A Strongly Non-Saturated Aronszajn Tree Without Weak Kurepa Trees

John Krueger; \v{S}\'arka Stejskalov\'a

REVIEW 2 major objections 5 minor 20 references

A Strongly Non-Saturated Aronszajn Tree Without Weak Kurepa Trees

T0 review · 2 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read A forcing construction proves that a strongly non-saturated Aronszajn tree can coexist with the complete absence of weak Kurepa trees.

desk verdict A genuinely new forcing construction that answers the motivating question, but the proof leans on an unproved transfer of side-condition amalgamation results from [Kru17] that deserves close scrutiny before the main theorem is accepted. read the letter →

arxiv 2506.06878 v2 pith:C3TN6K2V submitted 2025-06-07 math.LO

John Krueger , Šárka Stejskalová This is my paper

classification math.LO MSC 03E0503E3503E40

keywords stronglynon-saturatedAronszajntreeweakKurepasideconditionsadequatesetsY-properforcingomega_1-approximationpropertyequiconsistency

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 that a strongly non-saturated Aronszajn tree can coexist with the total absence of weak Kurepa trees, and that this coexistence is exactly as strong as the existence of an inaccessible cardinal. The proof builds a forcing poset with finite conditions whose working part approximates the tree and a family of pairwise strongly almost disjoint uncountable subtrees, while a side condition of finitely many countable models keeps the approximations coherent. The poset is proper and $\kappa$-c.c., collapses $\kappa$ to become $\omega_2$, and has quotients that remain Y-proper on a stationary set after further Y-proper forcing, which gives the $\omega_1$-approximation property and rules out new cofinal branches of $\omega_1$-trees. A separate c.c.c. forcing, available from the negation of Chang's conjecture via a weak $\rho$-function, adds the same kind of tree without the large cardinal, and combining it with a Mahlo cardinal gives a model where the negation of the Kurepa hypothesis is c.c.c. indestructible.

What carries the argument

The load-bearing object is a side-condition forcing $P$ whose conditions are quadruples $(T, W, D, A)$: $T$ is a finite approximation of the generic tree, $W$ assigns to finitely many indices $\eta < \kappa$ a downwards closed finite subtree, $D$ records pairs of indices whose subtrees should be finitely intersected, and $A$ is a finite adequate set of countable elementary substructures of $H(\kappa)$ from a club $X \subseteq [\kappa]^\omega$. The interaction is $A$-separation: whenever two indices lie in a model $M \in A$, the corresponding $W$-subtrees intersect only inside $M$. Amalgamating conditions over several models is handled by a lemma that combines split finite trees and an adequate union of model sets; the quotients are analyzed through projections $\pi_\theta$ to $P \cap \mathrm{Sk}(\theta)$, with a dense set $E_\theta$ designed so that the projection is a proper projection mapping and quotient forcings inherit Y-properness on a stationary set. Y-properness, a strengthening of properness introduced in [CZ15], is what transfers to the $\omega_1$-approximation property and blocks new cofinal branches.

What would settle it

Force with $P$ over an inaccessible $\kappa$ and look in $V[G]$ for an $\omega_1$-tree whose underlying set is $\omega_1$ and which has at least $\omega_2$ many cofinal branches; finding one would refute the theorem. Since the proof rules such trees out by showing that every quotient $P/H$ over an intermediate model $P_\theta$ has the $\omega_1$-approximation property, the decisive local check is whether that property holds for every such quotient.

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Extended reading notes

Core claim

The central claim is that from any inaccessible cardinal $\kappa$ there is a forcing poset $P$ which forces that $\kappa = \omega_2$ and adds a normal, infinitely splitting Aronszajn tree $T_G$ together with a family $\{W_G(\eta) : \eta < \kappa\}$ of uncountable downwards closed subtrees whose pairwise intersections are finitely generated, so $T_G$ is strongly non-saturated, while no weak Kurepa tree exists. The proof works by representing conditions as pairs of a finite approximation to the tree and its subtrees and a finite adequate set of countable models, with a separation condition tying the two parts together. The decisive property is that quotients of $P$ by projections to intermediate models are Y-proper on a stationary set, and remain so after any Y-proper forcing extension; Y-properness on a stationary set implies the $\omega_1$-approximation property, so no new cofinal branches of trees of height $\omega_1$ appear in the final model. The paper also derives an equiconsistency: the existence of a strongly non-saturated Aronszajn tree with no weak Kurepa tree is consistent if and only if an inaccessible cardinal is consistent.

