Lower consistency bounds for mutual stationarity with divergent cofinalities and limited covering

Dominik Adolf

REVIEW 3 major objections 4 minor 14 references

Lower consistency bounds for mutual stationarity with divergent cofinalities and limited covering

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Mutually stationary sequences with alternating cofinality blocks force the core-model sharp 0¶ to exist.

desk verdict A serious inner-model-theory paper with a potentially load-bearing gap in its countable-cofinality claims; worth refereeing, but the referee must check the weak-covering step against the cited [ACW]. read the letter →

arxiv 1908.01332 v1 pith:32LJBV36 submitted 2019-08-04 math.LO

Dominik Adolf This is my paper · ORCID

classification math.LO MSC 03E4503E5503E05

keywords mutualstationaritydivergentcofinalitiescoremodel0¶Jónssoncardinalweakcoveringconsistencystrengthinnertheory

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 proves new lower consistency bounds for mutual stationarity, a combinatorial property of stationary sets attached to the cardinals below ℵ_ω. It shows that if a mutually stationary sequence concentrates on points whose cofinalities follow an alternating-block pattern, allowing the same cofinality to be ω infinitely often, then the core model K below 0¶ must already exhibit large covering failures. From that failure it derives the existence of 0¶, the sharp for an inner model with a strong cardinal, both for such sequences and for any Jónsson cardinal κ with κ < ℵ_κ. The advance over earlier work is a reduced reliance on covering hypotheses, which is what lets countable cofinalities enter the pattern infinitely often.

What carries the argument

The core of the proof is the co-iteration of the core model K below 0¶ with K_X, the transitive collapse of X ∩ K for an elementary Skolem hull X of H_{ℵ_ω} or H_κ. The engine is a fine-structural lemma (Lemma 2.1) stating that a regular cardinal sitting between two projecta of a J-structure has the same cofinality as the lower projectum; repeated application converts the prescribed oscillation of the cofinalities μ_n into forced truncations and drops in the iteration. For the Jónsson theorem, a pseudo-drop construction provides a small iterable model whose fixed cofinalities on a club force infinitely many truncations, producing the contradiction.

What would settle it

A direct refutation would be a model of ZFC in which 0¶ does not exist but there is a stationary S ⊂ P(ℵ_ω) with alternating blocks of size 3 whose cofinality pattern uses ω infinitely often, and in which no κ_n < ℵ_n satisfies (κ_n^+)^K < ℵ_n with the stated Mitchell-order lower bound. Equally decisive would be a model with no 0¶ and a Jónsson cardinal κ < ℵ_κ.

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Extended reading notes

Core claim

The paper's central claim is a dichotomy below 0¶. If S ⊂ P(ℵ_ω) is stationary and has alternating blocks of size l — meaning its elements have fixed cofinalities μ_n and blocks of length l repeat with more than one cofinality appearing infinitely often — then for infinitely many n there is κ_n < ℵ_n with (κ_n^+)^K < ℵ_n and o^K(κ_n) ≥ max(ℵ_n, ($κ_n^{{+(l+1)}}$)^K). Theorem 1.3 strengthens this: if for some k < l every sequence choosing cofinalities from {k,l} on a tail is mutually stationary, then 0¶ exists. Theorem 1.4 uses the same co-iteration analysis to show that a Jónsson cardinal κ with κ < ℵ_κ already yields 0¶.

Load-bearing premise

The load-bearing premise is that the core model below 0¶ satisfies weak covering at singular cardinals of uncountable cofinality, and that this covering property remains true when K is collapsed to a Skolem hull K_X. If reflected weak covering failed in one of these hulls, the cofinality computations in Lemmas 3.3 and 4.2(a) — and with them the contradiction — would break.

Editorial extensions

If this is right

Below 0¶, a stationary set with alternating blocks of size l yields, for infinitely many n, a cardinal κ_n < ℵ_n whose K-successor is small and whose Mitchell order is at least ℵ_n and (κ_n^{+(l+1)})^K.
If for some k < l every choice of cofinalities from {k,l} on a tail is mutually stationary, then 0¶ exists, so that hypothesis has at least the consistency strength of a strong cardinal.
A Jónsson cardinal κ with κ < ℵ_κ implies 0¶, giving a new lower bound for such Jónsson cardinals.
The argument no longer requires all cofinalities to be uncountable, so the countable-cofinality cases carry the same covering-strength consequences as the uncountable ones.

