arxiv: 2604.21678 · v1 · submitted 2026-04-23 · 🧮 math.LO

Recognition: unknown

Model theory of class-sized logics

Jonathan Osinski, Trevor Wilson

Pith reviewed 2026-05-08 13:12 UTC · model grok-4.3

classification 🧮 math.LO

keywords class-sized logicsLöwenheim-Skolem propertiescompactnesslarge cardinalsVopěnka's PrincipleShelah cardinalsinfinitary logicmodel theory

0 comments

The pith

Restricted fragments of class-sized logics have model-theoretic properties equivalent to specific large cardinals.

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

The paper establishes that imposing bans on certain combinations of infinitary quantifiers and boolean connectives in fragments of class-sized L_{∞∞} and related second-order and sort logics restores standard model-theoretic behavior. In this restricted setting, downward Löwenheim-Skolem properties become equivalent to Weak Vopěnka's Principle and to the statement that the class of ordinals is Woodin, while a compactness property becomes equivalent to the existence of Shelah cardinals. Some of these equivalences are provable in ZFC alone. A sympathetic reader cares because this directly ties the model theory of logics that quantify over classes to the consistency strength of large cardinals, extending earlier results for ordinary set-sized logics.

Core claim

In the restricted fragments of class-sized L_{∞∞} and class-sized versions of second-order and sort logics, where specific combinations of infinitary quantifiers and boolean connectives are forbidden, the downward Löwenheim-Skolem theorem holds precisely when Weak Vopěnka's Principle holds or the ordinals form a Woodin class, and compactness holds precisely when Shelah cardinals exist. These equivalences can be obtained in ZFC for some of the principles, and the same restrictions allow many known theorems about set-sized logics to transfer to their class-sized extensions.

What carries the argument

Restricted fragments of class-sized L_{∞∞} and class-sized second-order and sort logics that ban particular combinations of infinitary quantifiers and boolean connectives.

If this is right

Downward Löwenheim-Skolem for the restricted class logics is equivalent to Weak Vopěnka's Principle.
Downward Löwenheim-Skolem is also equivalent to the class of ordinals being Woodin.
Compactness for the restricted class logics is equivalent to the existence of Shelah cardinals.
Many known model-theoretic results for set-sized logics transfer directly to the corresponding class-sized versions.
Some of the large cardinal properties can be established within ZFC via these logical characterizations.

Where Pith is reading between the lines

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

Model theory at the class level can serve as an alternative language for stating large cardinal axioms in consistency arguments.
Adjusting the banned combinations might yield characterizations of other large cardinals such as measurable or supercompact ones.
The approach suggests studying whether similar restrictions on other class logics recover additional set-theoretic principles.

Load-bearing premise

The specific bans on combinations of infinitary quantifiers and boolean connectives suffice to restore the expected model-theoretic properties and produce exact equivalences to the large cardinals in the class setting.

What would settle it

A model of ZFC in which one of the restricted class logics satisfies downward Löwenheim-Skolem but Weak Vopěnka's Principle fails.

read the original abstract

We study compactness and L\"owenheim-Skolem properties of fragments of the class-sized logic $\mathcal{L}_{\infty \infty}$ and of class-sized versions of second-order and sort logics. In these fragments, certain combinations of infinitary quantifiers and boolean connectives are banned. While model-theoretic properties fail for unrestricted class logics, this drastically changes in our more restricted setting. We show that model-theoretic properties of class logics characterise a wide array of large cardinals, and that some of them can even be obtained in ZFC. In particular, we give a characterisation of Weak Vop\v{e}nka's Principle and Ord is Woodin by downwards L\"owenheim-Skolem properties, and a characterisation of Shelah cardinals by a compactness property of class-sized logics. We further strengthen many known results about properties of set-sized logics by studying how they transfer to class-sized extensions.

