arxiv: 2605.09161 · v1 · submitted 2026-05-09 · 🧮 math.LO

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On the Intermediate Models of Strongly Compact Prikry Forcing

Ben-Zion Weltsch, Sebastiano Thei, Tom Benhamou

Pith reviewed 2026-05-12 03:32 UTC · model grok-4.3

classification 🧮 math.LO

keywords strongly compact Prikry forcingprojectionssupercompact cardinalsfine measuresintermediate modelsRudin-Keisler criteriastem projectionsdistributive forcings

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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.

The paper establishes that for a supercompact cardinal κ, a basic combinatorial condition on a poset completely identifies whether it arises as a projection of the strongly compact Prikry forcing constructed from κ-complete fine measures on P_κ(λ). This matters because it maps out the intermediate models between the ground model and the extension by this large-cardinal forcing, showing precisely which smaller posets can appear as quotients. When κ is 2^λ-strongly compact the same condition classifies every forcing of size at most λ that projects from the corresponding λ-strongly compact Prikry forcing. The work also isolates Rudin-Keisler-style criteria for projections relative to a fixed measure and proves that the projections of exact size λ are exactly those whose projection maps depend only on the stem of a Prikry condition.

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

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

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.

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 a combinatorial characterization of projections of strongly compact Prikry forcing via a fixed κ-complete fine measure, with a clean stem-only criterion for λ-sized cases that generalizes some earlier results.

read the letter

The main takeaway is that the authors supply an explicit combinatorial property, built from a κ-complete fine measure U on P_κ(λ), that identifies which posets arise as projections (and iterated projections) of the strongly compact Prikry forcing. They also prove that, when the forcing has size λ, the projection maps depend only on the stem, and they give Rudin-Keisler-style criteria for projections from a fixed U. When κ is 2^λ-strongly compact they characterize all projections of size at most λ. These pieces recover the Benhamou-Hayut-Gitik results and some folklore on κ-distributive embeddings as special cases.

Referee Report

2 major / 2 minor

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

2 responses · 0 unresolved

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

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
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

0 steps flagged

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

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on ZFC plus the existence of supercompact or strongly compact cardinals and their associated fine measures; no free parameters or invented entities are introduced beyond standard large-cardinal hypotheses.

axioms (2)

standard math ZFC set theory
Background framework for all forcing arguments.
domain assumption Existence of a supercompact cardinal κ (or 2^λ-strongly compact)
Required to define the strongly compact Prikry forcing and the fine measures used in the characterizations.

pith-pipeline@v0.9.0 · 5531 in / 1265 out tokens · 35401 ms · 2026-05-12T03:32:23.994649+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

214 extracted references · 214 canonical work pages

[1]

THE TREE PROPERTY AT _

Dima Sinapova , journal =. THE TREE PROPERTY AT _

work page
[2]

Proceedings of the American Mathematical Society , year=

Classical Namba forcing can have the weak countable approximation property , author=. Proceedings of the American Mathematical Society , year=

work page
[3]

Hugh Woodin , author=

The Stationary Tower: Notes on a Course by W. Hugh Woodin , author=. 2004 , publisher=

work page 2004
[4]

Forcing, Iterated Ultrapowers, and Turing Degrees , publisher =

John Steel , title =. Forcing, Iterated Ultrapowers, and Turing Degrees , publisher =. 2015 , doi =. https://www.worldscientific.com/doi/pdf/10.1142/9789814699952_0003 , abstract =

work page doi:10.1142/9789814699952_0003 2015
[5]

Aronszajn Trees and the Failure of the Singular Cardinals Hypothesis , volume =

Itay Neeman , journal =. Aronszajn Trees and the Failure of the Singular Cardinals Hypothesis , volume =

work page
[6]

On SCH and the Approachability Property , urldate =

Moti Gitik and Assaf Sharon , journal =. On SCH and the Approachability Property , urldate =

work page
[7]

Journal of Mathematical Logic , volume =

Benhamou, Tom and Gitik, Moti and Hayut, Yair , title =. Journal of Mathematical Logic , volume =. 2024 , doi =

work page 2024
[8]

Jech , abstract =

Thomas J. Jech , abstract =. More game-theoretic properties of boolean algebras , journal =. 1984 , issn =. doi:https://doi.org/10.1016/0168-0072(84)90038-1 , url =

work page doi:10.1016/0168-0072(84)90038-1 1984
[9]

Dimonte, Vincenzo and Poveda, Alejandro and Thei, Sebastiano , journal=. The

work page
[10]

Supercompact extender based Magidor–Radin forcing , journal =

Merimovich, Carmi , keywords =. Supercompact extender based Magidor–Radin forcing , journal =. 2017 , issn =. doi:https://doi.org/10.1016/j.apal.2017.02.006 , url =

work page doi:10.1016/j.apal.2017.02.006 2017
[11]

Forcing and

Eskew, Monroe , journal=. Forcing and

work page
[12]

Singular cardinals through the lens of

Thei, Sebastiano , year=. Singular cardinals through the lens of

work page
[13]

