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arxiv: 1907.11765 · v1 · pith:5SERJOVXnew · submitted 2019-07-26 · 🧮 math.LO

Ultrafilters on singular cardinals of uncountable cofinality

James Cummings , Charles Morgan This is my paper

Pith reviewed 2026-05-24 14:53 UTC · model grok-4.3

classification 🧮 math.LO
keywords ultrafilterssingular cardinalsuncountable cofinalityweakly inaccessible cardinalsuniform ultrafilterscharacter of ultrafiltersforcing constructions
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The pith

Consistently there exists a singular cardinal κ of uncountable cofinality such that 2^κ is weakly inaccessible and every regular cardinal between κ and 2^κ is the character of some uniform ultrafilter on κ.

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

The paper establishes the consistency of a model containing a singular cardinal κ with uncountable cofinality where the value of 2^κ is a weakly inaccessible cardinal. In this model every regular cardinal strictly larger than κ and smaller than 2^κ arises as the character of at least one uniform ultrafilter on κ. A sympathetic reader would care because the result shows that the possible characters of ultrafilters on singular cardinals of uncountable cofinality can fill an entire interval of regular cardinals up to a weakly inaccessible limit, extending earlier knowledge that was limited to regular cardinals or singular cardinals of countable cofinality.

Core claim

From the consistency of ZFC plus suitable large cardinal hypotheses, one can force a model in which there is a singular cardinal κ of uncountable cofinality, 2^κ is weakly inaccessible, and the set of characters of uniform ultrafilters on κ includes every regular cardinal in the open interval (κ, 2^κ).

What carries the argument

A forcing iteration that preserves the relevant cardinals, arranges the ultrafilter characters, and avoids collapsing 2^κ to a successor cardinal.

If this is right

  • The power set of such a κ is a limit cardinal that is not a successor of any cardinal.
  • There exist at least as many distinct characters of uniform ultrafilters on κ as there are regular cardinals below 2^κ.
  • The continuum function at κ can be arranged so that its value is a weakly inaccessible cardinal while controlling ultrafilter characters below it.
  • The result applies specifically to singular cardinals whose cofinality is uncountable, distinguishing them from the countable-cofinality case.

Where Pith is reading between the lines

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

  • The same forcing technique might be used to control additional combinatorial properties of ultrafilters on the same κ.
  • One could ask whether the conclusion can be strengthened so that 2^κ is a Mahlo cardinal or satisfies other large-cardinal properties while still realizing all the intermediate regular characters.

Load-bearing premise

The construction requires the consistency of ZFC plus large cardinal hypotheses strong enough to support a forcing iteration that preserves the cardinals and arranges the ultrafilter characters without collapsing 2^κ.

What would settle it

A proof inside ZFC that no singular cardinal of uncountable cofinality can have every regular cardinal between it and its power set realized as the character of a uniform ultrafilter on it.

read the original abstract

We prove that consistently there is a singular cardinal $\kappa$ of uncountable cofinality such that $2^\kappa$ is weakly inaccessible, and every regular cardinal strictly between $\kappa$ and $2^\kappa$ is the character of some uniform ultrafilter on $\kappa$.

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 / 2 minor

Summary. The manuscript proves the consistency of the existence of a singular cardinal κ of uncountable cofinality such that 2^κ is weakly inaccessible and every regular cardinal strictly between κ and 2^κ arises as the character of some uniform ultrafilter on κ. The argument proceeds by a forcing iteration that arranges the desired ultrafilter characters while preserving the singularity of κ and the inaccessibility of 2^κ.

