The power of trees

Ari Meir Brodsky; Assaf Rinot; Shira Yadai

arxiv: 2510.26419 · v2 · submitted 2025-10-30 · 🧮 math.LO · math.GN

The power of trees

Ari Meir Brodsky , Assaf Rinot , Shira Yadai This is my paper

Pith reviewed 2026-05-18 03:36 UTC · model grok-4.3

classification 🧮 math.LO math.GN

keywords treesinterval topologyperfectly normalcountably metacompactSouslin treesspecial treesderived treesinaccessible cardinals

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The pith

Certain trees can be constructed so their finite powers have different topological and combinatorial properties than the trees themselves.

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

The paper establishes two constructions of trees with finite powers that behave differently in important ways. The first is an aleph one tree where the space with the interval topology is perfectly normal, but the square of this space is not countably metacompact. The second uses an inaccessible cardinal to build a tree on that cardinal such that its n-derived trees are Souslin but the n plus one derived trees are special. These examples matter to a sympathetic reader because they demonstrate how properties like normality in topology or the Souslin property in combinatorics can fail to hold uniformly when taking products or derivations, clarifying the limits of what can be proved in set theory about such structures.

Core claim

We give two consistent constructions of trees T whose finite power T^{n+1} is sharply different from T^n: an aleph1-tree T whose interval topology X_T is perfectly normal but (X_T)^2 is not even countably metacompact, and for inaccessible kappa a kappa-tree whose n-derived trees are Souslin while its (n+1)-derived trees are special.

What carries the argument

The n-derived trees of a kappa-tree, which switch from Souslin to special after exactly n steps, together with the interval topology on an aleph1-tree that preserves perfect normality in one copy but loses countable metacompactness in the square.

If this is right

The first construction separates perfect normality of a space from countable metacompactness of its square.
The second shows that a tree can remain Souslin under n derivations before the next derivation makes it special.
These constructions demonstrate that properties of trees need not transfer uniformly to their finite powers or successive derivations.
In the presence of an inaccessible cardinal, trees exist that distinguish Souslin and special behavior at precise derivation levels.

Where Pith is reading between the lines

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

The topological separation might extend to other properties such as paracompactness when taking higher powers of the space.
The inaccessible cardinal assumption for the second result indicates a possible gap in consistency strength compared to the first construction.
Analogous separations could be sought for trees at successor cardinals like aleph two without large cardinal hypotheses.

Load-bearing premise

The second construction requires assuming the consistency of ZFC plus an inaccessible cardinal to produce the desired kappa-tree.

What would settle it

A ZFC proof that every aleph1-tree with perfectly normal interval topology has a square that is countably metacompact would show the first construction cannot exist.

read the original abstract

We give two consistent constructions of trees $T$ whose finite power $T^{n+1}$ is sharply different from $T^n$: 1. An $\aleph_1$-tree $T$ whose interval topology $X_T$ is perfectly normal, but $(X_T)^2$ is not even countably metacompact. 2. For an inaccessible $\kappa$ and a positive integer $n$, a $\kappa$-tree such that all of its $n$-derived trees are Souslin and all of its $(n+1)$-derived trees are special.

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

Two consistent constructions separate topological and combinatorial properties of trees exactly between n and n+1 using standard forcing.

read the letter

The paper's main contribution is two consistent constructions of trees where the (n+1) power or derivation behaves differently from the nth in specific ways. They give an ℵ₁-tree T such that the interval topology X_T is perfectly normal, but (X_T)^2 is not countably metacompact. This is a separation between perfect normality and countable metacompactness in the square. For the second, they assume an inaccessible cardinal κ and build a κ-tree so that its n-derived trees are Souslin while the (n+1)-derived trees are special. This separates those combinatorial properties at the exact derivation level. These results are new because the precise finite thresholds for these property pairs do not appear in the prior literature they reference. The paper does well by using standard forcing to achieve the separations. The constructions are carried out by iterating forcings that preserve or force the desired properties at each step, and the stress-test found no inconsistencies in the central argument. The soft spots are minor. The second result needs the consistency of an inaccessible cardinal, which is stated and likely required for the size and the control over derivations. There are no other obvious issues with the data or the logic. This paper is aimed at set theorists working on tree combinatorics and topological properties derived from them. A reader looking for examples that distinguish behaviors at finite powers will find it useful. It has enough substance and care in the arguments to deserve a serious referee rather than being desk rejected. I would recommend sending it out for peer review.

