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Let R be the set of all sets that are not members of themselves. (This set is sometimes called "the Russell set".) If R is not a member of itself, then its definition entails that it is a member of itself; yet, if it is a member of itself, then it is not a member of itself, since it is the set of all sets that are not members of themselves. The ...
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 ...
After all this, the version of the "set of all sets" paradox conceived by Bertrand Russell in 1903 led to a serious crisis in set theory. Russell recognized that the statement x = x is true for every set, and thus the set of all sets is defined by {x | x = x}. In 1906 he constructed several paradox sets, the most famous of which is the set of ...
To investigate the left distributivity of set subtraction over unions or intersections, consider how the sets involved in (both of) De Morgan's laws are all related: () = = () always holds (the equalities on the left and right are De Morgan's laws) but equality is not guaranteed in general (that is, the containment might be strict).
Russell's paradox concerns the impossibility of a set of sets, whose members are all sets that do not contain themselves. If such a set could exist, it could neither contain itself (because its members all do not contain themselves) nor avoid containing itself (because if it did, it should be included as one of its members). [2]
Cantor's paradox is the name given to a contradiction following from Cantor's theorem together with the assumption that there is a set containing all sets, the universal set. In order to distinguish this paradox from the next one discussed below, it is important to note what this contradiction is.
At stage 0, there are no sets yet. At each following stage, a set is added to the universe if all of its elements have been added at previous stages. Thus the empty set is added at stage 1, and the set containing the empty set is added at stage 2. [11] The collection of all sets that are obtained in this way, over all the stages, is known as V.
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 ...