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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.
Cantor defined cardinality in terms of bijective functions: two sets have the same cardinality if, and only if, there exists a bijective function between them. Between any two real numbers a < b , no matter how close they are to each other, there are always infinitely many other real numbers, and Cantor showed that they are as many as those ...
Besides the cardinality, which describes the size of a set, ordered sets also form a subject of set theory. The axiom of choice guarantees that every set can be well-ordered, which means that a total order can be imposed on its elements such that every nonempty subset has a first element with respect to that order.
Cantor gave two proofs that the cardinality of the set of integers is strictly smaller than that of the set of real numbers (see Cantor's first uncountability proof and Cantor's diagonal argument). His proofs, however, give no indication of the extent to which the cardinality of the integers is less than that of the real numbers.
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These curves are not a direct proof that a line has the same number of points as a finite-dimensional space, but they can be used to obtain such a proof. Cantor also showed that sets with cardinality strictly greater than c {\displaystyle {\mathfrak {c}}} exist (see his generalized diagonal argument and theorem ).