Reading between the lines

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

The value of the cofinalities appears immaterial; only the recurrence pattern of the blocks matters. This suggests mutual-stationarity strength is controlled by oscillation patterns, and one can test this by forcing sequences whose cofinalities come from very different intervals while keeping the same block pattern.
A natural endpoint is that the exact consistency strength of Theorem 1.2 is the existence of a cardinal κ with o(κ) ≥ κ^{+(l+1)}; constructing a model with such a cardinal in which the mutually stationary sequence exists would confirm the bound as optimal.
Theorem 1.4 raises the possibility that Jónsson cardinals below ℵ_κ are consistency-equivalent to a strong cardinal or stronger; the paper explicitly asks whether a Woodin cardinal follows, so this is an open direction rather than a paper claim.

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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

Quick take: if correct, this paper significantly improves the known lower bounds for mutual stationarity with divergent cofinalities, and the Jónsson result (Theorem 1.4) is new. But I have a concrete concern about the use of weak covering at countable cofinalities that may affect Theorems 1.2 and 1.3, the very results the paper advertises as its main improvement over [ACW].

What the paper does well: it reduces the reliance on covering assumptions, extends the earlier arguments to sequences where countable cofinalities occur infinitely often, and proves that a Jónsson cardinal with κ<ℵ_κ implies 0^¶ exists. The proof strategy is coherent: mutual stationarity is used to find hulls with good cofinality patterns, then the co-iteration of K and K_X is analyzed, with Mitchell's pseudo-drop idea applied in the Jónsson case. The writing is clear about where results come from, and the historical note at the end is a plus. The main external citation, [ACW], is the author's own prior work, which is standard in this research program, but it does make verification harder because Lemma 2.1 is cited rather than reproduced.

The soft spot: in Section 3, after defining (γ_{n_i+j})^* as the successor of β in K_X, the paper asserts cof((γ)^*) = μ_{n_i} even when μ_{n_i}=ω, citing [ACW, Obs. 25]. Standard weak covering below 0^¶ gives cf((λ^+)^K) = cf(λ) only for singular λ of uncountable cofinality; for countable cofinality it can fail (in L, cf(ℵ_ω^{+L}) = ℵ_{ω+1} ≠ ω). The paper gives no proof of the countable-cofinality version, and the reference is a to-appear paper. If [ACW, Obs. 25] does not actually cover this case, the convergence arguments in Lemmas 3.5 and 3.6 collapse whenever countable-cofinality blocks appear. That is load-bearing. The Jónsson theorem (Section 4) seems less exposed, because the cofinalities there are uncountable, but even there Lemma 4.2(a) invokes weak covering without spelling out why the relevant ordinal has uncountable cofinality.

My own view: the reader's UNVERDICTED verdict is right. I could not find a fatal flaw, but the fine-structure details are beyond what I can verify quickly, and the weak-covering gap is concrete enough that the claims about countable cofinalities should not be taken at face value. I would want a referee who works in core model theory to check exactly what [ACW, Obs. 25] proves.

Recommendation: send it to a good journal and get a serious referee. If the countable-cofinality step is fixed, or the cited lemma already covers it, this is a solid paper worth publishing. If not, Theorems 1.2 and 1.3 need significant revision, and Theorem 1.4 might survive alone. Either way, it deserves referee time, not a desk reject.

Referee Report

3 major / 4 minor

Summary. The paper studies the consistency strength of mutually stationary sequences with divergent cofinalities. It claims three main results: (1) Theorem 1.2, a lower bound in the core model K below 0¶ from the existence of a stationary set with alternating blocks of cofinalities of fixed length l, where countable cofinalities may appear infinitely often; (2) Theorem 1.3, a variant asserting that if all sequences alternating between two prescribed cofinalities are mutually stationary, then 0¶ exists; and (3) Theorem 1.4, that a Jónsson cardinal κ with κ < ℵ_κ implies 0¶ exists. The proofs analyze co-iterations of K with transitive collapses K_X of Skolem hulls, using fine structure, weak covering, and a pseudo-drop/quasi-iteration technique attributed to Mitchell. The paper explicitly states that it improves earlier work of Adolf, Cox, and Welch by reducing reliance on covering properties and by allowing countable cofinalities.

