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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 ...
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 ...
In mathematical logic, the diagonal lemma (also known as diagonalization lemma, self-reference lemma [1] or fixed point theorem) establishes the existence of self-referential sentences in certain formal theories of the natural numbers—specifically those theories that are strong enough to represent all computable functions.
Cantor's diagonal argument shows that the power set of a set (whether infinite or not) always has strictly higher cardinality than the set itself (or informally, the power set must be larger than the original set). In particular, Cantor's theorem shows that the power set of a countably infinite set is uncountably infinite.
Cantor's diagonal argument. set is smaller than its power set; uncountability of the real numbers; Cantor's first uncountability proof. uncountability of the real numbers; Combinatorics; Combinatory logic; Co-NP; Coset; Countable. countability of a subset of a countable set (to do) Angle of parallelism; Galois group. Fundamental theorem of ...
This is known as Cantor's theorem. The argument below is a modern version of Cantor's argument that uses power sets (for his original argument, see Cantor's diagonal argument). By presenting a modern argument, it is possible to see which assumptions of axiomatic set theory are used.
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