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.

Aliases (4)

Morse–Kelley set theory ×3
MK set theory ×1
Morse–Kelley class theory ×1
Morse–Kelley set theory (in proof-theoretic strength) ×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	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 →
isStrongerThan	Zermelo–Fraenkel set theory with Choice → von Neumann–Bernays–Gödel set theory →
isUsedIn	formalization of large mathematical structures → foundations of category theory → metamathematics →

Referenced by (7)

Subject (surface form when different)	Predicate
Morse–Kelley set theory by class–set distinction ("Morse–Kelley set theory") → Morse–Kelley set theory by class–set distinction ("MK set theory") → Morse–Kelley set theory by class–set distinction ("Morse–Kelley class theory") →	hasAlternativeName
Grothendieck universe ("Morse–Kelley set theory") →	alternativeTo
von Neumann–Bernays–Gödel set theory ("Morse–Kelley set theory") →	isRelatedTo
von Neumann–Bernays–Gödel set theory ("Morse–Kelley set theory (in proof-theoretic strength)") →	isWeakerThan
Burali-Forti paradox →	resolvedIn