Significance. If the results are correct, they constitute a genuine advance: the previous treatment in [ACW] required all cofinalities to be uncountable, and the new theorems remove that restriction. The Jónsson-cardinal result Theorem 1.4 is also a strong lower consistency bound, improving on earlier work. The paper contains serious technical work: it develops a pseudo-drop lemma (Lemma 4.4) and adapts comparison arguments to cases where the K_X side of the iteration is nontrivial. However, the paper relies on a to-appear citation [ACW] for a key lemma (Lemma 2.1) and for the observation about reflected weak covering, and several central proof steps are compressed. The most important issue is that the treatment of countable cofinalities in Section 3 appears to invoke weak covering in a regime where the standard theorem does not apply; this directly affects the paper's main advertised improvement.

major comments (3)

[§3, paragraph after Lemma 3.3] The claim that cof((γ^X_{n_i+j})*) = μ_{n_i+j} for all j < l, including the case μ_{n_i} = ω, is asserted to follow from 'weak covering reflected down to K_X' and cited to [ACW, Obs. 25]. Standard weak covering below 0¶ gives cf((λ^+)^K) = cf(λ) only when λ is singular of uncountable cofinality; for cf(λ) = ω it can fail, as in L where cf(ℵ_ω^{+L}) = ℵ_{ω+1} ≠ ω. Since the hypotheses of Theorem 1.2 and 1.3 explicitly allow μ_n = ω infinitely often, this equality is not established. The equality is used in Lemmas 3.5 and 3.6 to force a contradiction when μ_{n_i} ≠ ρ, so without it the convergence argument collapses for blocks with countable μ. Thus Theorems 1.2 and 1.3 are not yet supported in the advertised countable-cofinality case.
[Lemma 4.2(a)] The step 'It follows from weak covering that cof((α^+)^{K_X}) = (μ^X_0)^{+(n-1)}' needs verification that the relevant cardinal in K_X is singular of uncountable cofinality. The text does not spell this out. Since (μ^X_0)^{+(n-1)} is uncountable if μ^X_0 ≥ ℵ_2, this is likely repairable, but as written the reflected covering step is not justified and the lemma's proof is incomplete.
[Lemma 4.4] The proof of Lemma 4.4 is central to the quasi-iteration argument for Theorem 1.4, but it is too compressed. The 'easy induction' showing M^X_α = Hull^{M^X_α}_ω(κ^X_α ∪ {π_{2,α}(f̄_n) : n < ω}) is not shown, and the reflection claim '∀δ∀ξ_0...∀ξ_{m-1}∃β' is stated without a full derivation. Since this lemma substitutes for Lemma 2.1 in a case where the usual soundness argument fails, the proof should be expanded or the lemma should be stated with a pointer to a complete proof in the literature.

minor comments (4)

[Preliminaries, second paragraph] The text says 'In Section 5 we will have to consider special iterations', but the paper has no Section 5; the reference should be to Section 4.
[Proposition 3.2 proof] In the proof, 'we have some X_A ∈ S with X ≺ A' uses the variable X without introducing it; this should be 'with X_A ≺ A' to avoid confusion.
[Lemma 3.3] The expression Ult(K;σ_X↾(K||α^X_0)) uses the notation K||α^X_0 without definition; a brief reminder of the standard Zeman notation would improve readability.
[Throughout] There are numerous typographical errors and inconsistent cross-reference labels (for example, '1.2Introductionthm.1.2'). A careful proofreading pass is needed.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the paper derives consistency lower bounds from stated hypotheses using established core-model lemmas; cited prior work is independent support, not a re-labeled input.

full rationale

The derivation chain in Theorems 1.2–1.4 is genuinely conditional: assuming the nonexistence of 0¶ and the existence of a mutually stationary sequence with alternating blocks (or a small Jónsson cardinal), the paper obtains a contradiction from core-model comparison, truncations, cofinality computations, and Mitchell-order considerations. There is no fitted parameter that is later renamed a prediction, and no definitional equation by which the target conclusion is built into the hypothesis. Lemma 2.1 is imported from the author's prior work [ACW], but it is a fine-structural lemma used under explicit soundness and projectum hypotheses; it is not a restatement of Theorem 1.2 or 1.3. The invocations of weak covering reflected down to K_X in Section 3 and Lemma 4.2(a), including the citation to [ACW, Obs. 25], rely on established core-model covering theory rather than on the theorem being proved. Even if the skeptical concern about countable cofinality cases is a genuine correctness gap, that is a mathematical gap, not circularity: a failed or unsupported lemma does not mean the conclusion was assumed. The Jónsson proof likewise assumes ¬0¶ and derives a contradiction from the behavior of the co-iteration, so 0¶ is not used as an input. Self-citations occur, but they are load-bearing only as citations to published lemmas with independent content; they do not reduce the theorems to their hypotheses by construction. Accordingly, no circular step is exhibited.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claims rest on standard core model theory below 0¶: existence of K, iterability, weak covering, and fine structural soundness results. No new postulates or entities are introduced; the proof's burden is carried by the technical lemmas rather than by free parameters.

