tower of length $\nu^+$ even when $\mathrm{Depth}(\kappa^\kappa)=\nu^+$, which would isolate whether the depth parameter is essential.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague, here's my read on Matet's 'Towers and clubs'. The paper is exactly what it says: a systematic attempt to turn known non-saturation results for the nonstationary ideal into tower existence results for arbitrary κ-complete ideals extending NS, including non-normal ones. The main new statements—Theorem 3.9, Proposition 3.14, Observation 4.8, Theorem 5.12—are genuine generalizations of results by Rinot, Gitik-Shelah, and Džamonja, and the tower formulations are a useful unifying device. The proofs are mostly detailed adaptations of known arguments rather than a new framework, but that's fine; the paper is a solid piece of combinatorial set theory.

The soft spots, though, are real. The stress-test on Theorem 3.9 holds up. In the proof, after choosing θ<ν with |A^i_δ|=θ for δ in T, the recursion defines g:θ^+→τ\(γ+1). For this to make sense, every sup of fewer than θ^+ ordinals below τ must be below τ. Since τ is regular, that requires θ^+≤τ, i.e. θ<τ. Nothing in the hypotheses gives this: θ is bounded by ν, and τ is only bounded by Depth(κ^κ). The concrete counterexample with κ=ω_{ω+1}, τ=ω, θ=ω1 shows the construction fails at step ω. So Theorem 3.9, the central new result, is unproved as written in the case θ≥τ. This is not a minor typo; the recursion is the heart of the proof.

Second, Proposition 3.15 explicitly says the proof modifies a proof 'which we do not understand.' That is an honest admission, but it also means that step is not verified. The author may be right, but a referee should ask for a self-contained argument there.

The rest of the paper is in much better shape. The facts cited from the author's own prior work are used as background, not in a circular way. The generalizations in Section 4 and the Gitik-Shelah material in Section 5 look plausible and are carefully argued. I did not spot other load-bearing gaps.

Bottom line: this paper deserves a serious referee, not a desk reject, but it should not be accepted until Theorem 3.9 is either repaired (by adding a condition like θ<τ or by a different construction) and Proposition 3.15 is clarified. For a reading group, I'd say maybe—the ideas are worth discussing, but you'd want to flag the gap. I wouldn't cite it in the next year until the central theorem is settled.

Referee Report

3 major / 5 minor

Summary. The paper studies club principles (diamond, club, and their variants) and their connection to the non-saturation of ideals extending the nonstationary ideal on a regular uncountable cardinal κ. The main new result is Theorem 3.9, which asserts that if κ = ν^+, J is a κ-complete ideal on κ satisfying ♣cof,−_κ[J], and τ is a regular cardinal less than Depth(κ^κ), then there exists either an ascending (J,I_κ)-tower of length τ or a descending (J,J)-tower of length τ. The paper also revisits several known results, proves variants, and extends results on generalized club guessing and on Gitik–Shelah style non-saturation.

Significance. If Theorem 3.9 and its corollaries (notably Proposition 3.14) are correct, they provide a uniform framework for deriving tower non-saturation of ideals from weak club principles, covering ideals that are not necessarily normal. The paper is also useful as a compendium of related principles and contains many small observations. However, the proof of the central theorem contains a gap (see Major Comment 1), so the significance of the paper is conditional on repairing that argument.

major comments (3)

[§3.1, Theorem 3.9 proof] The recursive construction of g : θ^+ → τ \(γ+1) implicitly requires θ < τ. Since τ is regular, the supremum of fewer than τ ordinals below τ is below τ; for ζ < θ^+ this requires θ^+ ≤ τ, i.e. θ < τ. The hypotheses only give θ < ν, with no relation between ν and τ (τ < Depth(κ^κ) may be larger or smaller than ν). Concretely, take κ = ω_{ω+1}, ν = ω_ω, τ = ω, and θ = ω_1; at step ζ = ω the supremum sup{g(ξ)^* : ξ < ω} is already ω = τ, so g(ω) cannot be defined. The proof therefore depends on an unstated cardinality assumption that is load-bearing for Theorem 3.9, and Proposition 3.14 inherits the gap.
[§4, Theorem 4.7 proof] The same issue appears in the proof of Theorem 4.7: the function g : θ^+ → τ \(γ+1) is defined with θ < σ, but nothing ensures θ < τ. Since τ is regular and θ^+ may exceed τ, the recursion can break before reaching θ^+. Thus Theorem 4.7 is not proved as stated.
[§3.3, Proposition 3.15] The proof of Proposition 3.15(i) begins with the sentence 'We modify the proof (which we do not understand) of Theorem 3.5 in [36]'. This is an explicit admission that the author cannot vouch for the correctness of the argument being modified. Since Proposition 3.15 is used in Proposition 3.17 and in the proof of Fact 2.18, this is a missing-support concern in a chain leading to a stated proof. The author should either supply a fully understood proof or explicitly mark the result as dependent on [36].

minor comments (5)