Load-bearing premise

The proof leans on a gluing procedure for finite sets of countable models that it imports as an unproved tool; if that procedure fails for inaccessible cardinals of the kind being collapsed to omega-two, the construction of the forcing and of its quotients falls apart.

Editorial extensions

If this is right

An inaccessible cardinal is exactly the consistency strength needed for a strongly non-saturated Aronszajn tree to coexist with the negation of the weak Kurepa hypothesis.
In the final model, $\kappa = \omega_2$ and the generic tree is normal and infinitely splitting, with $\omega_2$ many subtrees whose pairwise intersections are finitely generated.
Every quotient $P/H$ over an intermediate generic filter on $P_\theta$ has the $\omega_1$-approximation property in $V[H]$, and this survives any further Y-proper forcing extension.
Under the negation of Chang's conjecture, the c.c.c. forcing $P'$ alone adds a strongly non-saturated Aronszajn tree; combined with a Mahlo cardinal it gives a model where the negation of the Kurepa hypothesis is c.c.c. indestructible.
From a supercompact cardinal, the construction is compatible with the indestructible guessing model principle IGMP.

Reading between the lines

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

Editorial extension: the quotient indestructibility at the heart of the proof is a transferable engine, so other two-cardinal tree principles that are forceable by Y-proper iterations should be combinable with a strongly non-saturated Aronszajn tree using the same template.
Editorial extension: the c.c.c. forcing $P'$ shows the strongly non-saturated tree itself carries no large-cardinal strength, so weakening the demand that weak Kurepa trees be absent might allow the same tree-building method to run from weaker hypotheses.
Editorial extension: the $A$-separation condition could plausibly be adapted to build other objects that require pairwise controlled intersections of many definable subsets, such as rigid Aronszajn trees or strongly almost disjoint families with prescribed structure.

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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

The main equiconsistency result is a real advance: from an inaccessible you get a strongly non-saturated Aronszajn tree plus the failure of the weak Kurepa hypothesis. The two finite-condition forcings are new, and the use of Y-proper quotients to rule out weak Kurepa trees is a genuinely novel mechanism. The c.c.c. forcing in Section 3 using a weak rho-function is also clean and gives a nice answer to the first question under Mahlo. The paper is serious and the central construction is worth engaging with.

The soft spots are real but localized. The biggest one is the transfer of the side-condition amalgamation theorems from [Kru17] to the present setting. The paper says the proofs work 'almost identically' with X = [κ]^ω instead of a thin stationary set, and that the thin set was only needed when CH fails. But the target model has CH failing, so that is exactly the case in which the thin set was needed. The stress-test note is right that this is not a proof. I am not saying the transfer is false; I suspect it is true, but the paper needs to show it, or at least point to the specific lemmas in [Kru17] that remain valid without the thin set and explain why.

The second sketch is the IGMP result from a supercompact cardinal. Theorem 13.5 is presented as a sketch, which is honest but means that part is not yet at the same standard as the rest of the paper. That is a minor issue if the main equiconsistency is the focus, but a referee should ask for a fuller write-up.

Lemma 2.11 being left as an exercise is fine; the proof is indeed straightforward. I also cannot machine-check the long case analysis in Lemma 7.1, but what I read is internally coherent, and the reliance on published theorems from [Kru17] is legitimate self-citation rather than circularity.

Who is this for? Set theorists working on Aronszajn trees, side conditions, and Y-proper forcing. It deserves a serious referee, but the referee should be instructed to verify the [Kru17] transfer or demand that the authors provide that verification in the paper.

Referee Report

2 major / 5 minor

Summary. The paper constructs two forcing posets for adding a strongly non-saturated Aronszajn tree. The first, P', is c.c.c. under the negation of Chang's conjecture and is used, over L after a Levy collapse, to obtain a strongly non-saturated Aronszajn tree with the negation of the Kurepa hypothesis c.c.c. indestructible. The second, P, uses countable models as side conditions and is intended to be proper, κ-c.c., and to collapse an inaccessible κ to ω2 while adding a strongly non-saturated Aronszajn tree and no weak Kurepa tree. The quotient forcings of P are shown to be Y-proper on a stationary set, which yields the ω1-approximation property and therefore the claimed non-existence of weak Kurepa trees. A supercompact cardinal is used to obtain a model with the indestructible guessing model principle.

