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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 ...
Cantor–Bernstein–Schroeder theorem (set theory, cardinal numbers) Cantor's intersection theorem (real analysis) Cantor's isomorphism theorem (order theory) Cantor's theorem (set theory, Cantor's diagonal argument) Carathéodory–Jacobi–Lie theorem (symplectic topology) Carathéodory's existence theorem (ordinary differential equations)
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.
B. Russell: The principles of mathematics I, Cambridge 1903. B. Russell: On some difficulties in the theory of transfinite numbers and order types, Proc. London Math. Soc. (2) 4 (1907) 29-53. P. J. Cohen: Set Theory and the Continuum Hypothesis, Benjamin, New York 1966. S. Wagon: The Banach–Tarski Paradox, Cambridge University Press ...
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 ...
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
This theorem is proved using Cantor's first uncountability proof, which differs from the more familiar proof using his diagonal argument. Cantor introduced fundamental constructions in set theory, such as the power set of a set A, which is the set of all possible subsets of A.