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Alternative set theory; Axiomatic set theory; Kripke–Platek set theory with urelements; Morse–Kelley set theory; Naive set theory; New Foundations; Positive set theory; Zermelo–Fraenkel set theory; Zermelo set theory; Set (mathematics) Simple theorems in the algebra of sets; Subset; Θ (set theory) Tree (descriptive set theory) Tree (set ...
The precise definition of "class" depends on foundational context. In work on Zermelo–Fraenkel set theory, the notion of class is informal, whereas other set theories, such as von Neumann–Bernays–Gödel set theory, axiomatize the notion of "proper class", e.g., as entities that are not members of another entity.
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects.Although objects of any kind can be collected into a set, set theory – as a branch of mathematics – is mostly concerned with those that are relevant to mathematics as a whole.
Download QR code; Print/export ... This page is a list of articles related to set theory. ... Class (set theory) Complement (set theory) ...
This article lists mathematical properties and laws of sets, involving the set-theoretic operations of union, intersection, and complementation and the relations of set equality and set inclusion. It also provides systematic procedures for evaluating expressions, and performing calculations, involving these operations and relations.
Kurt Gödel proved in 1940 that the negation of the continuum hypothesis, i.e., the existence of a set with intermediate cardinality, could not be proved in standard set theory. [2] The second half of the independence of the continuum hypothesis – i.e., unprovability of the nonexistence of an intermediate-sized set – was proved in 1963 by ...
In the foundations of mathematics, von Neumann–Bernays–Gödel set theory (NBG) is an axiomatic set theory that is a conservative extension of Zermelo–Fraenkel–choice set theory (ZFC). NBG introduces the notion of class , which is a collection of sets defined by a formula whose quantifiers range only over sets.
Every set is a projective object in Set (assuming the axiom of choice). The finitely presentable objects in Set are the finite sets. Since every set is a direct limit of its finite subsets, the category Set is a locally finitely presentable category. If C is an arbitrary category, the contravariant functors from C to Set are often an important ...