Morse–Kelley set theory by class–set distinction
E91147
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.
All labels observed (5)
| Label | Occurrences |
|---|---|
| Morse–Kelley set theory | 4 |
| 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]
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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.
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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.
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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.
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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.
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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 |
MK set theory
ⓘ
Morse–Kelley class theory ⓘ 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 class theory ⓘ
linked to:
von Neumann–Bernays–Gödel set theory
|
| isStrongerThan |
Zermelo–Fraenkel set theory with Choice
ⓘ
linked to:
Zermelo–Fraenkel set theory
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 (8)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Morse–Kelley set theory by class–set distinction
von Neumann–Bernays–Gödel set theory
→
isWeakerThan
→
Morse–Kelley set theory (in proof-theoretic strength)
ⓘ
linked to: Morse–Kelley set theory by class–set distinction
linked to: Morse–Kelley set theory by class–set distinction
linked to: Morse–Kelley set theory by class–set distinction
linked to: Morse–Kelley set theory by class–set distinction
linked to: Morse–Kelley set theory by class–set distinction
subject linked to:
Axiom of Extensionality in set theory
linked to: Morse–Kelley set theory by class–set distinction