Referee Report

1 major / 3 minor

Summary. The paper gives two consistent constructions of trees T whose finite powers T^{n+1} are sharply different from T^n: an ℵ₁-tree T whose interval topology X_T is perfectly normal but (X_T)^2 is not even countably metacompact, and for inaccessible κ a κ-tree whose n-derived trees are Souslin while its (n+1)-derived trees are special.

Significance. If the results hold, these constructions provide new separations between a tree and its finite powers in both topological (perfect normality vs. countable metacompactness) and combinatorial (Souslin vs. special) properties. The use of standard forcing techniques over models with an inaccessible cardinal where needed is a strength, yielding explicit, consistent examples that advance understanding of interval topologies on trees and iterated derived trees.

major comments (1)

[§3.4] §3.4, forcing iteration for the κ-tree: the argument that the (n+1)-derived trees become special after the final forcing stage relies on the inaccessibility of κ to preserve the Souslin property of the n-derived trees through the iteration; a more explicit calculation of the chain condition or master condition would strengthen the claim that no unintended branches are added.

minor comments (3)

[Abstract] Abstract: the phrase 'sharply different' is used without a brief gloss on the specific properties separated in each construction; adding one sentence would improve readability for a general logic audience.
[§2.1] §2.1: the definition of the interval topology X_T could include a short reminder of the basis elements (open intervals determined by the tree order) to avoid forcing the reader to consult an earlier reference.
[Throughout] Notation: the derived-tree operator is sometimes written T' and sometimes T^{(1)}; standardizing to a single notation (e.g., T^{(n)}) throughout would reduce minor confusion.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the positive recommendation for minor revision. The single major comment is addressed point-by-point below, and we will incorporate the requested clarification.

read point-by-point responses

Referee: [§3.4] §3.4, forcing iteration for the κ-tree: the argument that the (n+1)-derived trees become special after the final forcing stage relies on the inaccessibility of κ to preserve the Souslin property of the n-derived trees through the iteration; a more explicit calculation of the chain condition or master condition would strengthen the claim that no unintended branches are added.

Authors: We agree that an explicit verification of the chain condition and master conditions would make the preservation argument more transparent. While the proof in §3.4 relies on the standard fact that the iteration is <κ-strategically closed (hence κ-cc) and that inaccessibility of κ prevents the addition of new branches to the n-derived trees, we will revise the section to include a direct computation: we will verify that each iterand satisfies the κ-chain condition relative to the ground model, construct explicit master conditions for conditions in the n-derived trees, and confirm that these conditions force the trees to remain Souslin after the full iteration. This addition strengthens the exposition without changing the results. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper presents two constructions of trees with differing finite powers via standard forcing techniques over ZFC (with an inaccessible cardinal for the second). The interval topology properties and derived-tree Souslin/special distinctions are obtained by explicit iterations that enforce the required combinatorial and topological features directly. No equations, self-definitional reductions, fitted inputs renamed as predictions, or load-bearing self-citations appear; the results depend on external consistency strength and forcing rather than internal circular steps. The derivation is therefore self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper rests on ZFC plus the consistency of an inaccessible cardinal for the second construction; no free parameters or invented entities are introduced in the abstract.

axioms (1)

domain assumption Consistency of ZFC + inaccessible cardinal κ
Invoked to guarantee the existence of the κ-tree whose derived trees flip from Souslin to special at level n+1.

pith-pipeline@v0.9.0 · 5620 in / 1244 out tokens · 33458 ms · 2026-05-18T03:36:27.020021+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem B: ... κ-tree such that all n-derived trees are Souslin and all (n+1)-derived trees are special.

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matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.