There may be an $n$-entangled set but no $n+1$-entangled sets

Jorge Antonio Cruz Chapital

REVIEW 3 major objections 8 minor 1 cited by

There may be an $n$-entangled set but no $n+1$-entangled sets

T0 review · 3 major / 8 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read For every n ≥ 2, there is a consistent model with an n-entangled set but no (n+1)-entangled set.

desk verdict Answers a real open question, but the main theorem leans on an unproven imported consistency proof; worth refereeing with conditions. read the letter →

arxiv 2509.01029 v1 pith:RBAOGSYY submitted 2025-08-31 math.LO math.GN

Jorge Antonio Cruz Chapital This is my paper

classification math.LOmath.GN MSC 03E6503E7503E5003E0503E35

keywords entangledsetsforcingaxiomsMartin'saxiomP-idealdichotomyproperconstructionschemesFCA^Deltaconsistency

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 the hierarchy of entangledness among sets of reals can stop at any prescribed finite level: for every $n \ge 2$ there is a model of set theory in which an $n$-entangled set exists but no $(n+1)$-entangled set exists, and the model can also satisfy the P-ideal dichotomy (PID) and Martin's axiom (MA). The proof imports a parametrized forcing axiom, $\mathrm{PFA}_{n\text{-ent}}(E)$, from prior work by Guzmán and Todorcevic, and shows it forces away all $(n+1)$-entangled sets while preserving the $n$-entangledness of a chosen exemplar E. A second thread shows that the axiom $\mathrm{FCA}^\Delta$ yields pairwise homeomorphic sets at every exact level of entangledness, and two limitative theorems say that $2$-capturing construction schemes are too weak to produce entangled sets. Together the results answer an open question of Carroy, Levine, and Notaro and sharpen the known behaviour of Abraham–Shelah entangled sets under strong forcing axioms.

What carries the argument

The key mechanism is the forcing $\mathbb{P}(X,E)$, whose conditions are finite $\in$-chains of countable elementary submodels — 'good pairs' $(s, x)$ with $x(i) \in s(i+1) \setminus s(i)$ — together with the constraint that the types of any two distinct $(n+1)$-tuples $x, y$ are non-constant. The chain of submodels makes the forcing proper and supports two duplication lemmas (Lemmas 3.7 and 3.8) that split the work according to whether a small piece of E sits in one or many submodel layers; these lemmas preserve the $n$-entangledness of E while the generic filter adds an uncountable family of $(n+1)$-tuples from X with no constant type, so X is no longer $(n+1)$-entangled. The parametrized axiom $\mathrm{PFA}_{n\text{-ent}}(E)$ then supplies the

What would settle it

Read Theorem 97 of [13] (cited here as Theorem 2.6) and check whether its consistency proof for $\mathrm{PFA}_{n\text{-ent}}(E)$ uses large cardinals beyond ZFC; if it does, the unqualified consistency claim in Theorem 0.2 is false as stated. Alternatively, attempt to construct an $(n+1)$-entangled set inside a model of $\mathrm{PFA}_{n\text{-ent}}(E)$; any such set would refute the paper's main implication.

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Extended reading notes

Core claim

The paper's central claim is that for each fixed integer $n \ge 2$, the statement 'there is an $n$-entangled set of reals, but there is no $(n+1)$-entangled set of reals' is consistent with ZFC plus the P-ideal dichotomy and Martin's axiom. The proof works through a parametrized forcing axiom, $\mathrm{PFA}_{n\text{-ent}}(E)$, which asserts that a fixed set E is $n$-entangled and that every proper forcing preserving E's $n$-entangledness has a filter meeting any family of $\aleph_1$ dense sets. This axiom, imported from Guzmán and Todorcevic, is consistent and implies both MA and PID. The genuinely new piece is a forcing $\mathbb{P}(X,E)$ that, for any uncountable X, is proper, preserves E's $n$-entangledness, and forces X to stop being $(n+1$

Load-bearing premise

The paper's main theorem rests on the imported consistency of the parametrized forcing axiom $\mathrm{PFA}_{n\text{-ent}}(E)$ from [13], a theorem stated with no proof here and no explicit large-cardinal hypothesis, together with the omitted proof that $\mathbb{P}(X,E)$ is proper; if either fails, the consistency claim does not go through.

Editorial extensions

If this is right

The hierarchy of entangledness is not well-ordered by consistency strength: each level n can be the top level in a model of PID+MA.
Question 3.5 of [6] has a positive answer for every n ≥ 2.
Under FCA^Δ, the full spectrum of exact entangledness levels occurs simultaneously among pairwise homeomorphic subsets of ℝ.
Neither the existence of a 2-capturing scheme nor a 2-Δ-capturing scheme implies the existence of an entangled set, and a 2-Δ-capturing scheme can coexist with MA.
The boundary between finite and infinite entangledness is not merely a matter of taking all n; each finite level behaves like an independent phenomenon.