Significance. If the proof is completed, the paper would answer a question from [KMM24] and give the first consistency proof of a strongly non-saturated Aronszajn tree without weak Kurepa trees, with an equiconsistency at an inaccessible cardinal. The first forcing and the main properness/collapsing analysis are substantial and mostly self-contained. The paper also provides useful new tools: the finite-condition forcing with adequate side conditions, the projection mapping to P ∩ Sk(θ), and the quotient Y-properness analysis. The main caveat is that several load-bearing adequate-set lemmas are imported from [Kru17] with only a handwritten claim of transfer to the present context, and the final IGMP theorem is only sketched.

major comments (2)

[§5, p. 11] The proof of the main forcing theorem depends on adequate-set amalgamation lemmas (Proposition 5.8, Theorems 5.11 and 5.15, and Corollary 5.12) imported from [Kru17], where the ambient cardinal is ω2 and a fixed thin stationary subset of [ω2]^ω is used exactly when CH fails. The paper states only that 'everything works almost identically' for an inaccessible κ with X=[κ]^ω. This transfer is load-bearing: Lemma 7.1 invokes Corollary 5.12, Proposition 10.9 invokes Theorem 5.15, Lemma 11.9 invokes Lemma 5.16, and the properness, chain condition, and quotient Y-properness all rest on these facts. Moreover, the forcing P produces a model where CH fails, since a strongly non-saturated Aronszajn tree implies ¬CH, so the case that originally required the thin stationary set is exactly the relevant case. The manuscript should supply a proof of the transferred lemmas or a precise citation in [Kru17] covering arbitrary inaccessible κ and X=[κ]^ω; as written, this is an unverified black box in the central consistency proof.
[§13, Theorem 13.5] Theorem 13.5 is stated as a theorem but the proof is only a sketch. The final paragraph says 'We provide a sketch of a proof' and then uses 'standard methods', an elementary embedding j, and the forcing j(P)/G without verifying the stationarity of the relevant guessing models or the full IGMP conclusion. Since this theorem is advertised in the abstract and is one of the paper's main results, it needs either a complete proof or an explicit statement that it is a conjecture or an outlined argument. The main theorem about weak Kurepa trees does not depend on this sketch, but the advertised IGMP consistency does.

minor comments (5)

[§7, first paragraph] The sentence 'the general case d ≥ 2 is used in Section 7 to prove that P is Y-proper' appears to refer to Section 8, where Y-properness is actually proved.
[Proof of Lemma 2.15] The reference 'By Lemma 2.13(b,c)' should be 'By Lemma 2.14(b,c)', since the facts used are exactly Lemma 2.14(b,c).
[Lemma 6.7] In the definition of C in Lemma 6.7, the symbol A is used before it is introduced; it should be Ap, the side condition of the condition p.
[Proof of Theorem 12.1] After extending q to r, the tuple is written as wθ(q,N), but it should be wθ(r,N), since q may not belong to Dθ and only (r,N) is known to be in the domain of wθ.
[Lemma 2.11] Lemma 2.11 is stated with its proof left as an exercise, but it is used in Lemma 3.5 and Lemma 6.4; a short proof in the text would improve verifiability.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation: the forcing construction proves the consistency results from an inaccessible cardinal using independent published side-condition theorems; the [Kru17] transfer is a support gap, not a circular step.

full rationale

The paper does not assume or encode its target conclusion in its inputs. The first forcing P′ is shown c.c.c. from a weak ρ-function supplied by the negation of Chang's conjecture, and the tree it adds is proved strongly non-saturated from lemmas about finite trees and separation properties. The second forcing P is defined from finite working parts and adequate sets, and the claims that it is proper, κ-c.c., Y-proper, and adds a strongly non-saturated Aronszajn tree are each proved by explicit arguments in Sections 7 through 10. The only load-bearing external inputs are the adequate-set amalgamation results imported from [Kru17] (Proposition 5.8, Theorems 5.11 and 5.15) and the Y-c.c./Y-properness framework imported from [CZ15]. These are published, parameter-free results that do not contain the target consistency statement. The paper explicitly acknowledges the context shift from ω2 in [Kru17] to inaccessible κ here, asserting that 'everything works almost identically' and replacing the former thin stationary set by [κ]^ω. That assertion is an unverified portability claim and is a legitimate correctness risk because the thin set was originally needed in the CH-failure case, which is exactly the case that occurs in the intended model. However, a missing or shaky proof of a transferred lemma is not the same as circularity: the target theorem is not being used as a premise, no fitted parameters are relabeled as predictions, and no uniqueness or ansatz is smuggled in solely by self-citation. The proof that no weak Kurepa tree exists in the extension is a separate argument using the ω1-approximation property, not a restatement of the construction's definition. Accordingly, the correct circularity finding is 0.

