On the Intermediate Models of Strongly Compact Prikry Forcing
Pith reviewed 2026-05-12 03:32 UTC · model grok-4.3
The pith
A simple combinatorial property characterizes all projections of the strongly compact Prikry forcing using κ-complete fine measures.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
We exhibit a simple combinatorial property which, for a given supercompact cardinal κ, characterizes the projections of all projections of the strongly compact Prikry forcing using κ-complete fine measures. Considering level-by-level results, if κ is 2^λ-strongly compact, we characterize the forcings of size ≤λ which are projections of that λ-strongly compact Prikry forcing. Fixing a κ-complete fine measure U on P_κ(λ), we provide Rudin-Keisler-like criteria for the existence of projections from the strongly compact Prikry forcing with U. Finally, we prove that among all projections of the λ-strongly compact Prikry forcing, the class of forcings of cardinality λ are exactly those for which a
What carries the argument
the combinatorial property that identifies projections of the strongly compact Prikry forcing from κ-complete fine measures on P_κ(λ), together with stem-dependent projection maps
If this is right
- All κ-distributive forcings that embed into the supercompact Prikry forcing are captured by the same combinatorial condition.
- When κ is 2^λ-strongly compact every forcing of size at most λ that projects from the λ-version satisfies the property, and vice versa.
- Rudin-Keisler-type comparisons of measures yield explicit criteria for when a projection exists relative to a fixed U.
- Forcings of cardinality exactly λ that admit stem-only projection maps are precisely the projections of that size.
Where Pith is reading between the lines
- The stem-only characterization may simplify the computation of generic names and intermediate extensions in concrete applications of these forcings.
- The combinatorial property could be used to classify projections in related Prikry-type constructions such as Magidor or Gitik forcings.
- Level-by-level results suggest that the same technique may distinguish intermediate models when strong compactness holds only up to certain cardinals.
Load-bearing premise
The existence of a κ-complete fine measure on P_κ(λ) for the relevant λ, or that κ is 2^λ-strongly compact, together with the usual properties of Prikry conditions.
What would settle it
A concrete poset of size λ that satisfies the combinatorial property yet admits no projection map from the λ-strongly compact Prikry forcing built from any κ-complete fine measure, or conversely a projection that fails the property.
read the original abstract
We analyze the intermediate models of the strongly compact Prikry forcing. We exhibit a simple combinatorial property which, for a given supercompact cardinal $\kappa$, characterize the projections of all projections of the strongly compact Prikry forcing using $\kappa$-complete fine measures. Considering level-by-level results, if $\kappa$ is $2^\lambda$-strongly compact, we characterize the forcings of size $\leq\lambda$ which are projections of that $\lambda$-strongly compact Prikry forcing. Our characterization generalizes several known results, including those of Benhamou-Hayut-Gitik and folklore results regarding the class of $\kappa$-distributive forcing notions which are embedded into the supercompact Prikry forcing. Fixing a $\kappa$-complete fine measure $\mathcal{U}$ on $P_\kappa(\lambda)$, we also provide Rudin-Keisler like critiria for the existence projections from the strongly compact Prikry forcing with $\mathcal{U}$. Finally, we prove that among all projections of the $\lambda$-strongly compact Prikry forcing, the class of forcings of cardinality $\lambda$ are exactly those for which there is a projection map which depends only on the stem of the Prikry condition. We also give partial results regarding projections of arbitrary cardinality.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper analyzes intermediate models of strongly compact Prikry forcing. It exhibits a combinatorial property, defined from a fixed κ-complete fine measure U on P_κ(λ), that characterizes all projections (and iterated projections) of the strongly compact Prikry forcing for supercompact κ. Level-by-level, when κ is 2^λ-strongly compact, it characterizes all forcings of size ≤λ that arise as projections of the λ-strongly compact Prikry forcing. The results generalize Benhamou-Hayut-Gitik and folklore characterizations of κ-distributive posets embeddable into supercompact Prikry forcing. Additional contributions include Rudin-Keisler-style criteria for the existence of projections from the forcing with U, a proof that the cardinality-λ projections are precisely those admitting a stem-dependent projection map, and partial results for projections of arbitrary cardinality.