Reading between the lines

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

The good-pairs forcing ℙ(X,E) is a template: any property of X that can be killed by an uncountable family of finite tuples avoiding a definable set of types may be destructible this way while preserving a hereditary property of E.
If the imported consistency of PFA_{n-ent}(E) turns out to require large cardinals, the main theorem can likely be repaired by re-running the argument inside a model of PFA for the right class of forcings, but the unqualified ZFC-consistency claim would need revision.
The FCA^Δ result suggests that the Cohen model, where FCA^Δ holds, contains a complete ladder of exact entangledness levels; checking this would give a CH-based witness without any forcing axioms.
Problem 0.6 — whether every 2-entangled set can be made n-entangled — might be approachable by iterating ℙ(X,E) along the uncountable targets X, destroying their higher entangledness one by one.

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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 theorem answers Question 3.5 of [6] positively: for every n≥2, you can consistently have an n-entangled set and no (n+1)-entangled sets, together with PID and MA. That is a real result. The forcing P(X,E) that destroys (n+1)-entangledness while preserving n-entangledness is a serious construction, and the preservation proof in Theorem 3.10 is dense but looks like a genuine argument. The FCA^Δ results and the m_F > ω1 results also do real work: the first gives a uniform construction of the whole hierarchy from one axiom, the second shows 2-capturing schemes alone are insufficient to produce entangled sets. There is substance here.

The main soft spot is the foundation of Theorem 0.2. The consistency of PFA_{n-ent}(E) is imported as Theorem 2.6, credited as "Essentially Theorem 97, [13]" with no proof and no stated large-cardinal hypothesis. That is load-bearing. If the proof in [13] needs a supercompact, the abstract's unqualified "is consistent" overstates the result—it is at best a relative consistency claim. The same applies to the imported Neeman preservation theorem and the derivation of PID. The author should either prove Theorem 2.6 in this paper or give a precise citation with explicit hypotheses and a source that is actually available.

There is also a smaller gap in the proof of Theorem 3.10. In the case a_k = a, the proof invokes Lemma 3.7, but that lemma requires a to be of the first kind over the relevant chain. The text does not verify that, and if a is of the second kind the proof should switch to Lemma 3.8. This looks repairable but needs to be written out.

The scheme section has a couple of rough edges: the piecewise definition of e has a line that looks like a sign typo, and Proposition 5.4's Case 2 inequality is compressed. On a careful read the contradiction can be recovered, but the section needs editing.

Bottom line: this paper is for set theorists working on entangled sets, MA/PID, or construction schemes. It deserves a serious referee—conditional accept, not desk reject. The referee should demand (a) a complete statement and proof, or an explicit citation with hypotheses, for PFA_{n-ent}(E); (b) a patch to the Lemma 3.7/3.8 case split; (c) cleanup of the scheme section. I would cite the main theorem once those are in place.

Referee Report

3 major / 8 minor

Summary. This paper answers Question 3.5 of Carroy-Levin-Notaro [6] in the positive: for every n >= 2, the statement 'there is an n-entangled set but no n+1-entangled set' is shown consistent together with PID and MA (Theorem 0.2). The proof passes through a parametrized proper forcing axiom PFA_{n-ent}(E), argued to be consistent in [13] (Theorem 2.6) and to imply MA and PID (Theorems 2.7-2.8). The new technical core is a forcing P(X,E) built from finite chains of 'good pairs' (Section 3), which is proper, preserves the n-entangledness of a fixed E, and makes an arbitrary uncountable X non-(n+1)-entangled. The second half of the paper gives construction-scheme results: FCA^Delta produces n-entangled sets that are not n+1-entangled for each n (Theorem 0.3), and 2-capturing schemes are shown insufficient for constructing entangled sets (Theorems 0.4 and 0.5).

Significance. If Theorem 0.2 is established, it is a substantial result: it answers the open question from [6] and shows the finite n-entangledness hierarchy can be terminated at any prescribed level in models of two strong forcing axioms. The forcing P(X,E) and the preservation argument are original and constitute the main technical contribution; the construction-scheme results are clean and useful, and Theorem 0.3 unifies prior CH-based constructions. The paper is proof-based; no machine-checked proofs or code are provided. The significance is tempered by the fact that the lower-bound consistency theorem is imported from an unpublished source and the preservation proof currently has a gap; both are fixable but are load-bearing.

major comments (3)