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

The paper's consistency proofs assume large cardinals (inaccessible, Mahlo, supercompact) and several cited theorems. The side-condition framework from [Kru17] is the most substantial unproved input. There are no fitted numerical parameters and no invented physical entities.

assumptions (5)

domain assumption Existence of an inaccessible cardinal κ (or Mahlo or supercompact for specific theorems).
The forcing poset P in Sections 5-10 is defined for an inaccessible κ and collapses it to ω2; Theorem 4.3 uses a Mahlo cardinal; Theorem 13.5 uses a supercompact cardinal.
domain assumption The negation of Chang's conjecture is equivalent to the existence of a weak ρ-function e : (ω2)^2 → ω1 (Theorem 4.1, cited from [Tod91]).
Used in Section 4 to ensure P' is c.c.c. and adds a strongly non-saturated Aronszajn tree in the model of Theorem 4.3.
standard math Y-properness on a stationary set implies the ω1-approximation property (Theorem 8.2, cited from [CZ15]).
This is the key transfer from the quotient Y-properness to the absence of new cofinal branches in Theorem 13.1.
domain assumption Adequate sets satisfy the amalgamation properties stated in Proposition 5.8 and Theorems 5.11 and 5.15 of [Kru17].
These black boxes are used repeatedly, e.g., in Corollary 5.12, Lemma 7.1, and Proposition 10.9, to show unions of adequate sets are adequate. The paper does not prove them.
standard math Standard set-theoretic theorems: the Baumgartner-Malitz-Reinhardt theorem and Dushnik-Miller ω1→(ω1,ω)^2; the ∆-system lemma and pressing down.
Used in Lemma 2.15, Theorem 3.13, and Section 12 (Theorem 12.4) to find compatible conditions and incomparable elements.

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Cite this review

Pith. "Pith review of A Strongly Non-Saturated Aronszajn Tree Without Weak Kurepa Trees." pith.science (2026). https://pith.science/paper/C3TN6K2V

@misc{pith2026250606878,
  author       = {Pith},
  title        = {Pith review of: A Strongly Non-Saturated Aronszajn Tree Without Weak Kurepa Trees},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/C3TN6K2V}},
  note         = {Machine review of arXiv:2506.06878}
}

abstract

Assuming the negation of Chang's conjecture, there is a c.c.c. forcing which adds a strongly non-saturated Aronszajn tree. Using a Mahlo cardinal, we construct a model in which there exists a strongly non-saturated Aronszajn tree and the negation of the Kurepa hypothesis is c.c.c. indestructible. For any inaccessible cardinal $\kappa$, there exists a forcing poset which is Y-proper and $\kappa$-c.c., collapses $\kappa$ to become $\omega_2$, and adds a strongly non-saturated Aronszajn tree. The quotients of this forcing in intermediate extensions are indestructibly Y-proper on a stationary set with respect to any Y-proper forcing extension. As a consequence, we prove from an inaccessible cardinal that the existence of a strongly non-saturated Aronszajn tree is consistent with the non-existence of a weak Kurepa tree. Finally, we prove from a supercompact cardinal that the existence of a strongly non-saturated Aronszajn tree is consistent with two-cardinal tree properties such as the indestructible guessing model principle.

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Reference graph

Works this paper leans on

20 extracted references · 20 canonical work pages

[1]

J. E. Baumgarter, Almost-disjoint sets, the dense set problem and the partition calculus, Ann. Math. Logic 9 (1976), no. 4, 401--439

work page 1976
[2]

J. E. Baumgartner, Bases for A ronszajn trees , Tsukuba J. Math. 9 (1985), no. 1, 31--40

work page 1985
[3]

J. E. Baumgartner, J. Malitz, and W. Reinhardt, Embedding trees in the rationals, Proc. Nat. Acad. Sci. 67 (1970), no. 4, 1748--1753

work page 1970
[4]

Cox and J

S. Cox and J. Krueger, Quotients of strongly proper forcings and guessing models, J. Symb. Log. 81 (2016), no. 1, 264--283

work page 2016
[5]

, Indestructible guessing models and in the continuum, Fund. Math. 239 (2017), no. 3, 221--258

work page 2017
[6]

Chodounsk\' y and J

D. Chodounsk\' y and J. Zapletal, Why Y -c.c. , Ann. Pure Appl. Logic 166 (2015), no. 11, 1123--1149

work page 2015
[7]

