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In classical set theory, the membership of elements in a set is assessed in binary terms according to a bivalent condition—an element either belongs or does not belong to the set. By contrast, fuzzy set theory permits the gradual assessment of the membership of elements in a set; this is described with the aid of a membership function valued in the real unit interval [0, 1].
Generic filter – in set theory, given a collection of dense open subsets of a poset, a filter that meets all sets in that collectionPages displaying wikidata descriptions as a fallback. Ideal (set theory) – Non-empty family of sets that is closed under finite unions and subsets.
Naïve set theory defines a set as any well-defined collection of distinct elements, but problems arise from the vagueness of the term well-defined. Axiomatic set theory. In subsequent efforts to resolve these paradoxes since the time of the original formulation of naïve set theory, the properties of sets have been defined by axioms.
Zermelo set theory (sometimes denoted by Z- ), as set out in a seminal paper in 1908 by Ernst Zermelo, is the ancestor of modern Zermelo–Fraenkel set theory (ZF) and its extensions, such as von Neumann–Bernays–Gödel set theory (NBG). It bears certain differences from its descendants, which are not always understood, and are frequently ...
Set theory is a branch of mathematical logic that studies sets, which informally are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics. The language of set theory can be used to define nearly all mathematical objects .
Positive set theory. In mathematical logic, positive set theory is the name for a class of alternative set theories in which the axiom of comprehension holds for at least the positive formulas (the smallest class of formulas containing atomic membership and equality formulas and closed under conjunction, disjunction, existential and universal ...
Naive set theory is the original set theory developed by mathematicians at the end of the 19th century, treating sets simply as collections of things. Axiomatic set theory is a rigorous axiomatic theory developed in response to the discovery of serious flaws (such as Russell's paradox) in naive set theory. It treats sets as "whatever satisfies ...
