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The example mapping f happens to correspond to the example enumeration s in the picture above. A generalized form of the diagonal argument was used by Cantor to prove Cantor's theorem: for every set S, the power set of S—that is, the set of all subsets of S (here written as P(S))—cannot be in bijection with S itself. This proof proceeds as ...
Cantor's theorem and its proof are closely related to two paradoxes of set theory. 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 ...
Define two sets to have the same size by: there exists a bijection between the two sets (a one-to-one correspondence between the elements). Then a cardinal number is, by definition, a class consisting of all sets of the same size. To have the same size is an equivalence relation, and the cardinal numbers are the equivalence classes.
Diagonal argument can refer to: Diagonal argument (proof technique), proof techniques used in mathematics. A diagonal argument, in mathematics, is a technique employed in the proofs of the following theorems: Cantor's diagonal argument (the earliest) Cantor's theorem; Russell's paradox; Diagonal lemma. Gödel's first incompleteness theorem
In mathematics, Lawvere's fixed-point theorem is an important result in category theory. [1] It is a broad abstract generalization of many diagonal arguments in mathematics and logic, such as Cantor's diagonal argument, Cantor's theorem, Russell's paradox, Gödel's first incompleteness theorem, Turing's solution to the Entscheidungsproblem, and Tarski's undefinability theorem.
The graphical shape of Cantor's pairing function, a diagonal progression, is a standard trick in working with infinite sequences and countability. [b] The algebraic rules of this diagonal-shaped function can verify its validity for a range of polynomials, of which a quadratic will turn out to be the simplest, using the method of induction ...
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 Zermelo–Fraenkel set theory with the axiom of choice (ZFC), this is equivalent to the following equation in aleph numbers: =, or even shorter with beth numbers: =. The continuum hypothesis was advanced by Georg Cantor in 1878, [ 1 ] and establishing its truth or falsehood is the first of Hilbert's 23 problems presented in 1900.