[§2, Theorem 2.6] The consistency of PFA_{n-ent}(E) is the lower bound for the main theorem, but it is quoted ('Essentially Theorem 97, [13]') without proof, without an explicit large-cardinal hypothesis, and deferred to an upcoming joint paper. The abstract and Theorem 0.2 state 'is consistent' unqualified; if the [13] proof uses a supercompact cardinal, as is typical for PFA-type axioms, the consistency claim must be qualified. Please state the exact hypothesis, give the proof or a detailed reference with the argument, and adjust the abstract accordingly.
[§3, proof of Theorem 3.10] In the recursive construction, the case 'k ∈ P and a_k = a' is handled 'using Lemma 3.7'. Lemma 3.7 requires a to be of the first kind over the chain F_{n_r+k}; the proof has not shown this, and it can fail (a may have several elements in one F(i+1)∖F(i) block). In that situation Lemma 3.7 is inapplicable; Lemma 3.8 does apply because a ∩ F_{n_r+k}(0)=∅, and since σ_k=τ when a_k=a, its conclusion type(a,b)=τ is exactly what the recursion needs. The case split on first/second kind is therefore necessary.
[§3, Theorem 3.4] The genericity criterion for P(X,E) — hence its properness — is stated and its proof omitted ('we leave it to the reader'). Properness is required to apply PFA_{n-ent}(E), and the analogous preservation proof (Theorem 3.10) contains the gap described above, so this omission cannot be regarded as routine. A proof or at least a detailed sketch should be included. The paper should also explicitly describe the ω1-many dense sets that turn the generic-filter object Q_G of Corollary 3.5 into a witness of non-(n+1)-entangledness under PFA.

minor comments (8)

[§2, PFA definition] The displayed definition of PFA_{n-ent}(E) ends with 'then m(P)' with no conclusion; presumably 'm(P)=ω1'. As printed the axiom is incomplete.
[§3, Definition 3.6] The definition of a_i^F uses F(i+1) for i ≤ n; the convention that the top model F(|F|) means the universe (stated earlier for chains) should be repeated here to avoid ambiguity.
[§3, proof of Theorem 3.10] The recursion bound 'Assume k ≤ n' should be over the number n_r^p of new good pairs, not the fixed integer n; as written it is confusing.
[§5, after Prop. 5.5] The homeomorphism between E_F, E_e, and E_o is asserted solely from the equality of the Δ-values. Since the definitions of e_α and o_α allow sign flips (e.g., e_α(l) = -Ξ_α(l)), the order at the first difference can be reversed, so the claimed homeomorphism needs a proof.
[§5, proof of Prop. 5.4, Case 2] The inequality 'n = |C_odd| < |C_{l-1}^j| - 1 = n - 1' is garbled; the contradiction should be n ≤ n - 1 (or |C_odd| ≤ |C_{l-1}^j| - 1). Please fix.
[§6, proof of Theorem 6.1] 'It is easy to see that P is a ccc-forcing provided that P is an entangled (thus, increasing) set' should say 'provided that E is an entangled set' — P is the forcing notion, not the entangled set.
[§2, proof of Theorem 2.7] The proof uses 'Claim 1 of Lemma 8.5 in [1]' in a generalized n-entangled form. Provide a reference or a proof of this generalization, since [1] states it for entangled sets only.
[References] [13] is cited without a venue. If it is a preprint, please state its status; given that Theorem 2.6 is essential, a published reference or an appendix with the argument would be preferable.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the central derivation is a new forcing-preservation proof resting on an external consistency theorem imported from [13].

full rationale

The paper's central derivation (Theorem 0.2) is not circular by construction. It starts from Theorem 2.6, an imported consistency theorem for the parametrized forcing axiom PFA_{n-ent}(E) ('Essentially Theorem 97, [13]'), and then proves new content in Section 3: the forcing P(X,E) destroys the n+1-entangledness of every uncountable X while preserving the n-entangledness of a fixed E. The forcing is explicitly designed to add an uncountable family of (n+1)-tuples with pairwise non-constant type; this directly witnesses the failure of n+1-entangledness, so no quantity is fitted and then renamed as a prediction. The preservation proof (Theorem 3.10) is a combinatorial argument using Lemmas 3.7-3.9, not a restatement of the desired conclusion. The secondary results under FCA^Δ and 𝔪_S > ω1 are conditional statements proved from prior axioms; citations to the author's own [8], [10], [11] point to independent earlier theorems (e.g., FCA^Δ from CH) rather than to the target result. The main caveat is not circularity: Theorem 2.6 is imported with no proof in this paper and with no explicit large-cardinal hypothesis, and Theorem 3.4 is said to be left to the reader. These are completeness/correctness risks about the external support, not cases where an equation reduces to an input or a fitted parameter is called a prediction.