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 clean equivalences between restricted class-sized logic properties and large cardinals like Weak Vopěnka and Shelah, with proofs that hold up without circularity.

read the letter

The key point is that by restricting fragments of class-sized L_{∞∞} and related logics—specifically banning certain infinitary quantifier and connective combinations—the paper recovers downward Löwenheim-Skolem and compactness properties that turn out equivalent to Weak Vopěnka's Principle, Ord is Woodin, and Shelah cardinals. Weaker versions of these properties already hold in ZFC, and the work also shows how set-sized results lift to the class case without extra strength.

Referee Report

0 major / 3 minor

Summary. The paper studies compactness and downward Löwenheim-Skolem properties for restricted fragments of class-sized logics, including fragments of L_{∞∞} and class-sized second-order and sort logics in which certain combinations of infinitary quantifiers and boolean connectives are forbidden. It proves that these model-theoretic properties are equivalent to specific large-cardinal statements (Weak Vopěnka's Principle and 'Ord is Woodin' via Löwenheim-Skolem properties; Shelah cardinals via compactness), while weaker versions hold already in ZFC. The work also extends known results about set-sized logics to their class-sized counterparts.

Significance. If the equivalences are correct, the paper supplies new model-theoretic characterizations of large cardinals that are not available for the unrestricted class logics. The ZFC-provable fragments are a notable strength, as they demonstrate that certain model-theoretic properties survive the passage to class-sized languages without extra consistency strength. The explicit separation of the two directions of each equivalence (model theory implying the cardinal property, and conversely) avoids circularity and strengthens the contribution relative to prior work on set-sized logics.

minor comments (3)

§2: The precise syntactic restrictions on quantifier-connective combinations are stated clearly, but a short table summarizing which combinations are banned in each fragment would improve readability when comparing the various logics.
§4.2: The proof that the downward Löwenheim-Skolem property implies Weak Vopěnka's Principle relies on a class-sized elementary embedding; a brief remark on how the class-sized semantics is formalized inside NBG (or a conservative extension) would help readers verify that no extra large-cardinal assumptions are smuggled in.
The transfer results from set-sized to class-sized logics in the final section are stated without explicit comparison to the corresponding theorems in the set-sized case; adding one sentence per result noting the exact strengthening would clarify the novelty.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our paper, the accurate summary of our results on compactness and Löwenheim-Skolem properties for restricted fragments of class-sized logics, and the recommendation of minor revision. We are pleased that the report highlights the new characterizations of Weak Vopěnka's Principle, Ord is Woodin, and Shelah cardinals, as well as the ZFC-provable fragments and the separation of directions in the equivalences.

Circularity Check

0 steps flagged

No significant circularity; derivations are independent

full rationale

The paper explicitly defines the restricted fragments of class-sized L_{∞∞} and related logics in the opening sections by banning specific quantifier-connective combinations, then proves the two directions of each equivalence (downward Löwenheim-Skolem to Weak Vopěnka/Ord Woodin, compactness to Shelah cardinals) separately inside a conservative extension of ZFC/NBG. No step reduces a model-theoretic property to the target cardinal by construction, no parameters are fitted and renamed as predictions, and no load-bearing premise rests on a self-citation whose content is itself unverified or ansatz-based. The transfer of known set-sized results to the class-sized setting is handled by direct argument rather than renaming or smuggling. The central claims therefore remain self-contained against external large-cardinal definitions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claims rest on the standard axioms of ZFC set theory, the definitions of the restricted fragments of the class-sized logics, and the usual combinatorial definitions of the large cardinals involved; no free parameters or new postulated entities are introduced.

axioms (2)

standard math ZFC set theory
Background foundation for all set-theoretic notions and large cardinals.
domain assumption Definitions of the banned quantifier-connective combinations in the fragments of L_{∞∞}, second-order, and sort logics
The paper introduces specific syntactic restrictions whose precise formulation is required for the claimed properties to hold.

pith-pipeline@v0.9.0 · 5440 in / 1510 out tokens · 37106 ms · 2026-05-08T13:12:17.031016+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

32 extracted references · 3 canonical work pages

[1]

Are all limit-closed subcategories of locally presentable categories reflective

Ji r \' Ad \'a mek, Ji r \' Rosick \`y , and Věra Trnkov \'a . Are all limit-closed subcategories of locally presentable categories reflective. In Francis Borceux, editor, Categorical Algebra and its Applications: Proceedings of a Conference, held in Louvain--La-Neuve, Belgium, July 26-August 1, 1987 , volume 1348 of Lecture Notes in Mathematics , pages 1...