assumptions (5)

standard math ZFC (or appropriate large cardinal axioms) plus '0¶ does not exist' where assumed
The proofs are within standard set theory, using the core model K below 0¶. The assumption that 0¶ does not exist is explicit in the theorem statements.
domain assumption Existence and basic iterability of the core model K below 0¶, including comparison with Skolem hulls K_X
Used throughout; the paper draws on [Zem01] and [Cox09] for fine structure and iterability. Section 2 and Lemma 3.3.
domain assumption Weak covering for K below 0¶ at singular cardinals of uncountable cofinality
Invoked in Lemma 3.3 (weak covering reflected down to K_X) and in Lemma 4.2(a) to control cofinalities of successors. This is a known theorem for K below 0¶, not reproved here.
standard math Known fine structural lemmas, e.g., Lemma 2.1 (soundness above λ implies cofinality equals projectum), taken from [ACW]
Lemma 2.1 is imported from the author's earlier paper and used repeatedly in Sections 3-4.
domain assumption Mitchell's pseudo-drop construction and the notion of quasi-iteration from [Mit99]
Section 4 adapts Mitchell's construction; relies on the reader accepting the framework and its properties.

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Cite this review

Pith. "Pith review of Lower consistency bounds for mutual stationarity with divergent cofinalities and limited covering." pith.science (2026). https://pith.science/paper/32LJBV36

@misc{pith2026190801332,
  author       = {Pith},
  title        = {Pith review of: Lower consistency bounds for mutual stationarity with divergent cofinalities and limited covering},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/32LJBV36}},
  note         = {Machine review of arXiv:1908.01332}
}

abstract

We improve previous work on the consistency strength of mutually stationary sequences of sets concentrating on points with divergent cofinality building on previous work by Adolf, Cox and Welch. Specifically, we have greatly reduced our reliance on covering properties in the proof. This will allow us to handle sequences in which sets concentrating on points of countable cofinality appear infinitely often. Furthermore we will show that if $\kappa$ is a J\'onsson cardinal with $\kappa < \aleph_\kappa$ then the sharp for a model with a strong cardinal exists.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

14 extracted references · 14 canonical work pages

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Pith tools

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

[1] [1]

Lower consistency bounds for mutual stationarity with divergent uncountable cofinalities

Dominik Adolf, Sean Cox, and Philip Welch. Lower consistency bounds for mutual stationarity with divergent uncountable cofinalities. to appear, online preprint: https://arxiv.org/abs/1607.04790

work page Pith review arXiv

[2] [2]

Baumgartner

J.E. Baumgartner. On the size of closed unbounded sets. Annals of Pure and Applied Logic , 54:195 -- 227, 1991

work page 1991

[3] [3]

On singular stationarity i

Omer Ben-Neria. On singular stationarity i. submitted

work page

[4] [4]

Canonical structure in the universe of set theory: part two

James Cummings, Matthew Foreman, and Menachem Magidor. Canonical structure in the universe of set theory: part two. Annals of Pure and Applied Logic , 142:55 -- 75, 2006

work page 2006

[5] [5]

Covering theorems for the core model, and an application to stationary set reflection

Sean Cox. Covering theorems for the core model, and an application to stationary set reflection. Annals of Pure and Applied Logic , 161:66 -- 93, 2009

work page 2009

[6] [6]

Large cardinals and definable counterexamples to the continuum hypothesis

Matthew Foreman and Menachem Magidor. Large cardinals and definable counterexamples to the continuum hypothesis. Annals of Pure and Applied Logic , 76(1):47 -- 97, 1995

work page 1995

[7] [7]

Mutually stationary sequences of sets and the non-saturation of the non-stationary ideal on _ ( )

Matthew Foreman and Menachem Magidor. Mutually stationary sequences of sets and the non-saturation of the non-stationary ideal on _ ( ) . Acta Mathematica , 186(2):271 -- 300, 2001

work page 2001

[8] [8]

Set Theory - The Third Millenium Edition,revised and expanded

Thomas Jech. Set Theory - The Third Millenium Edition,revised and expanded . Springer Monographs in Mathematics. Springer-Verlag, Berlin, 3rd edition, 2006

work page 2006