A general tool for consistency results related to

Dimonte, Vincenzo and Wu, Liuzhen , journal=. A general tool for consistency results related to. 2016 , publisher=

work page 2016
[14]

Archive for Mathematical Logic , volume=

I0 and combinatorics at ^+ , author=. Archive for Mathematical Logic , volume=. 2017 , publisher=

work page 2017
[15]

Suitable extender models

Woodin, W Hugh , journal=. Suitable extender models. 2011 , publisher=

work page 2011
[16]

Shi, Xianghui , journal=. Axiom. 2015 , publisher=

work page 2015
[17]

Jech, Thomas , journal=. A. 1978 , volume=

work page 1978
[18]

Games Played on Boolean Algebras , volume =

Matthew Foreman , journal =. Games Played on Boolean Algebras , volume =

work page
[19]

Journal of the American Mathematical Society , pages=

When does almost free imply free?(For groups, transversals, etc.) , author=. Journal of the American Mathematical Society , pages=. 1994 , publisher=

work page 1994
[20]

, title=

Sharon, A. , title=. Ph.D. Thesis, Tel Aviv University , year=

work page
[21]

Archive for Mathematical Logic , year=

Ben-Neria, Omer , title=. Archive for Mathematical Logic , year=

work page
[22]

2019 , volume=

Ben-Neria, Omer , journal=. 2019 , volume=

work page 2019
[23]

Benhamou, Tom and Gitik, Moti , title=. Ann. Pure App. Log. , volume=. doi:10.1016/j.apal.2022.103107 , year=

work page doi:10.1016/j.apal.2022.103107 2022
[24]

Annals of Mathematics , volume=

Maharam, Dorothy , title=. Annals of Mathematics , volume=. 1947 , pages=

work page 1947
[25]

Marimovich, Carmi , title=. Ann. Pure Appl. Log. , volume=

work page
[26]

Fuchs, Gunther , journal=

work page
[27]

Perlmutter, Norman Lewis , TITLE =. Arch. Math. Logic , FJOURNAL =. 2015 , NUMBER =. doi:10.1007/s00153-014-0410-y , URL =

work page doi:10.1007/s00153-014-0410-y 2015
[28]

Failures of square in

Larson, Paul B and Sargsyan, Grigor , journal=. Failures of square in

work page
[29]

Comptes Rendus Mathematique , Volume=

Shimon Garti , title=. Comptes Rendus Mathematique , Volume=. 2018 , Pages=

work page 2018
[30]

Weak diamond and Galvin’s property , author=. Period. Math. Hung. , year=

work page
[31]

Archive for Mathematical Logic , volume=

Fuchs, Gunther , title=. Archive for Mathematical Logic , volume=

work page
[32]

Journal of Symbolic Logic , year=

Velleman, Dan , title=. Journal of Symbolic Logic , year=

work page
[33]

Abraham, Uri , title =. Ann. Pure Appl. Logic , year =. doi:10.1016/0168-0072(83)90006-4 , fjournal =

work page doi:10.1016/0168-0072(83)90006-4
[34]

, author=

Some exact equiconsistency results in set theory. , author=. Notre Dame Journal of Formal Logic , volume=. 1985 , publisher=

work page 1985
[35]

and Shelah, Saharon , title =

Apter, Arthur W. and Shelah, Saharon , title =. Trans. Amer. Math. Soc. , year =. doi:10.1090/S0002-9947-97-01531-6 , fjournal =

work page doi:10.1090/s0002-9947-97-01531-6
[36]

1998 , edition=

Shelah, Saharon , title=. 1998 , edition=

work page 1998
[37]

Annals of Pure and Applied Logic , year =

Shai Ben-David and Saharon Shelah , title =. Annals of Pure and Applied Logic , year =

work page
[38]

Archive for Mathematical Logic , volume=

Raghavan, Dilip and Shelah, Saharon , title=. Archive for Mathematical Logic , volume=. 2020 , doi=

work page 2020
[39]

arXiv: Logic , year=

Iteration of Semiproper Forcing Revisited , author=. arXiv: Logic , year=

work page
[40]

Guessing models and the approachability ideal , author=. J. Math. Log. , year=

work page
[41]

Mitchell , title=

William J. Mitchell , title=. Notre Dame Journal of Formal Logic , volume=

work page
[42]

Notre Dame J

Forcing with Sequences of Models of Two Types , author=. Notre Dame J. Formal Log. , year=

work page
[43]

Friedman, SD. , year=. Forcing with. Set Theory. Trends in Mathematics. , place=

work page
[44]

Proper forcing remastered , DOI=

Velickovic, Boban and Venturi, Giorgio , editor=. Proper forcing remastered , DOI=. Appalachian Set Theory: 2006–2012 , publisher=. 2012 , pages=

work page 2006
[45]

On the powersets of singular cardinals in HOD , author=

work page
[46]

and Hirshberg, I

Farah, I. and Hirshberg, I. and Vignati, A. , title=. Israel J. Math. , year=

work page
[47]

Naimark, M. A. , title=. Uspehi Matem. Nauk , volume=. 1951 , pages=

work page 1951
[48]