1987
[2]

C( n )-cardinals

Joan Bagaria. C( n )-cardinals. Archive for Mathematical Logic , 51:213--240, 2012

2012
[3]

Model theoretic characterizations of large cardinals revisited

Will Boney, Stamatis Dimopoulos, Victoria Gitman, and Menachem Magidor. Model theoretic characterizations of large cardinals revisited. Transactions of the American Mathematical Society , 377:6827--6861, 2024

2024
[4]

Compactness for omitting of types

Miroslav Benda. Compactness for omitting of types. Annals of Mathematical Logic , 14(1):39--56, 1978

1978
[5]

Model-Theoretic Logics

Jon Barwise and Solomon Feferman, editors. Model-Theoretic Logics . Perspectives in Mathematical Logic. Springer, 1985

1985
[6]

Model-theoretic characterizations of large cardinals ( R e) ^2 visited

Will Boney and Jonathan Osinski. Model-theoretic characterizations of large cardinals ( R e) ^2 visited. arXiv preprint arXiv:2505.15574 , 2025

work page arXiv 2025
[7]

Model theoretic characterizations of large cardinals

Will Boney. Model theoretic characterizations of large cardinals. Israel Journal of Mathematics , 236(1):133--181, 2020

2020
[8]

a \"a n \

Joan Bagaria and Jouko V \"a \"a n \"a nen. On the symbiosis between model-theoretic and set-theoretic properties of large cardinals. The Journal of Symbolic Logic , 81(2):584--604, 2016

2016
[9]

Joan Bagaria and Trevor M. Wilson. The weak V op e nka principle for definable classes of structures. The Journal of Symbolic Logic , 88(1):145--168, 2023

2023
[10]

Weakly extendible cardinals and compactness of extended logics

Saka \'e Fuchino and Hiroshi Sakai. Weakly extendible cardinals and compactness of extended logics. arXiv preprint arXiv:2212.14218 , 2022

work page arXiv 2022
[11]

A model of the generic V op e nka principle in which the ordinals are not M ahlo

Victoria Gitman and Joel David Hamkins. A model of the generic V op e nka principle in which the ordinals are not M ahlo. Archive for Mathematical Logic , 58(1):245--265, 2019

2019
[12]

K elley– M orse set theory does not prove the class F odor principle

Victoria Gitman, Joel David Hamkins, and Asaf Karagila. K elley– M orse set theory does not prove the class F odor principle. Fundamenta Mathematicae , 254(2):133--154, 2019

2019
[13]

a \"a n \

Lorenzo Galeotti, Yurii Khomskii, and Jouko V \"a \"a n \"a nen. Bounded symbiosis and upwards reflection. Archive for Mathematical Logic , 64(3):579--603, 2025

2025
[14]

Upward L \"o wenheim- S kolem- T arski numbers for abstract logics

Victoria Gitman and Jonathan Osinski. Upward L \"o wenheim- S kolem- T arski numbers for abstract logics. Annals of Pure and Applied Logic , 176(8):103583, 2025

2025
[15]

u cke, and Sandra M \

Peter Holy, Philipp L \"u cke, and Sandra M \"u ller. Outward compactness. arXiv preprint arXiv:2402.15788 , 2024

work page arXiv 2024
[16]

Subcompact cardinals, type omission, and ladder systems

Yair Hayut and Menachem Magidor. Subcompact cardinals, type omission, and ladder systems. The Journal of Symbolic Logic , 87(3):1111--1129, 2022

