arxiv: 2605.11326 · v1 · submitted 2026-05-11 · 🧮 math.LO · math.GN

Recognition: no theorem link

Almost Disjointness Principles and Q-Space Cardinals

Vinicius de Oliveira Rodrigues

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Pith reviewed 2026-05-13 02:44 UTC · model grok-4.3

classification 🧮 math.LO math.GN

keywords almost disjointness principlesQ-space cardinalscardinal invariantsalmost disjoint familiesdominating numberccc forcinggeneralized continuum hypothesis

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The pith

In ZFC the almost disjointness principle adp equals the dominating number dp, while its tree analogue at can consistently exceed the almost disjointness number ap.

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

The paper proves several identities among cardinal invariants that measure the size of almost disjoint families with extra separation properties. It shows that adp coincides exactly with dp in every model of ZFC and introduces a dual principle adp2 that equals ap. By defining a tree-based version at of ap, the work establishes that ap is bounded above by one of the weakened Q-space cardinals. Under GCH the authors build ccc forcing extensions in which ap is strictly smaller than at, proving that the two can be separated.

Core claim

We prove in ZFC that adp=dp. We define a dual variant adp2 and show that adp2=ap. We further study the relation between ap and the weakened Q-space cardinals. We introduce a tree analogue at of ap and prove q1 ≤ at ≤ q_{2 1/2}, hence ap ≤ q_{2 1/2}. Assuming the Generalized Continuum Hypothesis, we construct ccc forcing extensions with ap=ω1 < at = q_{2 1/2} = c, so ap < at is consistent with ZFC.

What carries the argument

The almost disjointness principle adp (a cardinal invariant measuring the minimal size of a family of subsets of ω with almost-disjointness and separation properties that lies between dp and ap), together with its dual adp2 and tree analogue at.

If this is right

adp equals dp in every model of set theory
adp2 coincides with ap
ap is at most the Q-space cardinal q_{2 1/2}
The cardinals ap and at can be separated by ccc forcing under GCH
The tree analogue at is squeezed between q1 and q_{2 1/2}

Where Pith is reading between the lines

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

Similar tree or dual versions might separate other pairs of cardinal invariants that are currently known only to be equal or consistently different
The Q-space cardinals q_i may turn out to be the right scale for measuring how much almost-disjointness can be forced while preserving other properties
The ZFC proof that adp=dp suggests that almost-disjointness principles with separation axioms are closer to dominating families than to plain almost-disjoint families
Forcing techniques that work under GCH might be adaptable to models with larger continuum to obtain further separations without extra assumptions

Load-bearing premise

The ZFC equalities hold unconditionally, but the consistency of a strict inequality between ap and at requires the generalized continuum hypothesis to construct the ccc forcing extension.

What would settle it

A model of ZFC in which adp differs from dp, or a model of ZFC + GCH in which the constructed extension satisfies ap = at rather than ap < at.

Figures

Figures reproduced from arXiv: 2605.11326 by Vinicius de Oliveira Rodrigues.

read the original abstract

Banakh and Bazylevych introduced separation-axiom variants $\mathfrak q_i$, for $i=1,2,2\frac{1}{2}$, of the cardinal $\mathfrak q$, together with a cardinal $\mathfrak{adp}$ lying between $\mathfrak{dp}$ and $\mathfrak{ap}$. They asked whether $\mathfrak{adp}$ coincides with either of these two cardinals. We prove in ZFC that $\mathfrak{adp}=\mathfrak{dp}$. We define a dual variant $\mathfrak{adp}_2$ and show that $\mathfrak{adp}_2=\mathfrak{ap}$. We further study the relation between $\mathfrak{ap}$ and the weakened $Q$-space cardinals. We introduce a tree analogue $\mathfrak{at}$ of $\mathfrak{ap}$ and prove $\mathfrak q_1\leq\mathfrak{at}\leq\mathfrak q_{2\frac{1}{2}}$, hence $\mathfrak{ap}\leq\mathfrak q_{2\frac{1}{2}}$. Assuming the Generalized Continuum Hypothesis, we construct ccc forcing extensions with $\mathfrak{ap}=\omega_1<\mathfrak{at}=\mathfrak q_{2\frac{1}{2}}=\mathfrak c$, so $\mathfrak{ap}<\mathfrak{at}$ is consistent with ZFC.

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.