[Abstract] The abstract contains the typo 'n onsaturation'; it should read 'non-saturation'.
[Definition 2.21] The definition of ♣_κ[J] contains an extra closing parenthesis: '♣κ[J])' should be '♣κ[J]'.
[Observation 2.27 proof] In the proof, 'fix i < j < κ' should be 'fix i < j < ρ', since the almost disjoint family is indexed by ρ.
[Theorem 4.7, Claim 4] The text 'fg(ξ(j)' appears to be a typo for 'f_{g(ξ)}(j)', and the following inequality has mismatched parentheses, making the claim hard to read.
[Various proofs] Several proofs are only sketched or left to the reader, including the descending case in Theorem 2.31, part (i) of Theorem 4.7, and part (ii) of Observation 4.8; at least brief indications would improve accessibility.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Theorem 3.9 and the related tower results are proved from stated hypotheses by internal arguments, and cited prior work is used as independent background lemmas rather than as a substitute for the conclusions.

full rationale

The paper's central new result, Theorem 3.9, derives from club-cofinality guessing and tau < Depth(kappa^kappa) the existence of either an ascending (J,I_kappa)-tower or a descending (J,J)-tower of length tau. The proof is a genuine contradiction argument: it assumes the absence of both towers, uses Lemma 3.8 to obtain the sets A^i_delta, defines the sets S_alpha and S_alphabeta, and then constructs g:theta^+ -> tau\ (gamma+1] from the assumption that no descending tower exists. The tower is produced from the guessing data and the ideal J; the conclusion is not assumed in the hypotheses or in the cited facts. The paper does cite several results from the author's own prior work (e.g., Facts 2.10, 2.12, 2.13, 2.16, 5.16, 5.18), but these are background facts about density numbers, diamond, and good points. None of them states or presupposes the tower theorems proved here, and several are jointly attributed to external authors such as Jensen, Shelah, Kojman, and Cummings, Foreman, and Magidor. A possible concern is that the recursive definition of g in Theorem 3.9 requires theta < tau for the successor step (sup{g(xi)^* : xi < zeta}) + 1 to remain below the regular cardinal tau, while the hypotheses only give theta < nu and tau < Depth(kappa^kappa). That is a cardinality gap in the proof, not a circularity: it concerns the size of the guessing sets |A^i_delta| = theta, not an equivalence between input and output. No fitted parameter is renamed as a prediction, no uniqueness theorem is imported from the author's prior work to force a choice, and no known result is relabeled as a new principle. The derivation chain is therefore self-contained apart from standard external lemmas.

Assumptions & free parameters 0 free parameters · 10 assumptions · 0 invented entities

The paper is a ZFC set theory article that imports many established theorems as facts. The axioms listed capture the main external results used without proof. There are no free parameters in the empirical sense and no invented entities; the various club principles are definitions, not entities.

assumptions (10)

standard math ZFC (standard axioms of set theory) is the background theory.
All theorems are proved in ZFC, with additional hypotheses stated in each result.
standard math Fact 2.10: various computations of d(tau,sigma) and m(tau,sigma) from [21],[27],[29].
Used in Sections 2 and 3 to bound density numbers.
standard math Fact 2.12: Shelah's Strong Hypothesis equivalence from [31].
Used in the discussion of when d(theta,nu)=nu holds.
standard math Fact 2.13: Shelah's bounds on d(tau,sigma) from [45],[27].
Used to show d(theta,nu)=nu in many cases.
standard math Fact 2.35: characterization of b_kappa via closed unbounded sets from [1].
Used in Observation 2.36 and Proposition 5.4.
standard math Fact 2.47: inequalities relating b_kappa, Depth(C_kappa), and Depth(kappa^kappa) from [46].
Used to compare tower lengths and depths.
standard math Fact 2.55: Shelah's PCF result on Depth(C_rho^+) from [46].
Used in Section 2.5 to show Depth depends on lower depths.
standard math Fact 3.12: Rinot's club-guessing lemma from [37].
Used in Proposition 3.14 and Corollary 3.27.
standard math Fact 5.11: Krueger's full reflection equivalence from [23].
Used in Theorem 5.12.
standard math Facts 5.16 to 5.18: PCF and good points from [4],[28],[26].
Used in Proposition 5.19.

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Pith. "Pith review of Towers and clubs." pith.science (2026). https://pith.science/paper/IF7OCWWY

@misc{pith2026190805336,
  author       = {Pith},
  title        = {Pith review of: Towers and clubs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IF7OCWWY}},
  note         = {Machine review of arXiv:1908.05336}
}

abstract

We revisit several results concerning club principles and nonsaturation of the nonstationary ideal, attempting to improve them in various ways. So we typically deal with a (non necessarily normal) ideal $J$ extending the nonstationary ideal on a regular uncountable (non necessarily successor) cardinal $\kappa$, our goal being to witness the nonsaturation of $J$ by the existence of towers (of length possibly greater than $\kappa^+$).

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [36]

PRIMA VESI, Guessing Axioms, Invariance and Suslin Trees , Doctoral dissertation, University of East Anglia, 2011

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B. BALCAR and P. SIMON, Disjoint reﬁnement , in : Handbook of Boolean Algebras (J.D. Monk and R. Bonnet, eds.), vol. 2, North-Holland, 19 89, pp. 333-388

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