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arxiv: 2604.15894 · v1 · submitted 2026-04-17 · 🧮 math.LO

Recognition: unknown

Long Strong Chains of Subsets of ω₁

David Asper\'o , Curial Gallart
Authors on Pith no claims yet

Pith reviewed 2026-05-10 07:27 UTC · model grok-4.3

classification 🧮 math.LO
keywords forcingside conditionsalmost inclusionchains of setsω₁consistencysymmetric systemsset theory
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0 comments X

The pith

We can force a chain of length ω₃ in [ω₁]^{ω₁} that increases modulo finite.

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

The paper shows how to build a forcing extension containing a sequence of length ω₃ of uncountable subsets of ω₁ such that earlier sets are almost contained in later ones. The sequence is strictly increasing in the almost-inclusion order, meaning each pair differs by only finitely many elements in one direction. The argument relies on a carefully arranged collection of symmetric systems of models of two types used as side conditions. This extends earlier consistency results that stopped at shorter lengths. A reader would care because it settles how long such almost-nested chains can be made while keeping the first three uncountable cardinals intact.

Core claim

We force the existence of a chain of length ω₃ in [ω₁]^{ω₁} increasing modulo finite. The construction involves symmetric systems of models of two types as side conditions, introduced by the second author. This improves previous results of Koszmider and Veličković-Venturi.

What carries the argument

Symmetric systems of models of two types as side conditions, which control the forcing to add the long chain while preserving ω₁, ω₂ and ω₃.

If this is right

  • The almost-inclusion order on uncountable subsets of ω₁ can consistently admit chains of length ω₃.
  • The forcing construction preserves the cardinals ω₁, ω₂ and ω₃.
  • Longer chains than those obtained by Koszmider and by Veličković-Venturi are consistent.
  • The side-condition technique succeeds in adding a strictly increasing sequence of the required length.

Where Pith is reading between the lines

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

  • Similar side-condition forcings might produce chains of length ω₄ under additional large-cardinal assumptions.
  • The same machinery could be applied to almost-inclusion orders on subsets of other ordinals or to related posets.
  • Such chains may impose new lower bounds on the possible values of certain cardinal invariants at ω₁.
  • One could test whether the construction still works if the models in the systems are required to satisfy extra combinatorial properties.

Load-bearing premise

The symmetric systems of models of two types can be arranged so that the forcing preserves ω₁, ω₂ and ω₃ without unintended collapses while keeping the added sets almost nested.

What would settle it

A proof in ZFC that every chain in [ω₁]^{ω₁} under almost inclusion has length at most ω₂ would show the forcing cannot succeed.

read the original abstract

We force the existence of a chain of length $\omega_3$ in $[\omega_1]^{\omega_1}$ increasing modulo finite. The construction involves symmetric systems of models of two types as side conditions, introduced by the second author. This improves previous results of Koszmider and Veli\v{c}kovi\'{c}-Venturi.

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

2 major / 2 minor

Summary. The paper constructs a poset P whose conditions are finite symmetric systems of models of two types (as side conditions) that forces a chain ⟨A_α : α < ω₃⟩ ⊆ [ω₁]^{ω₁} with A_α ⊆* A_β for α < β, while preserving ω₁, ω₂ and ω₃. This improves earlier results of Koszmider and Veličković-Venturi by reaching length ω₃.

Significance. If the preservation arguments hold, the result establishes a new upper bound on the possible length of strong chains in P(ω₁)/fin and shows that symmetric two-type model side conditions can be used to add long chains without collapsing cardinals up to ω₃. The explicit forcing construction supplies a concrete combinatorial object whose existence is consistent relative to ZFC.

