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

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

The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.

Instruction

You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10.

# Requirements
- If you don't know the subject at all, return an empty list.
- If the subject is not a named entity, return an empty list.
- Include at least one triple where predicate is "instanceOf".
- Do not get too wordy.
- Separate several objects into multiple triples with one object.

Input

Subject: Morse–Kelley set theory by class–set distinction
Description of subject: 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.

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