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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 ...
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.
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 ...
Also, the union of a sequence of positive or negative sets is also positive or negative; more formally, if ,, … is a sequence of positive sets, then = is also a positive set; the same is true if the word "positive" is replaced by "negative". A set which is both positive and negative is a -null set, for if is a measurable subset of a positive ...
In set theory, is also used to indicate 'not in the set of': is the set of all members of U that are not members of A. Regardless how it is notated or symbolized , the negation ¬ P {\displaystyle \neg P} can be read as "it is not the case that P ", "not that P ", or usually more simply as "not P ".
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.
The axiom of constructibility implies the axiom of choice (AC), given Zermelo–Fraenkel set theory without the axiom of choice (ZF). It also settles many natural mathematical questions that are independent of Zermelo–Fraenkel set theory with the axiom of choice (ZFC); for example, the axiom of constructibility implies the generalized continuum hypothesis, the negation of Suslin's hypothesis ...
In mathematics, Suslin's problem is a question about totally ordered sets posed by Mikhail Yakovlevich Suslin () and published posthumously.It has been shown to be independent of the standard axiomatic system of set theory known as ZFC; Solovay & Tennenbaum (1971) showed that the statement can neither be proven nor disproven from those axioms, assuming ZF is consistent.