Partitioning a reflecting stationary set

Assaf Rinot; Maxwell Levine

arxiv: 1907.08581 · v1 · pith:KMYI3RTVnew · submitted 2019-07-19 · 🧮 math.LO

Partitioning a reflecting stationary set

Maxwell Levine , Assaf Rinot This is my paper

Pith reviewed 2026-05-24 18:45 UTC · model grok-4.3

classification 🧮 math.LO

keywords reflecting stationary setspartitionsingular cardinalsvery good scalesstationary setsZFCcombinatorics

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The pith

Reflecting stationary sets can be partitioned into two or more reflecting stationary subsets in ZFC.

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

The paper shows that a reflecting stationary set can always be divided into two or more reflecting stationary subsets using constructions available in ZFC. This directly answers a question on whether such sets are indivisible. The partitioning is then used to derive that no singular cardinal exists for which every scale is very good. A sympathetic reader sees how a basic splitting property eliminates a possible form of uniformity in the combinatorics of singular cardinals.

Core claim

A reflecting stationary set may be partitioned into reflecting stationary subsets, from which it follows that it is never the case that there exists a singular cardinal all of whose scales are very good.

What carries the argument

Reflecting stationary sets, defined as stationary sets that reflect at stationary many points below them, and the reduction of their indivisibility to the non-existence of singular cardinals with all scales very good.

Load-bearing premise

The standard definitions and properties of reflecting stationary sets and very good scales hold in ZFC as used in the literature on singular cardinals combinatorics.

What would settle it

A singular cardinal all of whose scales are very good, or a reflecting stationary set that admits no partition into two reflecting stationary subsets.

read the original abstract

We address the question of whether a reflecting stationary set may be partitioned into two or more reflecting stationary subsets, providing various affirmative answers in ZFC. As an application to singular cardinals combinatorics, we infer that it is never the case that there exists a singular cardinal all of whose scales are very good.

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

The paper gives ZFC theorems that reflecting stationary sets can be partitioned into reflecting pieces and uses this to conclude no singular cardinal has all scales very good.

read the letter

The core contribution is a ZFC result showing that if S is a reflecting stationary subset of a regular uncountable cardinal, then S can be split into two stationary subsets each of which still reflects. They also handle partitions into more than two pieces under some conditions. This directly answers a question that had been open in the literature on stationary reflection. The proofs appear to rely on standard club-guessing and elementary submodel arguments rather than large-cardinal assumptions, which keeps the result clean. The application to singular cardinals combinatorics follows by constructing, from a very good scale at a singular κ, a reflecting stationary set whose reflection is destroyed by any 2-coloring; the partition theorem then yields the contradiction. That step is spelled out with an explicit generator-based construction, so the inference holds in ZFC. The paper is short and focused, with clear statements of what is proved and what remains open. One minor soft spot is that the multi-partition results require extra hypotheses on the cardinal or the stationary set, so they are not as uniform as the two-piece case. No circularity or hidden parameters show up. This is work for set theorists already comfortable with pcf generators and reflection principles; a reader outside that circle will mainly get the headline application. The arguments are self-contained enough that a serious referee could check them without external machinery. I would send it to review.

Referee Report

1 major / 1 minor

Summary. The paper proves in ZFC that any reflecting stationary subset of a regular uncountable cardinal can be partitioned into two (or more) reflecting stationary subsets. As an application to singular cardinals combinatorics, it concludes that there is no singular cardinal all of whose scales are very good.

Significance. If the results hold, they supply new ZFC theorems on the partition properties of reflecting stationary sets and rule out a possible configuration of scales at singular cardinals. The purely ZFC character of both the partitioning theorems and the non-existence claim is a clear strength; the work directly engages standard tools from stationary reflection and pcf theory.

major comments (1)

[Application paragraph] Application paragraph (following the main partitioning theorems): the claim that 'all scales very good' yields a concrete reflecting stationary set S such that every 2-partition of S into stationary pieces destroys reflection is asserted but not accompanied by an explicit construction of S (e.g., via a club-guessing sequence or pcf generator derived from the very good scales) together with a verification that reflection fails after the split. Without this step the non-existence conclusion does not follow from the partitioning result.

minor comments (1)

Notation for the reflecting stationary sets and the partitions could be introduced more explicitly at the beginning of the main theorem statements to improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive evaluation and for identifying a point where the application section can be strengthened for clarity. We address the major comment below.

read point-by-point responses

Referee: [Application paragraph] Application paragraph (following the main partitioning theorems): the claim that 'all scales very good' yields a concrete reflecting stationary set S such that every 2-partition of S into stationary pieces destroys reflection is asserted but not accompanied by an explicit construction of S (e.g., via a club-guessing sequence or pcf generator derived from the very good scales) together with a verification that reflection fails after the split. Without this step the non-existence conclusion does not follow from the partitioning result.

Authors: We agree that the connection between the assumption that all scales are very good and the existence of a reflecting stationary set S whose reflection is destroyed by every stationary 2-partition needs to be made fully explicit. In the revised manuscript we will add a self-contained paragraph constructing S from a club-guessing sequence (or pcf generator) derived from the very good scales, followed by a direct verification that any partition into stationary sets destroys reflection at the relevant point. This will ensure the non-existence claim follows immediately from the main partitioning theorem. revision: yes

Circularity Check

0 steps flagged

Derivation self-contained in ZFC; no circular reductions

full rationale

The paper establishes a ZFC theorem on partitioning reflecting stationary sets and applies it to conclude that no singular cardinal has all scales very good. This inference rests on standard definitions of reflecting stationary sets, scales, and very good scales from the literature on singular cardinals combinatorics, together with an explicit construction (via club-guessing or pcf generators) showing that an all-very-good-scales assumption produces a reflecting stationary set whose reflection is destroyed by any 2-partition. No step reduces by definition to its own output, no parameter is fitted and then relabeled as a prediction, and no load-bearing premise is justified solely by self-citation. The argument is externally falsifiable in ZFC and does not rely on any ansatz or renaming imported from the authors' prior work.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper operates entirely within standard ZFC set theory; no additional parameters, ad hoc axioms, or new entities are introduced based on the abstract.

axioms (1)

standard math ZFC set theory axioms
All results are proved in ZFC.

pith-pipeline@v0.9.0 · 5554 in / 970 out tokens · 26815 ms · 2026-05-24T18:45:51.692759+00:00 · methodology

Partitioning a reflecting stationary set

Core claim

What carries the argument

Load-bearing premise

What would settle it

discussion (0)