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Besides the cardinality, which describes the size of a set, ordered sets also form a subject of set theory. The axiom of choice guarantees that every set can be well-ordered, which means that a total order can be imposed on its elements such that every nonempty subset has a first element with respect to that order.
In mathematical logic, Russell's paradox (also known as Russell's antinomy) is a set-theoretic paradox published by the British philosopher and mathematician, Bertrand Russell, in 1901. [ 1 ] [ 2 ] Russell's paradox shows that every set theory that contains an unrestricted comprehension principle leads to contradictions. [ 3 ]
Cantor's paradox: The set of all sets would have its own power set as a subset, therefore its cardinality would be at least as great as that of its power set. But Cantor's theorem proves that power sets are strictly greater than the sets they are constructed from. Consequently, the set of all sets would contain a subset greater than itself.
In set theory, Cantor's paradox states that there is no set of all cardinalities.This is derived from the theorem that there is no greatest cardinal number.In informal terms, the paradox is that the collection of all possible "infinite sizes" is not only infinite, but so infinitely large that its own infinite size cannot be any of the infinite sizes in the collection.
The intuitive approach tacitly assumes that a set may be formed from the class of all objects satisfying any particular defining condition. This assumption gives rise to paradoxes, the simplest and best known of which are Russell's paradox and the Burali-Forti paradox. Axiomatic set theory was originally devised to rid set theory of such paradoxes.
In naive set theory, a set is described as a well-defined collection of objects. These objects are called the elements or members of the set. Objects can be anything: numbers, people, other sets, etc. For instance, 4 is a member of the set of all even integers. Clearly, the set of even numbers is infinitely large; there is no requirement that a ...
A set of polygons in an Euler diagram This set equals the one depicted above since both have the very same elements.. In mathematics, a set is a collection of different [1] things; [2] [3] [4] these things are called elements or members of the set and are typically mathematical objects of any kind: numbers, symbols, points in space, lines, other geometrical shapes, variables, or even other ...
In set theory, a paradoxical set is a set that has a paradoxical decomposition. A paradoxical decomposition of a set is two families of disjoint subsets, along with appropriate group actions that act on some universe (of which the set in question is a subset), such that each partition can be mapped back onto the entire set using only finitely ...