Total Failure of Approachability at Successors of Singulars of Countable Cofinality
Pith reviewed 2026-05-19 00:27 UTC · model grok-4.3
The pith
Relative to class many supercompact cardinals, a model of ZFC+GCH exists where for every singular cardinal δ of countable cofinality and every regular uncountable μ<δ there are stationarily many non-approachable points of cofinality μ in δ⁺
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
Relative to class many supercompact cardinals, we construct a model of ZFC+GCH where for every singular cardinal δ of countable cofinality and every regular uncountable μ<δ there are stationarily many non-approachable points of cofinality μ in δ⁺.
What carries the argument
The forcing construction over class many supercompact cardinals that adds stationary sets of non-approachable points while preserving GCH and the relevant cardinals' properties.
If this is right
- For each such δ the approachability ideal on δ⁺ fails to contain all clubs through the non-approachable cofinality-μ points.
- The failure occurs uniformly for every singular cardinal of countable cofinality.
- GCH holds in the same model as this total failure of approachability.
- The result gives an affirmative answer to Mitchell's question and a decisive answer to the question of Foreman and Shelah.
Where Pith is reading between the lines
- The construction may be adaptable to produce simultaneous failures of other combinatorial principles at the same cardinals.
- It raises the question of whether the same total failure can be obtained from weaker large-cardinal hypotheses than class many supercompacts.
- The model provides a setting in which to test the interaction between non-approachability and scales or pcf generators at these successors.
Load-bearing premise
Class many supercompact cardinals exist and can be used to define a forcing that produces the desired stationary sets of non-approachable points without violating GCH.
What would settle it
A proof in ZFC alone that for some singular δ of countable cofinality and some regular uncountable μ<δ every set of cofinality-μ ordinals in δ⁺ contains an approachable point would show no such model exists.
read the original abstract
Relative to class many supercompact cardinals, we construct a model of $\ZFC+\GCH$ where for every singular cardinal $\delta$ of countable cofinality and every regular uncountable $\mu<\delta$ there are stationarily many non-approachable points of cofinality $\mu$ in $\delta^+$. This answers a question of Mitchell and provides a decisive answer to a question of Foreman and Shelah.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript proves, relative to the existence of class-many supercompact cardinals, the consistency of ZFC + GCH together with the statement that for every singular cardinal δ of countable cofinality and every regular uncountable μ < δ, the set of non-approachable points of cofinality μ is stationary in δ⁺. This is achieved via a class-length forcing iteration that enforces the desired failure of approachability at all such δ⁺ simultaneously while preserving GCH, thereby answering questions of Mitchell and of Foreman-Shelah.
Significance. If the central construction and its preservation arguments hold, the result is significant for the study of the approachability ideal and singular cardinal combinatorics. It establishes a model in which approachability fails in the strongest possible uniform way at all successors of singulars of countable cofinality, without violating GCH. The use of class forcing from supercompacts to obtain this global failure constitutes a technically substantial contribution.
minor comments (2)
- In the introduction, the precise formulation of Mitchell's question could be quoted or restated verbatim to make the connection between the theorem and the open problem fully explicit.
- The support conditions in the definition of the class iteration (likely in the section introducing the poset) would benefit from an additional sentence clarifying how the Easton support interacts with the supercompact embeddings at limit stages of countable cofinality.
Simulated Author's Rebuttal
We thank the referee for their positive assessment of the manuscript, the recognition of its significance for the approachability ideal and singular cardinal combinatorics, and the recommendation of minor revision. No specific major comments were raised in the report.
Circularity Check
No significant circularity; consistency result from external large-cardinal hypothesis
full rationale
The paper establishes a consistency statement relative to class-many supercompact cardinals by means of a class forcing construction that simultaneously enforces total failure of approachability at all relevant δ⁺ while preserving GCH. The derivation supplies explicit poset definitions, support conditions, and stationarity-preservation lemmas for the non-approachable sets of each cofinality μ. These steps are independent of the target conclusion and rest on standard forcing techniques plus the external large-cardinal assumption; no self-definitional reductions, fitted inputs renamed as predictions, or load-bearing self-citations appear in the central argument.
Axiom & Free-Parameter Ledger
axioms (1)
- domain assumption Existence of class many supercompact cardinals
discussion (0)
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