Jensen and K

R. Jensen and K. Schlechta, Results on the generic K urepa hypothesis , Arch. Math. Logic 30 (1990), 13--27

work page 1990
[8]

K\" o nig, P

B. K\" o nig, P. Larson, J. T. Moore, and B. Veli c kovi\' c , Bounding the consistency strength of a five element linear basis, Israel J. Math. 164 (2008), 1--18

work page 2008

Show all 20 references

[9]

Krueger and E

J. Krueger and E. Martinez Mendoza, An almost K urepa S uslin tree with strongly non-saturated square , 2024, preprint

2024
[10]

Krueger, Forcing with adequate sets of models as side conditions, Math

J. Krueger, Forcing with adequate sets of models as side conditions, Math. Log. Q. 63 (2017), no. 1-2, 124--149

2017
[11]

, Guessing models imply the singular cardinal hypothesis, Proc. Amer. Math. Soc. 147 (2019), no. 12, 5427--5434

2019
[12]

Krueger and S

J. Krueger and S . Stejskalov\' a , Forcing over a free S uslin tree , 2024, preprint

2024
[13]

Mitchell, Aronszajn trees and the independence of the transfer property, Ann

W. Mitchell, Aronszajn trees and the independence of the transfer property, Ann. Math. Logic 5 (1972/73), 21--46

1972
[14]

J. T. Moore, A five element basis for the uncountable linear orders, Ann. of Math. (2) 163 (2006), no. 2, 669--688

2006
[15]

Notes Log., vol

, Structural analysis of A ronszajn trees , Lect. Notes Log., vol. 28, Assoc. Symbol. Logic, Urbana, IL, 2008, pp. 85--106

2008
[16]

Silver, The independence of K urepa's conjecture and two-cardinal conjectures in model theory , Axiomatic Set Theory (D

J. Silver, The independence of K urepa's conjecture and two-cardinal conjectures in model theory , Axiomatic Set Theory (D. S. Scott, ed.), Proc. Sympos. Pure Math., vol. XIII, Part 1, Amer. Math. Soc., Providence, R.I., 1971, pp. 383--390

1971
[17]

Todor c evi\' c , Partitioning pairs of countable ordinals, Acta Math

S. Todor c evi\' c , Partitioning pairs of countable ordinals, Acta Math. 159 (1987), no. 3-4, 261--294

1987
[18]

, Remarks on M artin's axiom and the continuum hypothesis , Can. J. Math. 43 (1991), no. 4, 832--851

1991
[19]

Viale and C

M. Viale and C. Weiss, On the consistency strength of the proper forcing axiom, Adv. Math. 228 (2011), no. 5, 2672--2687

2011
[20]

Weiss, The combinatorial essence of supercompactness, Ann

C. Weiss, The combinatorial essence of supercompactness, Ann. Pure Appl. Logic 163 (2012), no. 11, 1710--1717

2012

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

J. E. Baumgarter, Almost-disjoint sets, the dense set problem and the partition calculus, Ann. Math. Logic 9 (1976), no. 4, 401--439

work page 1976

[2] [2]

J. E. Baumgartner, Bases for A ronszajn trees , Tsukuba J. Math. 9 (1985), no. 1, 31--40

work page 1985

[3] [3]

J. E. Baumgartner, J. Malitz, and W. Reinhardt, Embedding trees in the rationals, Proc. Nat. Acad. Sci. 67 (1970), no. 4, 1748--1753

work page 1970

[4] [4]

Cox and J

S. Cox and J. Krueger, Quotients of strongly proper forcings and guessing models, J. Symb. Log. 81 (2016), no. 1, 264--283

work page 2016

[5] [5]

, Indestructible guessing models and in the continuum, Fund. Math. 239 (2017), no. 3, 221--258

work page 2017

[6] [6]

Chodounsk\' y and J

D. Chodounsk\' y and J. Zapletal, Why Y -c.c. , Ann. Pure Appl. Logic 166 (2015), no. 11, 1123--1149

work page 2015

[7] [7]

Jensen and K

R. Jensen and K. Schlechta, Results on the generic K urepa hypothesis , Arch. Math. Logic 30 (1990), 13--27

work page 1990

[8] [8]

K\" o nig, P

B. K\" o nig, P. Larson, J. T. Moore, and B. Veli c kovi\' c , Bounding the consistency strength of a five element linear basis, Israel J. Math. 164 (2008), 1--18

work page 2008