, title=

Gitik, M. , title=. Proc. Am. Math. Sc. , volume=. 2017 , pages=

work page 2017
[49]

Kurepa trees and the failure of the Galvin property , DOI=. Proc. am. math. sc. , author=

work page
[50]

2023 , pages=

Tom Benhamou , title=. 2023 , pages=

work page 2023
[51]

and Hajnal, A

Baumgartner, J. and Hajnal, A. and Mate, A. , TITLE =. Infinite and finite sets (. 1975 , MRCLASS =

work page 1975
[52]

Brown, L. G. Brown and Douglas, R. G. and Fillmore, P. A. , title=. Ann. of Math. , volume=. 1977 , number=

work page 1977
[53]

Nik Weaver , title=. Bull. Symbolic Logic , volume=

work page
[54]

and Weaver, N

Akemann, C. and Weaver, N. , title=. Proc. Natl. Acad. Sci. USA , volume=. 2004 , number=

work page 2004
[55]

Vaccaro , title=

A. Vaccaro , title=. J. Funct. Anal. , volume=. 2018 , number=

work page 2018
[56]

Annals of Mathematics , year=

ALL AUTOMORPHISMS OF THE CALKIN ALGEBRA ARE INNER , author=. Annals of Mathematics , year=

work page
[57]

Extender based forcings , journal=

Gitik, Moti and Magidor, Menachem , year=. Extender based forcings , journal=

work page
[58]

The Journal of Symbolic Logic , volume=

Sinapova, Dima and Spencer, Unger , title=. The Journal of Symbolic Logic , volume=. 2016 , pages=

work page 2016
[59]

Stevo Todorcevic

Dow, Alan , year=. Stevo Todorcevic. Partition problems in topology. Contemporary mathematics, vol. 84. American Mathematical Society, Providence1989, xi 116 pp. , volume=. Journal of Symbolic Logic , publisher=. doi:10.2307/2275490 , number=

work page doi:10.2307/2275490
[60]

Ben-David, Shai and Shelah, Saharon , title =. Ann. Pure Appl. Logic , year =. doi:10.1016/0168-0072(86)90020-5 , fjournal =

work page doi:10.1016/0168-0072(86)90020-5
[61]

and Friedman, Sy-David , title =

Brooke-Taylor, Andrew D. and Friedman, Sy-David , title =. Israel J. Math. , year =. doi:10.1007/s11856-013-0007-x , fjournal =

work page doi:10.1007/s11856-013-0007-x
[62]

Hayut, Yair and Eskew, Monroe , journal=

work page
[63]

1990 , author =

Model theory , publisher =. 1990 , author =

work page 1990
[64]

Cummings, James and Foreman, Matthew , title =. Adv. Math. , year =. doi:10.1006/aima.1997.1680 , fjournal =

work page doi:10.1006/aima.1997.1680 1997
[65]

Cummings, James and Foreman, Matthew and Magidor, Menachem , title =. J. Math. Log. , year =. doi:10.1142/S021906130100003X , fjournal =

work page doi:10.1142/s021906130100003x
[66]

The Journal of Symbolic Logic , year =

James Cummings and Sy-David Friedman , title =. The Journal of Symbolic Logic , year =

work page
[67]

Devlin and Ronald Bjork Jensen , title =

Keith I. Devlin and Ronald Bjork Jensen , title =. Lecture Notes in Mathematics , year =

work page
[68]

, title =

Easton, W. , title =. 1970 , volume =

work page 1970
[69]

2014 , owner =

Eskew, Monroe Blake , title =. 2014 , owner =

work page 2014
[70]

Handbook of set theory , publisher =

Foreman, Matthew , title =. Handbook of set theory , publisher =. 2010 , pages =

work page 2010
[71]

Foreman, Matthew and Laver, Richard , title =. Adv. in Math. , year =. doi:10.1016/0001-8708(88)90041-2 , fjournal =

work page doi:10.1016/0001-8708(88)90041-2
[72]

Israel Journal of Mathematics , year =

Moti Gitik , title =. Israel Journal of Mathematics , year =

work page
[73]

Hayut, Yair and Golshani, Mohammad , title =

work page
[74]

1985 , author =

Building models by games , publisher =. 1985 , author =

work page 1985
[75]

, title =

Jech, Thomas J. , title =. Ann. Math. Logic , year =

work page
[76]

Annals of Mathematical Logic , year =

Jensen, Ronald Bjorn , title =. Annals of Mathematical Logic , year =

work page
[77]

Fundamenta Mathemasicae , year =

John Krueger , title =. Fundamenta Mathemasicae , year =

work page
[78]

1980 , publisher=

Introduction to Independence Proofs , author=. 1980 , publisher=

work page 1980
[79]

Kunen, Kenneth , title =. J. Symbolic Logic , year =. doi:10.2307/2271949 , fjournal =

work page doi:10.2307/2271949
[80]

Sprawozd

Kurepa, Georges , title =. Sprawozd. Towarz. Nauk. Warszaw. Mat.-Fiz. , year =

work page

Showing first 80 references.