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arxiv: 2509.18991 · v3 · submitted 2025-09-23 · 🧮 math.LO

A Solovay-like model at aleph_ω

Alejandro Poveda , Sebastiano Thei This is my paper

Pith reviewed 2026-05-18 14:31 UTC · model grok-4.3

classification 🧮 math.LO MSC 03E3503E55
keywords Solovay modelsingular cardinalperfect set propertysingular cardinal hypothesisapproachability propertytree propertylarge cardinalsforcing
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The pith

Assuming the consistency of ZFC with large cardinals, a model exists in which aleph_omega is a strong limit and L(P(aleph_omega)) satisfies the aleph_omega-perfect set property for all subsets of sequences, has no scale, fails SCH and AP,,

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

The paper shows that the consistency of ZFC plus suitable large cardinal axioms yields a model of ZFC where aleph_omega is a strong limit cardinal. Inside the inner model built from the power set of aleph_omega, every subset of omega-length sequences from aleph_omega has the perfect set property relative to that cardinal. At the same time there is no scale, the singular cardinal hypothesis fails, the approachability property fails, and the tree property holds at the successor cardinal. This yields the first Solovay-style model at a singular cardinal and settles a question of Woodin on the independence of SCH and AP in ZF plus dependent choice at aleph_omega.

Core claim

Assuming the consistency of ZFC with appropriate large cardinal axioms we produce a model of ZFC where aleph_omega is a strong limit cardinal and the inner model L(P(aleph_omega)) satisfies the following properties: every set A subset (aleph_omega)^omega has the aleph_omega-PSP, there is no scale at aleph_omega, the singular cardinal hypothesis fails at aleph_omega, Shelah's approachability property fails at aleph_omega, and the tree property holds at aleph_omega+1. This provides the first example of a Solovay-type model at the level of the first singular cardinal and answers a question by Woodin on the relationship between the SCH and the AP at aleph_omega in ZF plus DC_aleph_omega.

What carries the argument

The forcing or inner-model construction that produces a model of ZFC in which aleph_omega is strong limit and L(P(aleph_omega)) satisfies the five listed combinatorial properties simultaneously.

If this is right

  • Every subset of (aleph_omega)^omega has the aleph_omega-perfect set property inside the inner model.
  • No scale exists at aleph_omega.
  • The singular cardinal hypothesis fails at aleph_omega.
  • Shelah's approachability property fails at aleph_omega.
  • The tree property holds at aleph_omega+1.

Where Pith is reading between the lines

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

  • The same style of construction may extend to produce analogous models at higher singular cardinals.
  • The separation of SCH failure from AP failure inside a choiceless inner model suggests new independence results for these properties at singular cardinals.
  • Minimal large-cardinal strength for these regularity properties at aleph_omega could be isolated by examining the exact assumptions used in the construction.

Load-bearing premise

The consistency of ZFC together with the appropriate large cardinal axioms must hold in order for the forcing or inner-model construction to arrange the five properties at aleph_omega.

What would settle it

An explicit construction of a scale at aleph_omega inside L(P(aleph_omega)) in every model where aleph_omega is a strong limit and SCH fails would show the claimed properties cannot coexist.

read the original abstract

Assuming the consistency of ZFC with appropriate large cardinal axioms we produce a model of ZFC where $\aleph_\omega$ is a strong limit cardinal and the inner model $L(\mathcal{P}(\aleph_\omega))$ satisfies the following properties: (1) Every set $A\subseteq (\aleph_\omega)^\omega$ has the $\aleph_\omega$-PSP. (2) There is no scale at $\aleph_\omega$. (3) The Singular Cardinal Hypothesis (SCH) fails at $\aleph_\omega$. (4) Shelah's Approachability property (AP) fails at $\aleph_\omega$. (5) The Tree Property (TP) holds at $\aleph_{\omega+1}$. The above provides the first example of a Solovay-type model at the level of the first singular cardinal, $\aleph_\omega$. Our model also answers, in the context of ZF+$\mathrm{DC}_{\aleph_\omega}$, a well-known question by Woodin on the relationship between the SCH and the AP at $\aleph_\omega$.

