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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 ...
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 ...
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.
The presentation of the non-constructive proof without mentioning Cantor's constructive proof appears in some books that were quite successful as measured by the length of time new editions or reprints appeared—for example: Oskar Perron's Irrationalzahlen (1921; 1960, 4th edition), Eric Temple Bell's Men of Mathematics (1937; still being ...
This defines an injection from the quadratic irrationals to quadruples of integers, so their cardinality is at most countable; since on the other hand every square root of a prime number is a distinct quadratic irrational, and there are countably many prime numbers, they are at least countable; hence the quadratic irrationals are a countable set.
Being a G-delta set—i.e., a countable intersection of open subsets—in a complete metric space, the space of irrationals is completely metrizable: that is, there is a metric on the irrationals inducing the same topology as the restriction of the Euclidean metric, but with respect to which the irrationals are complete.
Cantor used a cardinality argument to show that there are only countably many algebraic numbers, and hence almost all numbers are transcendental. Transcendental numbers therefore represent the typical case; even so, it may be extremely difficult to prove that a given number is transcendental (or even simply irrational).
1896 Schröder announces a proof (as a corollary of a theorem by Jevons). [11] 1897 Bernstein, a 19-year-old student in Cantor's Seminar, presents his proof. [12] [13] 1897 Almost simultaneously, but independently, Schröder finds a proof. [12] [13] 1897 After a visit by Bernstein, Dedekind independently proves the theorem a second time.