Search results
Results from the WOW.Com Content Network
Georg Cantor published this proof in 1891, [1] [2]: 20– [3] but it was not his first proof of the uncountability of the real numbers, which appeared in 1874. [ 4 ] [ 5 ] However, it demonstrates a general technique that has since been used in a wide range of proofs, [ 6 ] including the first of Gödel's incompleteness theorems [ 2 ] and ...
As a consequence, the cardinality of the real numbers, which is the same as that of the power set of the integers, is strictly larger than the cardinality of the integers; see Cardinality of the continuum for details. The theorem is named for Georg Cantor, who first stated and proved it at the end of the 19th century.
Symbolically, if the cardinality of is denoted as , the cardinality of the continuum is c = 2 ℵ 0 > ℵ 0 . {\displaystyle {\mathfrak {c}}=2^{\aleph _{0}}>\aleph _{0}.} This was proven by Georg Cantor in his uncountability proof of 1874, part of his groundbreaking study of different infinities.
The proof of Cantor's second theorem came from Dedekind. However, it omits Dedekind's explanation of why the limits a ∞ and b ∞ exist. [45] Cantor restricted his first theorem to the set of real algebraic numbers. The proof he was using demonstrates the countability of the set of all algebraic numbers. [20]
The completion of the theory of complex numbers in the 19th century entailed the differentiation of irrationals into algebraic and transcendental numbers, the proof of the existence of transcendental numbers, and the resurgence of the scientific study of the theory of irrationals, largely ignored since Euclid.
The best known example of an uncountable set is the set of all real numbers; Cantor's diagonal argument shows that this set is uncountable. The diagonalization proof technique can also be used to show that several other sets are uncountable, such as the set of all infinite sequences of natural numbers (see: (sequence A102288 in the OEIS)), and the set of all subsets of the set ...
[2] His new proof uses his diagonal argument to prove that there exists an infinite set with a larger number of elements (or greater cardinality) than the set of natural numbers N = {1, 2, 3, ...}. This larger set consists of the elements ( x 1 , x 2 , x 3 , ...), where each x n is either m or w . [ 3 ]
His proof used an argument with nested intervals, but in an 1891 paper, he proved the same result using his ingenious and much simpler diagonal argument. The new cardinal number of the set of real numbers is called the cardinality of the continuum and Cantor used the symbol c {\displaystyle {\mathfrak {c}}} for it.