Significance. If the forcing construction and its preservation lemmas are correct, the result supplies a new consistency statement about the possible character spectra of uniform ultrafilters on singular cardinals of uncountable cofinality, relating them to the continuum function in a way that extends known results for regular cardinals and for singulars of countable cofinality. The construction would demonstrate that the interval (κ, 2^κ) can be densely realized by ultrafilter characters without collapsing 2^κ to a successor cardinal.

major comments (2)
  1. [Introduction / Theorem 1.1] The abstract and introduction do not state the precise large-cardinal hypotheses from which the consistency is derived. Because the iteration must simultaneously preserve singularity of κ, inaccessibility of 2^κ, and the assignment of all intermediate regular characters, the exact strength (e.g., supercompact or measurable cardinals with indestructibility) is load-bearing for verifying that no unintended collapses occur; this information is required to evaluate the result.
  2. [Section 3 (forcing construction)] The preservation arguments for the cofinality of κ and the regularity of 2^κ under the iteration are not detailed in the provided abstract. Without explicit lemmas showing that the iteration does not collapse cardinals in (κ, 2^κ) or change the cofinality of κ to ω, it is impossible to confirm that conditions (i)–(iii) listed in the stress-test note hold simultaneously.
minor comments (2)
  1. [Section 2] Notation for the character of an ultrafilter should be defined at first use and used consistently throughout.
  2. The paper would benefit from a diagram or table summarizing the large-cardinal assumptions, the forcing poset, and the preserved properties.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address each major comment below and will revise the manuscript to improve clarity on the large cardinal assumptions and preservation arguments.

read point-by-point responses

  1. Referee: [Introduction / Theorem 1.1] The abstract and introduction do not state the precise large-cardinal hypotheses from which the consistency is derived. Because the iteration must simultaneously preserve singularity of κ, inaccessibility of 2^κ, and the assignment of all intermediate regular characters, the exact strength (e.g., supercompact or measurable cardinals with indestructibility) is load-bearing for verifying that no unintended collapses occur; this information is required to evaluate the result.

    Authors: We agree that the large cardinal hypotheses should be stated explicitly in the abstract and introduction. The proof assumes the existence of a supercompact cardinal with suitable indestructibility properties to ensure the forcing iteration works as intended. We will update the abstract and the introduction to include this information. revision: yes

  2. Referee: [Section 3 (forcing construction)] The preservation arguments for the cofinality of κ and the regularity of 2^κ under the iteration are not detailed in the provided abstract. Without explicit lemmas showing that the iteration does not collapse cardinals in (κ, 2^κ) or change the cofinality of κ to ω, it is impossible to confirm that conditions (i)–(iii) listed in the stress-test note hold simultaneously.

    Authors: The full manuscript in Section 3 provides detailed lemmas establishing the preservation of the cofinality of κ (ensuring it remains singular of uncountable cofinality) and the regularity of 2^κ. These lemmas also ensure no cardinals are collapsed in (κ, 2^κ). We will add a summary paragraph in the introduction referencing these lemmas to address this concern. revision: partial

Circularity Check

0 steps flagged

No circularity: consistency result via forcing construction with no self-referential reductions.

full rationale

The paper establishes a consistency statement by exhibiting a forcing iteration that arranges ultrafilter characters on a singular cardinal κ while preserving its singularity, the inaccessibility of 2^κ, and the desired character spectrum. This is a standard set-theoretic construction relying on large-cardinal hypotheses and preservation lemmas external to the target statement. No equations, definitions, or predictions reduce to their own inputs by construction; there are no fitted parameters renamed as predictions, no self-definitional loops, and no load-bearing self-citations that collapse the argument. The derivation is self-contained against external forcing techniques and does not invoke uniqueness theorems or ansatzes from the authors' prior work in a circular manner.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper is a pure consistency result in ZFC set theory. No free parameters are fitted to data. The construction likely relies on standard ZFC axioms plus the consistency of large cardinals sufficient for the forcing; no invented entities are introduced.

axioms (1)
  • standard math ZFC
    The ambient foundation for all set-theoretic arguments in the paper.

pith-pipeline@v0.9.0 · 5554 in / 1180 out tokens · 21222 ms · 2026-05-24T14:53:07.135026+00:00 · methodology

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Reference graph

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