Significance. If the characterizations hold, the work supplies a clean, measure-based criterion that unifies several scattered results on Prikry-type projections and intermediate models. The stem-dependence theorem for |P|=λ and the level-by-level analysis for 2^λ-strong compactness are technically useful for constructing models with controlled distributivity and for iterating large-cardinal forcings. The Rudin-Keisler criteria and the generalization of the Benhamou-Hayut-Gitik theorem add concrete tools for the study of projections in the presence of fine measures.
major comments (2)
- [§4] §4 (Rudin-Keisler criteria): the statement that the criteria are 'Rudin-Keisler like' is not accompanied by an explicit comparison showing which direction of the RK-ordering is preserved or reversed under the projection map; without this, it is unclear whether the criterion is strictly stronger or weaker than the classical RK relation on the underlying measures.
- [Theorem 5.3] Theorem 5.3 (stem-dependent projections for |P|=λ): the proof assumes that the stem determines the projection when |P|=λ, but the argument does not address the case in which two distinct stems generate the same projected condition after the measure-one set is fixed; this case must be ruled out or handled separately to establish the 'exactly those' direction.
minor comments (2)
- [§2] The notation for iterated projections (e.g., π_{U,V}) is introduced without an explicit inductive definition; a short recursive clause would clarify the level-by-level statements.
- [final section] The partial results for arbitrary-cardinality projections (final section) are stated without an example showing why the stem-dependence criterion fails when |P|>λ; adding one concrete counter-example would make the limitation transparent.
Simulated Author's Rebuttal
We thank the referee for the careful reading and the helpful comments on our manuscript. The suggestions will improve the clarity of the Rudin-Keisler criteria and the stem-dependent characterization. We respond to each major comment below and indicate the revisions planned for the next version.
read point-by-point responses
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Referee: [§4] §4 (Rudin-Keisler criteria): the statement that the criteria are 'Rudin-Keisler like' is not accompanied by an explicit comparison showing which direction of the RK-ordering is preserved or reversed under the projection map; without this, it is unclear whether the criterion is strictly stronger or weaker than the classical RK relation on the underlying measures.
Authors: We agree that an explicit comparison is needed. In the revised manuscript we will add a short paragraph in §4 that directly compares our projection criterion with the classical Rudin-Keisler ordering on the underlying κ-complete fine measures, specifying the direction preserved or reversed by the projection map and clarifying the relative strength of the two notions. revision: yes
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Referee: [Theorem 5.3] Theorem 5.3 (stem-dependent projections for |P|=λ): the proof assumes that the stem determines the projection when |P|=λ, but the argument does not address the case in which two distinct stems generate the same projected condition after the measure-one set is fixed; this case must be ruled out or handled separately to establish the 'exactly those' direction.
Authors: The referee is right that this case must be treated explicitly to secure the 'exactly those' direction. We will revise the proof of Theorem 5.3 by adding a short sublemma showing that, under the κ-completeness and fineness of the measure, distinct stems cannot produce the same projected condition once the measure-one set is fixed; this rules out the problematic case and completes the argument. revision: yes
Circularity Check
No significant circularity in the derivation chain
full rationale
The paper establishes a combinatorial characterization of projections (and iterated projections) of strongly compact Prikry forcing via a property defined from a fixed κ-complete fine measure U on P_κ(λ). The argument proceeds from the standard Prikry condition, the Rudin-Keisler ordering on measures, and the fact that stems determine projections when |P|=λ. These are external, independently established ingredients of Prikry-type forcing. The central results generalize prior work (including Benhamou-Hayut-Gitik) but do not reduce any new claim to a self-citation, a fitted parameter renamed as a prediction, or an ansatz smuggled via citation. The derivation remains self-contained against the external large-cardinal assumption and standard forcing facts; no step equates a derived object to its input by construction.
Axiom & Free-Parameter Ledger
axioms (2)
- standard math ZFC set theory
- domain assumption Existence of a supercompact cardinal κ (or 2^λ-strongly compact)
discussion (0)
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