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arxiv: 2507.17503 · v2 · submitted 2025-07-23 · 🧮 math.LO · math.CO

Some questions on entangled linear orders

Rapha\"el Carroy , Maxwell Levine , Lorenzo Notaro This is my paper

Pith reviewed 2026-05-19 03:37 UTC · model grok-4.3

classification 🧮 math.LO math.CO
keywords entangled linear orderscontinuum hypothesisPi1^1 setsdiamond principlehomeomorphic realsnon-separable ordersset theory
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The pith

Under the continuum hypothesis there exist n-entangled linear orders that are not n+1-entangled for every n.

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

This paper proves conditional existence results for entangled linear orders of varying degrees. Assuming the continuum hypothesis, the authors construct, for each positive integer n, a linear order that is entangled exactly up to level n. They also show that homeomorphism between sets of reals does not preserve entanglement, that an entangled set of reals can be Pi_1^1 definable if all reals are constructible, and that a 2-entangled order need not be separable if the diamond principle holds. These findings clarify the independence and compatibility of entanglement with other set-theoretic and topological properties.

Core claim

The central discovery is that the property of being n-entangled can be realized without being (n+1)-entangled under CH, and that entanglement can coexist with homeomorphism distinctions, Pi_1^1 complexity under V=L, and non-separability under diamond.

What carries the argument

n-entangled linear orders, which the constructions show can have their entanglement level precisely controlled by the choice of underlying set theory axioms.

If this is right

  • If CH holds then entanglement is not an indivisible property but admits a hierarchy of finite degrees.
  • Two homeomorphic subsets of the reals can differ in whether they are 2-entangled.
  • An entangled set of reals can be Pi_1^1 definable under the assumption that the reals are constructible.
  • A 2-entangled linear order can fail to be separable when diamond holds.

Where Pith is reading between the lines

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

  • The results suggest that similar separations might hold for other combinatorial properties of orders under the same axioms.
  • These conditional constructions leave open whether ZFC alone suffices for such examples or if they are independent of the extra assumptions.
  • Neighbouring problems include whether entanglement can be separated from other definability classes or from metrizability in linear orders.

Load-bearing premise

The definitions of n-entangled linear orders and Pi_1^1 sets are fixed as in prior work, and the universe satisfies CH or V=L or diamond as stated.

What would settle it

A construction under CH of an n-entangled linear order that turns out to be (n+1)-entangled as well, or a model of CH where all entangled orders have infinite entanglement level.

read the original abstract

Entangled linear orders were first introduced by Abraham and Shelah. Todor\v{c}evi\'c showed that these linear orders exist under $\mathsf{CH}$. We prove the following results: (1) If $\mathsf{CH}$ holds, then, for every $n > 0$, there is an $n$-entangled linear order which is not $(n+1)$-entangled. (2) If $\mathsf{CH}$ holds, then there are two homeomorphic sets of reals $A, B \subseteq \mathbb{R}$ such that $A$ is entangled but $B$ is not $2$-entangled. (3) If $\mathbb{R}\subseteq \mathrm{L}$, then there is an entangled $\Pi_1^1$ set of reals. (4) If $\diamondsuit$ holds, then there is a $2$-entangled non-separable linear order.

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.

Referee Report

0 major / 4 minor

Summary. The manuscript claims to prove four results on entangled linear orders: Under CH, n-entangled but not (n+1)-entangled orders exist for each n>0; homeomorphic real sets with different entanglement degrees under CH; an entangled Π₁¹ set under R ⊆ L; and a 2-entangled non-separable order under diamond. These are established via explicit constructions that enumerate dense subsets using the respective axioms and verify the entanglement conditions directly.

Significance. If the results hold, they significantly refine our understanding of the spectrum of entanglement properties in linear orders by separating consecutive degrees of entanglement and relating them to separability and descriptive complexity. The explicit nature of the constructions, which build on Todorcevic's CH examples and use standard diamond and constructibility enumerations to control crossing properties, provides reproducible combinatorial arguments that can serve as a basis for further investigations in set theory of the reals.

minor comments (4)
  1. §1: The recall of the Abraham-Shelah definition of n-entangled linear order is helpful, but adding a sentence on how the crossing property is checked in the constructions would aid readers unfamiliar with the area.
  2. §2: In the proof of the first result, the inductive step for increasing n could reference the specific lemma or equation where the density of the chosen sets is verified.
  3. §3: Clarify whether 'homeomorphic' refers to order-isomorphism or topological homeomorphism for the sets A and B, as this affects how the differing entanglement is interpreted.
  4. References: Ensure all cited works, including Todorcevic's CH construction, have complete bibliographic details.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary of our results, the assessment of their significance, and the recommendation for minor revision. We will prepare a revised version of the manuscript.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper establishes four conditional existence results for n-entangled linear orders and related objects under independent axioms (CH, diamond, V=L) whose consistency is external. It recalls the Abraham-Shelah definition of n-entanglement in §1, cites Todorcevic's prior CH construction only as a base case, and supplies explicit inductive or recursive constructions that enumerate dense sets or clubs and directly verify the required crossing and density properties at each step. No step reduces a conclusion to a fitted parameter, self-definition, or unverified self-citation chain; all load-bearing arguments are self-contained combinatorial checks against the stated axioms.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The paper invokes standard set-theoretic assumptions that are independent of ZFC and relies on prior definitions without introducing new entities or free parameters.

axioms (3)
  • domain assumption Continuum Hypothesis (CH)
    Invoked for results (1) and (2) to guarantee the existence of the stated entangled orders and homeomorphic pairs.
  • domain assumption V = L (all reals are constructible)
    Invoked for result (3) to obtain an entangled Pi1^1 set.
  • domain assumption Diamond principle (♦)
    Invoked for result (4) to construct a 2-entangled non-separable linear order.

pith-pipeline@v0.9.0 · 5683 in / 1555 out tokens · 74372 ms · 2026-05-19T03:37:14.892048+00:00 · methodology

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Reference graph

Works this paper leans on

9 extracted references · 9 canonical work pages

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