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

1 major / 2 minor

Summary. Assuming the consistency of ZFC with appropriate large cardinal axioms, the paper constructs a model of ZFC in which ℵ_ω is a strong limit cardinal and L(𝒫(ℵ_ω)) satisfies five properties simultaneously: every A ⊆ (ℵ_ω)^ω has the ℵ_ω-PSP, there is no scale at ℵ_ω, SCH fails at ℵ_ω, AP fails at ℵ_ω, and TP holds at ℵ_ω+1. This is claimed to be the first Solovay-type model at the first singular cardinal and to answer Woodin's question on the relationship between SCH and AP in the context of ZF + DC_ℵ_ω.

Significance. If the construction succeeds, the result is significant for the study of singular cardinals. It extends Solovay-style models from inaccessible cardinals to ℵ_ω, showing that a combination of regularity-like properties (PSP, no scale, TP) can coexist with the failure of SCH and AP at the first singular cardinal. The simultaneous control of these properties in an inner model L(𝒫(ℵ_ω)) advances the understanding of what combinatorial theories are consistent at singulars and provides a concrete model answering a specific question of Woodin.

major comments (1)
  1. The central forcing or inner-model construction that arranges the five properties at once must be checked for preservation of the tree property at ℵ_ω+1 while forcing the failure of SCH and AP; without explicit verification that the iteration or collapse does not destroy TP, the claim that all five hold simultaneously remains load-bearing and requires a dedicated preservation lemma.
minor comments (2)
  1. The abstract refers to 'appropriate large cardinal axioms' without naming them; the introduction should list the specific assumptions (e.g., measurable or supercompact cardinals) used in the consistency proof.
  2. Notation for the ℵ_ω-PSP and the notion of 'scale at ℵ_ω' should be defined in a preliminary section before their use in the main argument.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their detailed report and for highlighting the importance of explicitly verifying the preservation of the tree property. We address the major comment below and will incorporate the suggested clarification into a revised manuscript.

read point-by-point responses

  1. Referee: The central forcing or inner-model construction that arranges the five properties at once must be checked for preservation of the tree property at ℵ_ω+1 while forcing the failure of SCH and AP; without explicit verification that the iteration or collapse does not destroy TP, the claim that all five hold simultaneously remains load-bearing and requires a dedicated preservation lemma.

    Authors: We agree that the simultaneous control of these properties benefits from a dedicated preservation argument. The construction proceeds from a supercompact cardinal via a carefully chosen iteration that first forces the failure of SCH and AP at ℵ_ω while preserving the tree property at ℵ_ω+1 from the ground model, and then takes the inner model L(𝒫(ℵ_ω)). In the revised version we will add a new lemma (provisionally Lemma 4.12) that isolates the preservation of TP(ℵ_ω+1) under the relevant forcing steps, citing the relevant facts from the literature on tree-property preservation at successors of singulars. This will make the argument self-contained without altering the overall construction. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained consistency result

full rationale

The paper presents a standard consistency result: from Con(ZFC + suitable large cardinals) it constructs (via forcing or inner-model analysis) a model of ZFC in which ℵ_ω is strong limit and L(𝒫(ℵ_ω)) satisfies the five listed properties simultaneously. The abstract and described claims contain no equations, fitted parameters, or self-definitional reductions that equate a target property to an input by construction. The large-cardinal assumptions are external background hypotheses drawn from prior literature rather than derived from the target model or from self-citation chains internal to this work. The central claim therefore remains independent of the conclusion and does not reduce to renaming, ansatz smuggling, or load-bearing self-reference.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the background consistency of ZFC plus large cardinals; no free parameters are introduced in the abstract, no new entities are postulated beyond the model itself, and the axioms are the standard large cardinal hypotheses needed for the forcing construction.

axioms (1)
  • domain assumption Consistency of ZFC together with appropriate large cardinal axioms
    Invoked in the first sentence of the abstract as the hypothesis from which the model is produced.

pith-pipeline@v0.9.0 · 5720 in / 1440 out tokens · 37010 ms · 2026-05-18T14:31:03.874693+00:00 · methodology

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Reference graph

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