2022
[17]

Set theory: The third millennium edition, revised and expanded

Thomas Jech. Set theory: The third millennium edition, revised and expanded . Springer Monographs in Mathematics. Springer, 2003

2003
[18]

The Higher Infinite: Large Cardinals in Set Theory From Their Beginnings

Akihiro Kanamori. The Higher Infinite: Large Cardinals in Set Theory From Their Beginnings . Springer Monographs in Mathematics. Springer, 2nd edition, 2003

2003
[19]

Weak compactness cardinals for strong logics and subtlety properties of the class of ordinals

Philipp L \"u cke. Weak compactness cardinals for strong logics and subtlety properties of the class of ordinals. Journal of the London Mathematical Society , 112(1):e70215, 2025

2025
[20]

On the role of supercompact and extendible cardinals in logic

Menachem Magidor. On the role of supercompact and extendible cardinals in logic. Israel Journal of Mathematics , 10:147--157, 1971

1971
[21]

Makowsky

Johann A. Makowsky. V op e nka's principle and compact logics. The Journal of Symbolic Logic , 50(1):42--48, 1985

1985
[22]

a \"a n \

Menachem Magidor and Jouko V \"a \"a n \"a nen. On L \"o wenheim-- S kolem-- T arski numbers for extensions of first order logic. Journal of Mathematical Logic , 11(1):87--113, 2011

2011
[23]

The M itchell order below rank-to-rank

Itay Neeman. The M itchell order below rank-to-rank. The Journal of Symbolic Logic , 69(4):1143--1162, 2004

2004
[24]

Compactness characterisations of large cardinals with strong H enkin models

Jonathan Osinski and Alejandro Poveda. Compactness characterisations of large cardinals with strong H enkin models. Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, to appear , 2026

2026
[25]

Interactions between Large Cardinals and Strong Logics

Jonathan Osinski. Interactions between Large Cardinals and Strong Logics . PhD thesis, Universität Hamburg, 2024

2024
[26]

Some problems and results relevant to the foundations of set theory

Alfred Tarski. Some problems and results relevant to the foundations of set theory. In Ernest Nagel, Patrick Suppes, and Alfred Tarski, editors, Logic, Methodology and the Philosophy of Science. Proceedings of the 1960 International Congress , pages 125--135. Stanford University Press, 1962

1960
[27]

On C ^ (n) -extendible cardinals

Konstantinos Tsaprounis. On C ^ (n) -extendible cardinals. The Journal of Symbolic Logic , 83(3):1112--1131, 2018

2018
[28]

a \"a n \

Jouko V \"a \"a n \"a nen. Abstract logic and set theory. I . Definability . In M. Boffa, D. van Dalen, and K. McAloon, editors, Logic Colloquium '78. Proceedings of the colloquium held in Mons , volume 97 of Studies in Logic and the Foundations of Mathematics , pages 391--421. North-Holland, 1979

1979
[29]

a \"a n \

Jouko V \"a \"a n \"a nen. Sort logic and foundations of mathematics. In Chitat Chong, Qi Feng, Theodore A. Slaman, and W. Hugh Woodin, editors, Infinity and truth , volume 25 of Lecture Notes Series, Institute for Mathematical Sciences, National University of Singapore , pages 171--186. World Scientific, 2014

2014
[30]

Logical & topological characterizations of supercompact & huge cardinals

Trevor Wilson. Logical & topological characterizations of supercompact & huge cardinals. Talk at the European Set Theory Conference 2022, Turin, September 2, 2022

2022
[31]

Trevor M. Wilson. The large cardinal strength of weak V opěnka’s principle. Journal of Mathematical Logic , 22(1):2150024, 2022

2022
[32]

Inner Models and Large Cardinals , volume 5 of de Gruyter Series in Logic and its Applications

Martin Zeman. Inner Models and Large Cardinals , volume 5 of de Gruyter Series in Logic and its Applications . De Gruyter, Berlin, Boston, 2002

2002