Knowra Von Neumann–Bernays–Gödel set theory Von Neumann–Bernays–Gödel set theory Von Neumann–Bernays–Gödel set theory is an axiomatic foundation distinguishing sets from proper classes. It formalizes class reasoning while allowing sets to be treated as classes.
Zermelo–Fraenkel set theory : An axiomatic set theory whose objects are sets and whose axioms govern their existence and relations. NBG shares much of ZF’s set-theoretic foundation, while explicitly admitting proper classes.
Class comprehension : A principle specifying classes by formulas, subject to restrictions that prevent paradoxes. NBG permits class formation using formulas with set quantifiers, avoiding unrestricted comprehension.
Morse–Kelley set theory : An axiomatic set theory with sets and proper classes that allows broader class comprehension than NBG. MK’s stronger comprehension distinguishes it from NBG, whose class existence is more restricted.
Category theory : The mathematical study of objects and structure-preserving maps between them. Proper classes allow large categories, such as the category of all sets, to be discussed directly.
First-order logic : A formal logic with quantified variables, predicates, and rules for valid inference. NBG’s axioms are expressed in a first-order language with class and set variables.
Replacement axiom : An axiom asserting that the image of a set under a definable function is a set. Its set-level role in NBG matches the corresponding strength of ZF.
Zermelo–Fraenkel set theory with Choice : Zermelo–Fraenkel set theory augmented by the axiom of choice. NBG is a class theory, but its set theorems align closely with ZFC.
Ordinals : Order types of well-ordered sets, extending the natural numbers into transfinite sequence. The class of all ordinals is a proper class in NBG, not a set.
Set : A collection treated as a single mathematical object, whose members are also sets in standard set theories. In NBG, sets are precisely the classes that belong to some class.
Global choice : A principle selecting one element from every nonempty set in a class-sized family of sets. NBG can state a global choice function on all nonempty sets as a class.
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