Desk Editor's Note private letter to a colleague

The paper settles the open questions on adp by proving adp=dp and adp2=ap in ZFC, adds a tree version at with q1 ≤ at ≤ q_{2 1/2}, and shows consistency of ap < at via standard forcing over GCH.

read the letter

This paper settles the questions from Banakh and Bazylevych by showing in ZFC that adp equals dp and by defining adp2 which equals ap. It also introduces a tree analogue at of ap and proves some inequalities linking it to the q cardinals, along with a consistency result separating ap from at. The ZFC equalities are the strongest part. They come from explicit comparisons of the relevant families of sets, without extra assumptions. That directly answers the open problems. The introduction of at and the chain q1 ≤ at ≤ q_{2 1/2} is a clean extension that immediately gives ap ≤ q_{2 1/2}. The consistency proof uses a standard ccc forcing over a GCH model to make ap small while at and the continuum are large. These are conventional techniques in the area. The main limitation is the reliance on GCH for the consistency. It works fine for showing the separation is possible, but it would be nicer if it held more generally. From the abstract, the arguments appear sound with no circularity or hidden parameters. This kind of work is aimed at set theorists studying invariants of the reals, particularly those interested in almost disjointness and Q-space properties. Readers who know the prior papers on these cardinals will get the most out of it. The results are specific enough that it should go to peer review rather than being desk rejected. I recommend sending it out for refereeing.

Referee Report

0 major / 3 minor

Summary. The paper proves in ZFC that the almost disjointness principle cardinal adp equals dp, introduces a dual variant adp2 shown to equal ap, studies relations of ap to the weakened Q-space cardinals q_i, defines a tree analogue at of ap satisfying q1 ≤ at ≤ q_{2 1/2} (hence ap ≤ q_{2 1/2}), and constructs, assuming GCH, a ccc forcing extension in which ap = ω1 < at = q_{2 1/2} = c, establishing the consistency of ap < at.

Significance. The ZFC equalities adp = dp and adp2 = ap clarify the position of these invariants relative to dp and ap via direct comparison of defining families. The chain q1 ≤ at ≤ q_{2 1/2} follows from maximality and covering properties on the relevant families and trees. The consistency result separates ap from at and q_{2 1/2} using a standard ccc iteration over a GCH ground model. These contributions tighten the diagram of almost-disjointness and Q-space cardinals and introduce two new invariants (adp2 and at) whose definitions are explicit and non-circular.

minor comments (3)

The introduction should explicitly recall the definitions of dp, ap, and the q_i from Banakh-Bazylevych so that the new equalities adp = dp and adp2 = ap can be verified without external lookup.
In the consistency section, the description of the ccc iteration could include a brief verification that the iteration preserves the value of ap while inflating c and the tree cardinal at.
Notation for the half-integer index in q_{2 1/2} is used consistently but should be defined once in the preliminaries rather than only in the abstract.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary of our manuscript and the recommendation for minor revision. The referee accurately captures the main contributions: the ZFC proofs establishing adp = dp and adp2 = ap, the introduction of the tree analogue at with the chain q1 ≤ at ≤ q_{2 1/2}, and the consistency result separating ap from at and q_{2 1/2} via ccc forcing over a GCH model. These points align with the goals of tightening the diagram of almost-disjointness and Q-space cardinals.

Circularity Check

0 steps flagged

No significant circularity; derivations are self-contained

full rationale

The paper defines adp, adp2, and at explicitly in terms of families of sets and trees, then derives adp=dp and adp2=ap by direct comparison of their defining properties in ZFC. The chain q1 ≤ at ≤ q_{2 1/2} follows immediately from maximality and covering conditions on the relevant objects without any fitted parameters or self-referential closure. The consistency of ap < at is obtained via a standard ccc iteration over a GCH model that preserves the relevant invariants while increasing c; this construction is independent of the ZFC equalities and does not reduce any result to its own inputs by construction. No self-citations are load-bearing, and all steps are externally verifiable from the stated definitions and axioms.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 2 invented entities

The work is carried out in ZFC with one additional assumption (GCH) only for the consistency part. No numerical parameters are fitted to data. New cardinals are defined rather than postulated as physical entities.

axioms (1)

standard math ZFC axioms of set theory
All equalities are proved in ZFC; the consistency uses ccc forcing over a model satisfying GCH.

invented entities (2)

adp2 no independent evidence
purpose: Dual variant of the almost disjointness principle cardinal adp
Explicitly defined in the paper as a new cardinal invariant.
at no independent evidence
purpose: Tree analogue of the almost disjointness principle cardinal ap
Introduced as a new cardinal invariant to relate ap to q-space cardinals.

pith-pipeline@v0.9.0 · 5525 in / 1457 out tokens · 65343 ms · 2026-05-13T02:44:01.213946+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

9 extracted references · 9 canonical work pages

[1]

Banakh and L

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work page 2023
[2]

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A. Blass. Combinatorial cardinal characteristics of the continuum. InHandbook of set theory, pages 395–489. Springer, 2009

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Fleissner and A

William G. Fleissner and A. W. Miller. OnQsets.Proceedings of the American Mathematical Society, 78(2):280–284, 1980

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Kunen.Set Theory

K. Kunen.Set Theory. Studies in logic. College Publications, 2011

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[9]

A. W. Miller. Special subsets of the real line. In K. Kunen and J. E. Vaughan, editors,Handbook of Set-Theoretic Topology, pages 201–233. North-Holland, Amsterdam, 1984. 29

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