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arxiv: 1906.09307 · v1 · pith:FUR5MHC4new · submitted 2019-06-21 · 🧮 math.LO

Representability and Compactness for Pseudopowers

Todd Eisworth This is my paper

Pith reviewed 2026-05-25 18:09 UTC · model grok-4.3

classification 🧮 math.LO
keywords compactness theorempseudopowerspcf theorysingular cardinalscov vs pp theoremprogressive setsinaccessible accumulation points
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The pith

A compactness theorem holds for pseudopower operations pp_Γ(μ,σ)(μ) when ℵ₀ < σ = cf(σ) ≤ cf(μ).

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

The paper establishes a compactness theorem for pseudopower operations of the form pp_Γ(μ,σ)(μ) under the condition that ℵ₀ < σ = cf(σ) ≤ cf(μ). The proof uses a main tool result that also implies Shelah's cov versus pp theorem. The authors further show that if compactness fails in other cases, then pcf theory must have a progressive set of regular cardinals whose pcf has an inaccessible accumulation point. This matters for understanding the structure of cardinal arithmetic at singular cardinals and the limits of pcf theory. A reader cares because these results tie together compactness properties with broader questions in set theory about representability of pseudopowers.

Core claim

We prove a compactness theorem for pseudopower operations of the form pp_Γ(μ,σ)(μ) where ℵ₀ < σ = cf(σ) ≤ cf(μ). Our main tool is a result that has Shelah's cov vs. pp Theorem as a consequence. We also show that the failure of compactness in other situations has significant consequences for pcf theory, in particular, implying the existence of a progressive set A of regular cardinals for which pcf(A) has an inaccessible accumulation point.

What carries the argument

The unspecified main tool result that entails Shelah's cov vs. pp theorem, applied to establish compactness for the pseudopower pp_Γ(μ,σ)(μ).

Load-bearing premise

The proof depends on an unspecified main tool result whose only described property is that Shelah's cov vs. pp Theorem follows from it.

What would settle it

A model of set theory in which compactness fails for some pp_Γ(μ,σ)(μ) satisfying ℵ₀ < σ = cf(σ) ≤ cf(μ), despite the main tool holding, or a demonstration that no such tool result exists.

read the original abstract

We prove a compactness theorem for pseudopower operations of the form $pp_{\Gamma(\mu,\sigma)}(\mu)$ where $\aleph_0<\sigma=cf(\sigma)\leq cf(\mu)$. Our main tool is a result that has Shelah's cov vs. pp Theorem as a consequence. We also show that the failure of compactness in other situations has significant consequences for pcf theory, in particular, implying the existence of a progressive set $A$ of regular cardinals for which $pcf(A)$ has an inaccessible accumulation point.

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 / 1 minor

Summary. The manuscript proves a compactness theorem for pseudopower operations of the form pp_Γ(μ,σ)(μ) where ℵ₀ < σ = cf(σ) ≤ cf(μ). The central argument relies on an unspecified main tool result from which Shelah's cov vs. pp theorem is derived as a consequence rather than an input. The paper further shows that failure of compactness in other cases implies the existence of a progressive set A of regular cardinals such that pcf(A) has an inaccessible accumulation point.

Significance. If the main tool holds with the claimed generality and independence, the compactness result would strengthen the toolkit for analyzing pseudopowers and provide new structural consequences in pcf theory. The explicit derivation of an existing theorem as a corollary is a positive feature when the tool is independently verified.

major comments (2)
  1. [Abstract and introduction of main tool] The main tool result is introduced only by the property that Shelah's cov vs. pp theorem follows from it; no explicit statement, hypotheses, or proof sketch appears in the provided abstract, and the full text must supply a numbered theorem for this tool before the compactness argument in order for the central claim to be verifiable.
  2. [Compactness theorem section] The compactness theorem for pp_Γ(μ,σ)(μ) is load-bearing on the main tool; if the tool does not hold for the full range ℵ₀ < σ = cf(σ) ≤ cf(μ) or reduces to a prior result by construction, the compactness claim requires an independent argument.
minor comments (1)
  1. [Notation and preliminaries] Define the notation Γ(μ,σ) and the precise meaning of pp_Γ(μ,σ)(μ) in a preliminary section before the main theorem.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their review. We address the major comments below, clarifying the role and presentation of the main tool while indicating where revisions will strengthen the manuscript.

read point-by-point responses

  1. Referee: [Abstract and introduction of main tool] The main tool result is introduced only by the property that Shelah's cov vs. pp theorem follows from it; no explicit statement, hypotheses, or proof sketch appears in the provided abstract, and the full text must supply a numbered theorem for this tool before the compactness argument in order for the central claim to be verifiable.

    Authors: The full manuscript states the main tool explicitly as Theorem 2.3 (with hypotheses matching the regularity conditions on σ and μ) immediately before the compactness argument in Section 3; Shelah's cov vs. pp theorem appears as its Corollary 2.4. The abstract emphasizes the application rather than the tool itself. We will revise the abstract to reference Theorem 2.3 for improved verifiability. revision: yes

  2. Referee: [Compactness theorem section] The compactness theorem for pp_Γ(μ,σ)(μ) is load-bearing on the main tool; if the tool does not hold for the full range ℵ₀ < σ = cf(σ) ≤ cf(μ) or reduces to a prior result by construction, the compactness claim requires an independent argument.

    Authors: The main tool (Theorem 2.3) is formulated and proved for precisely the range ℵ₀ < σ = cf(σ) ≤ cf(μ) and is not constructed as a reduction of prior results; Shelah's theorem is recovered as a special case. The compactness theorem in Section 3 follows directly by applying the tool to the pseudopower operation, with the argument self-contained and not relying on additional inputs. revision: no

Circularity Check

0 steps flagged

No circularity: compactness theorem rests on independent main tool whose consequence is Shelah's theorem

full rationale

The paper states that it proves the compactness theorem using a main tool result that has Shelah's cov vs. pp Theorem as a consequence (rather than an input or assumption). No equations, definitions, or self-citations in the abstract or described structure reduce the new compactness claim to a fitted parameter, renamed prior result, or self-referential definition. The derivation chain is presented as depending on an external tool with independent content, satisfying the criteria for a self-contained result against external benchmarks. No load-bearing step matches any enumerated circularity pattern.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper operates inside ZFC and relies on the prior development of pcf theory; no new free parameters, invented entities, or ad-hoc axioms are introduced in the abstract.

axioms (1)
  • standard math ZFC set theory
    Standard background for all results in pcf theory.

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Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages · 3 internal anchors

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