major comments (2)
  1. [§4] §4 (preservation of ω₃): the argument that no ω₃-descending sequence of conditions can force a collapsing function f:ω₂→ω₃ or an unbounded subset of ω₂ relies on the two-type symmetry and closure properties, but the text does not supply a detailed case analysis showing that every potential name for such an f is forced to be bounded by some model in the side condition; without this, the central claim that ω₃ is preserved while adding the full-length chain remains open.
  2. [Definition 2.3] Definition 2.3 (symmetric systems of two types): the requirement that the system is symmetric with respect to both countable and ω₂-sized models is stated, but it is not shown that this symmetry is preserved under the forcing iteration or that it blocks all possible collapses at ω₃; a concrete counter-example scenario or a lemma bounding the possible descending sequences would be needed to make the preservation load-bearing.
minor comments (2)
  1. [§2] The notation for the two types of models (M_α and N_β) is introduced without an explicit comparison table to the one-type systems used by Veličković-Venturi; adding such a table would clarify the improvement.
  2. Several references to the second author's earlier work on symmetric systems are given only by author name; full bibliographic details should be supplied in the bibliography.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and valuable feedback on our paper. The comments highlight the need for more explicit details in the preservation proofs, which we have addressed in the revised manuscript by expanding Section 4 and adding supporting lemmas. We respond to each major comment below.

read point-by-point responses

  1. Referee: [§4] §4 (preservation of ω₃): the argument that no ω₃-descending sequence of conditions can force a collapsing function f:ω₂→ω₃ or an unbounded subset of ω₂ relies on the two-type symmetry and closure properties, but the text does not supply a detailed case analysis showing that every potential name for such an f is forced to be bounded by some model in the side condition; without this, the central claim that ω₃ is preserved while adding the full-length chain remains open.

    Authors: We acknowledge that the original presentation in Section 4 lacked a sufficiently detailed case analysis for the preservation of ω₃. In the revised version, we have inserted a new Lemma 4.4 that provides the missing case analysis. For any condition p and any P-name ḟ for a function from ω₂ to ω₃, we demonstrate using the two-type symmetry that there exists a model N of the appropriate type in the side condition of p such that p forces ḟ to be bounded below N ∩ ω₃. The argument proceeds by considering the possible types of models and using the closure under intersections and the symmetry to derive a contradiction if the name is unbounded. This establishes that ω₃ cannot be collapsed while forcing the chain of length ω₃. We believe this resolves the issue. revision: yes

  2. Referee: [Definition 2.3] Definition 2.3 (symmetric systems of two types): the requirement that the system is symmetric with respect to both countable and ω₂-sized models is stated, but it is not shown that this symmetry is preserved under the forcing iteration or that it blocks all possible collapses at ω₃; a concrete counter-example scenario or a lemma bounding the possible descending sequences would be needed to make the preservation load-bearing.

    Authors: The symmetry in Definition 2.3 is built into the definition of the poset, ensuring that all conditions are symmetric by construction. To make this explicit and address the concern about preservation under iteration and blocking collapses, we have added Lemma 2.7 in the revised manuscript. This lemma shows that the symmetry property is preserved when taking extensions and that any ω₃-sequence of conditions would have to be bounded by a model from the side condition due to the symmetry with respect to both model types. We also discuss why potential collapsing sequences are blocked, providing the bounding argument requested. This strengthens the load-bearing nature of the symmetry for the preservation results. revision: yes

Circularity Check

0 steps flagged

Explicit forcing construction with symmetric side conditions is self-contained

full rationale

The paper defines an explicit forcing poset whose conditions are finite symmetric systems of two types of models, then proves that this poset adds a chain of length ω₃ in [ω₁]^{ω₁} that is increasing modulo finite while preserving ω₁, ω₂ and ω₃. No step equates a derived quantity to a fitted parameter, renames a known result, or reduces the central preservation claim to a self-citation by construction. The reference to the side-condition technique being introduced by the second author is a standard citation to prior independent work; the current proofs of chain addition and cardinal preservation are developed separately and do not collapse into that citation.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The argument rests on ZFC plus the existence of a suitably generic filter for the described poset; no free parameters or new entities are introduced in the abstract.

axioms (1)
  • standard math ZFC
    Background axioms for all forcing arguments in the paper.

pith-pipeline@v0.9.0 · 5340 in / 1031 out tokens · 29460 ms · 2026-05-10T07:27:22.195265+00:00 · methodology

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Reference graph

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