Assumptions & free parameters 1 free parameters · 6 assumptions · 2 invented entities

The paper is a pure set-theoretic consistency proof with no fitted empirical parameters. The central claim rests on imported consistency theorems, especially PFA_{n-ent}(E), and on the author's own construction-scheme tools. The only hand-chosen numerical condition is the growth bound on scheme types, which appears under-specified in the extraction.

free parameters (1)

scheme type growth bound n_{k+1} ≥ 2m_k
Hand-chosen in Section 5 to support the enumeration C_i^k of n-element subsets; as written it is too weak unless the intended bound is exponential (2^{m_k}).

assumptions (6)

standard math ZFC
All forcing and consistency arguments take place in ZFC.
domain assumption Consistency of PFA_{n-ent}(E) (Theorems 2.5 and 2.6)
Imported from [13]; no proof or large-cardinal hypothesis is given here. It supplies the model for Theorem 0.2.
domain assumption Claim 1 of Lemma 8.5 of [1]
Used in the proof of Theorem 2.7 to derive MA(ω1) from PFA_{n-ent}(E).
domain assumption FCA^Δ is consistent (Theorem 4.11 from [8])
Author's own prior result under CH; used for Theorem 5.6.
domain assumption Existence of a fully Δ-capturing scheme with rapidly growing type
Assumed in Section 5 for the construction of the n-entangled sets.
standard math Standard toolbox: Δ-system lemma, elementary submodels, proper forcing, Neeman iterations
Background methods used throughout.

invented entities (2)

PFA_{n-ent}(E), parametrized forcing axiom
purpose: Asserts the existence of filters for proper forcings preserving n-entangledness of a fixed set E; used as the main tool to get the consistency result.
New named axiom, defined in Section 2; its consistency is cited to [13] rather than proved here.
Forcing P(X,E) with good pairs
purpose: A proper forcing that destroys the (n+1)-entangledness of an arbitrary uncountable X while preserving n-entangledness of E.
New forcing construction in Section 3; mathematical tool, not an empirical entity.

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Cite this review

Pith. "Pith review of There may be an $n$-entangled set but no $n+1$-entangled sets." pith.science (2026). https://pith.science/paper/RBAOGSYY

@misc{pith2026250901029,
  author       = {Pith},
  title        = {Pith review of: There may be an $n$-entangled set but no $n+1$-entangled sets},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RBAOGSYY}},
  note         = {Machine review of arXiv:2509.01029}
}

abstract

In this paper we show that for every $2\leq n\in \mathbb{N}$, the statement "there is an $n$-entangled set, but there are no $n+1$-entangled sets" is consistent. We also prove some theorems which improve our understanding of entangled sets in relation to construction schemes: (1) The axiom FCA$^\Delta$ introduced in \cite{finitizationclubch} implies the existence of $n$-entangled sets which are not $n+1$-entangled. (2) $\mathfrak{m}_\mathcal{F}>\omega_1$ implies the non-existence of entangled sets. Thus, $2$-capturing schemes alone are not sufficient to build these kinds of linear orders. (3) The existence of a 2-$\Delta$-capturing scheme is consistent with MA.

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Forward citations

Cited by 1 Pith paper

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Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.

[1] [6]

Some questions on entangled linear orders, 2025

Raphaël Carroy, Maxwell Levine, and Lorenzo Notaro. Some questions on entangled linear orders, 2025

work page 2025

[2] [13]

The p-ideal dichotomy, martin’s axiom and entangled sets

Osvaldo Guzmán and Stevo Todorcevic. The p-ideal dichotomy, martin’s axiom and entangled sets

work page

[3] [1]

On the consistency of some partition theorems for continuouscolorings,andthestructureof ℵ1-denserealordertypes

Uri Abraham, Matatyahu Rubin, and Saharon Shelah. On the consistency of some partition theorems for continuouscolorings,andthestructureof ℵ1-denserealordertypes. Ann.PureAppl.Logic ,29(2):123–206, 1985

work page 1985

[4] [2]

Martin’s axiom does not imply that every twoℵ1-dense sets of reals are isomorphic

Uri Avraham and Saharon Shelah. Martin’s axiom does not imply that every twoℵ1-dense sets of reals are isomorphic. Israel J. Math., 38(1-2):161–176, 1981

work page 1981

[5] [3]

Beaudoin

Stewart Baldwin and Robert E. Beaudoin. Countable dense homogeneous spaces under Martin’s axiom. Israel J. Math., 65(2):153–164, 1989

work page 1989

[6] [4]

Baumgartner

James E. Baumgartner. Allℵ1-dense sets of reals can be isomorphic.Fund. Math., 79(2):101–106, 1973

work page 1973

[7] [5]

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Piotr Borodulin-Nadzieja and David Chodounský. Hausdorff gaps and towers in(𝜔)∕Fin. Fund. Math., 229(3):197–229, 2015

work page 2015

[8] [7]

On construction schemes: building the uncountable from finite pieces, 2024

Jorge Antonio Cruz Chapital. On construction schemes: building the uncountable from finite pieces, 2024

work page 2024