Morse–Kelley set theory by class–set distinction

E91147

axiomatic set theory foundational system for mathematics second-order set theory

Morse–Kelley set theory by class–set distinction is a foundational system that avoids certain set-theoretic paradoxes by rigorously distinguishing between sets and proper classes within a powerful axiomatic framework.

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All labels observed (5)

Label	Occurrences
Morse–Kelley set theory	3
MK set theory	1
Morse–Kelley class theory	1
Morse–Kelley set theory (in proof-theoretic strength)	1
Morse–Kelley set theory by class–set distinction canonical	1

How this entity was disambiguated

This entity first appeared as the object of triple T713827 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.

NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07

Target entity: Morse–Kelley set theory by class–set distinction
Context triple: [Burali-Forti paradox, resolvedIn, Morse–Kelley set theory by class–set distinction]

A. von Neumann–Bernays–Gödel set theory
Von Neumann–Bernays–Gödel set theory is an axiomatic set theory extending Zermelo–Fraenkel set theory by formally distinguishing between sets and classes, widely used in foundational studies of mathematics.
B. Zermelo set theory
Zermelo set theory is an early axiomatic system for set theory, introduced by Ernst Zermelo to rigorously formalize the concept of sets and avoid known paradoxes.
C. Zermelo–Fraenkel set theory
Zermelo–Fraenkel set theory is the standard axiomatic framework for modern set theory, designed to avoid paradoxes and provide a rigorous foundation for much of mathematics.
D. von Neumann paradox in set theory
The von Neumann paradox in set theory is a foundational result showing that, under certain group-theoretic conditions, a set can be decomposed and reassembled into paradoxical subsets of equal “size,” illustrating the counterintuitive consequences of the axiom of choice.
E. Remarks on the Foundations of Mathematics
Remarks on the Foundations of Mathematics is a posthumously published collection of Ludwig Wittgenstein’s later writings that critically examines the nature of mathematical truth, proof, and practice from a philosophical and language-centered perspective.
F. None of above. chosen
G. Unsure - the case is ambiguous/there is not enough information to decide.

NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07

Target entity: Morse–Kelley set theory by class–set distinction
Target entity description: Morse–Kelley set theory by class–set distinction is a foundational system that avoids certain set-theoretic paradoxes by rigorously distinguishing between sets and proper classes within a powerful axiomatic framework.

A. von Neumann–Bernays–Gödel set theory
Von Neumann–Bernays–Gödel set theory is an axiomatic set theory extending Zermelo–Fraenkel set theory by formally distinguishing between sets and classes, widely used in foundational studies of mathematics.
B. Zermelo set theory
Zermelo set theory is an early axiomatic system for set theory, introduced by Ernst Zermelo to rigorously formalize the concept of sets and avoid known paradoxes.
C. Zermelo–Fraenkel set theory
Zermelo–Fraenkel set theory is the standard axiomatic framework for modern set theory, designed to avoid paradoxes and provide a rigorous foundation for much of mathematics.
D. von Neumann paradox in set theory
The von Neumann paradox in set theory is a foundational result showing that, under certain group-theoretic conditions, a set can be decomposed and reassembled into paradoxical subsets of equal “size,” illustrating the counterintuitive consequences of the axiom of choice.
E. Remarks on the Foundations of Mathematics
Remarks on the Foundations of Mathematics is a posthumously published collection of Ludwig Wittgenstein’s later writings that critically examines the nature of mathematical truth, proof, and practice from a philosophical and language-centered perspective.
F. None of above. chosen

Statements (47)

Predicate	Object
instanceOf	axiomatic set theory ⓘ foundational system for mathematics ⓘ second-order set theory ⓘ
avoidsParadox	Burali-Forti paradox ⓘ Russell’s paradox ⓘ
distinguishesBetween	proper classes ⓘ sets ⓘ
formalizesIn	first-order language with two sorts of variables ⓘ
hasAlternativeName	Morse–Kelley set theory by class–set distinction ⓘ surface form: MK set theory Morse–Kelley set theory by class–set distinction ⓘ surface form: Morse–Kelley class theory Morse–Kelley set theory by class–set distinction ⓘ surface form: Morse–Kelley set theory
hasAxiom	axioms for sets similar to ZFC ⓘ class comprehension schema ⓘ extensionality for classes ⓘ global choice (in some formulations) ⓘ
hasComponent	class variables ⓘ set variables ⓘ
hasConsequence	ability to define many large classes not available as sets ⓘ existence of a universal class of all sets ⓘ
hasDomain	universe of sets and proper classes ⓘ
hasKeyFeature	ability to quantify over classes ⓘ powerful comprehension schema for classes ⓘ proper classes are not members of any class ⓘ rigorous distinction between sets and proper classes ⓘ treatment of sets as classes that are members of some class ⓘ use of classes as primitive objects ⓘ
hasMethod	allowing unrestricted comprehension for classes with set parameters (in standard formulations) ⓘ restricting membership to sets only ⓘ
hasMotivation	formal treatment of collections too large to be sets ⓘ provide a framework for talking about the totality of all sets ⓘ
hasProperty	conservative over ZFC for first-order statements about sets (under usual assumptions) ⓘ every set is a class ⓘ not every class is a set ⓘ proper classes cannot be elements of any class ⓘ
hasPurpose	avoidance of set-theoretic paradoxes ⓘ providing a strong foundation for mathematics ⓘ
hasTypicalExampleOfProperClass	class of all cardinals ⓘ class of all ordinals ⓘ class of all sets ⓘ
isRelatedTo	Zermelo–Fraenkel set theory ⓘ class–set distinction in axiomatic set theories ⓘ von Neumann–Bernays–Gödel set theory ⓘ surface form: von Neumann–Bernays–Gödel class theory
isStrongerThan	Zermelo–Fraenkel set theory ⓘ surface form: Zermelo–Fraenkel set theory with Choice von Neumann–Bernays–Gödel set theory ⓘ
isUsedIn	formalization of large mathematical structures ⓘ foundations of category theory ⓘ metamathematics ⓘ

How these facts were elicited

Referenced by (7)

Full triples — surface form annotated when it differs from this entity's canonical label.

Burali-Forti paradox → resolvedIn → Morse–Kelley set theory by class–set distinction ⓘ

von Neumann–Bernays–Gödel set theory → isRelatedTo → Morse–Kelley set theory by class–set distinction ⓘ

this entity surface form: Morse–Kelley set theory

von Neumann–Bernays–Gödel set theory → isWeakerThan → Morse–Kelley set theory by class–set distinction ⓘ

this entity surface form: Morse–Kelley set theory (in proof-theoretic strength)

Morse–Kelley set theory by class–set distinction → hasAlternativeName → Morse–Kelley set theory by class–set distinction ⓘ

this entity surface form: Morse–Kelley set theory

Morse–Kelley set theory by class–set distinction → hasAlternativeName → Morse–Kelley set theory by class–set distinction ⓘ

this entity surface form: MK set theory

Morse–Kelley set theory by class–set distinction → hasAlternativeName → Morse–Kelley set theory by class–set distinction ⓘ

this entity surface form: Morse–Kelley class theory

Grothendieck universe → alternativeTo → Morse–Kelley set theory by class–set distinction ⓘ

this entity surface form: Morse–Kelley set theory

All labels observed (5)

How this entity was disambiguated Show

Statements (47)

How these facts were elicited Show

Referenced by (7)

How this entity was disambiguated

How these facts were elicited