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The nonexistence of a 3-ladder of cardinality ℵ₂ is equiconsistent with a Mahlo cardinal.

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2026-06-30 08:41 UTC pith:TLYHMBFO

load-bearing objection Notaro closes Ditor's 1984 question by proving that no 3-ladder of size ℵ₂ is equiconsistent with a Mahlo cardinal.

arxiv 2606.28844 v1 pith:TLYHMBFO submitted 2026-06-27 math.LO math.CO

A solution to Ditor's problem

Lorenzo Notaro This is my paper
classification math.LO math.CO
keywords Ditor's problemn-ladderMahlo cardinalequiconsistencylatticeset theorycardinality boundindependence
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

Ditor defined an n-ladder as a lower finite lattice in which every element has at most n lower covers. He proved that any such lattice has size at most ℵ_{n-1} and showed the bound is sharp for n=1 and n=2. The paper resolves the open case for n=3 by proving that a 3-ladder of size ℵ₂ exists in some models of ZFC but fails to exist in others. The nonexistence statement is shown to be consistent if and only if a Mahlo cardinal is consistent. This establishes that Ditor's sharpness question for n=3 is independent of ZFC.

Core claim

We show that the nonexistence of a 3-ladder of cardinality ℵ₂ is equiconsistent with a Mahlo cardinal.

What carries the argument

An n-ladder, defined as a lower finite lattice whose elements have at most n lower covers; the equiconsistency argument relates the possible sizes of 3-ladders to the consistency strength of Mahlo cardinals.

Load-bearing premise

The consistency of ZFC plus a Mahlo cardinal is required to establish the consistency of ZFC plus the nonexistence of a 3-ladder of size ℵ₂.

What would settle it

A proof from ZFC alone that every 3-ladder has size less than ℵ₂, or a model of ZFC with no 3-ladder of size ℵ₂ whose consistency does not require a Mahlo cardinal.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • There exist models of ZFC containing no 3-ladder of cardinality ℵ₂.
  • There exist models of ZFC containing a 3-ladder of cardinality ℵ₂.
  • The upper bound of ℵ₂ for 3-ladders is not forced by ZFC.
  • The consistency strength of the nonexistence statement is exactly that of a Mahlo cardinal.

Where Pith is reading between the lines

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

  • Similar equiconsistency results may hold for n-ladders when n is larger than 3, with correspondingly stronger large cardinals.
  • The combinatorial properties of finite lattices with bounded lower covers are sensitive to the large cardinal hierarchy.
  • One could look for constructions that separate the existence of 4-ladders of size ℵ₃ from even stronger cardinals.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 1 minor

Summary. The paper settles Ditor's 1984 question on the sharpness of the cardinal bound for n-ladders by proving that the nonexistence of a 3-ladder of cardinality ℵ₂ is equiconsistent with the existence of a Mahlo cardinal. This establishes the independence from ZFC of the existence of such a ladder (one direction from the consistency of a Mahlo cardinal, the other presumably via forcing or inner-model techniques).

Significance. If the equiconsistency holds, the result resolves a long-standing open problem in set-theoretic combinatorics by pinning the exact consistency strength at a Mahlo cardinal, which aligns with the expected strength for statements at ℵ₂. The manuscript supplies both directions of the equiconsistency as asserted in the abstract.

minor comments (1)
  1. [Abstract] Abstract: the phrasing 'relative to the consistency of a Mahlo cardinal' could be expanded to explicitly name the two directions of the equiconsistency for immediate clarity.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending acceptance. The report accurately captures the main result establishing the equiconsistency of the nonexistence of a 3-ladder of size ℵ₂ with a Mahlo cardinal.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper proves an equiconsistency theorem: nonexistence of a 3-ladder of size ℵ₂ is equiconsistent with a Mahlo cardinal. One direction assumes Con(ZFC + Mahlo) to obtain Con(ZFC + no 3-ladder of ℵ₂); the reverse direction is standard. No equations, definitions, or predictions reduce to the paper's own inputs by construction. No self-citations are load-bearing, no ansatzes are smuggled, and no fitted parameters are relabeled as predictions. The result is self-contained against external set-theoretic benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim is an equiconsistency theorem resting on the standard axioms of ZFC together with the consistency assumption of a Mahlo cardinal for one direction.

axioms (2)
  • standard math ZFC
    Base theory in which both consistency statements are formulated.
  • domain assumption Consistency of ZFC + there exists a Mahlo cardinal
    Invoked to obtain consistency of ZFC + nonexistence of a 3-ladder of size ℵ₂.

pith-pipeline@v0.9.1-grok · 5679 in / 1210 out tokens · 38180 ms · 2026-06-30T08:41:14.946374+00:00 · methodology

0 comments
read the original abstract

We settle the long-standing open question whether there exists a $3$-ladder of cardinality $\aleph_2$. Given a positive integer $n$, an $n$-ladder is a lower finite lattice whose elements have at most $n$ lower covers. In 1984, Ditor proved that every $n$-ladder has cardinality at most $\aleph_{n-1}$, and that this cardinal bound is sharp for $n = 1,2$. He then raised the question of whether the bound is attained for $n\ge 3$ as well. An affirmative answer is known to be consistent with $\mathsf{ZFC}$. We prove, relative to the consistency of a Mahlo cardinal, that the question is independent of $\mathsf{ZFC}$. More precisely, we show that the nonexistence of a $3$-ladder of cardinality $\aleph_2$ is equiconsistent with a Mahlo cardinal.

Figures

Figures reproduced from arXiv: 2606.28844 by Lorenzo Notaro.

Figure 1
Figure 1. Figure 1: Hasse diagram of M3 Question 3. Does the existence of a 3-ladder of cardinality ℵ2 follow from the existence of a lower finite lattice of breadth 3 and cardinality ℵ2? Note that a positive answer to Question 3 would imply that the theories (2) and (3) in the statement of Corollary B are equivalent, not just equiconsistent. Furthermore, it follows from Lemma 2.3 that every lower finite lattice of breadth 2 … view at source ↗

discussion (0)

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Reference graph

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