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Georg Cantor, c. 1870. Cantor's first set theory article contains Georg Cantor's first theorems of transfinite set theory, which studies infinite sets and their properties. One of these theorems is his "revolutionary discovery" that the set of all real numbers is uncountably, rather than countably, infinite. [1]
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 ...
An illustration of Cantor's diagonal argument (in base 2) for the existence of uncountable sets. The sequence at the bottom cannot occur anywhere in the enumeration of sequences above. An infinite set may have the same cardinality as a proper subset of itself, as the depicted bijection f(x)=2x from the natural to the even numbers demonstrates ...
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As the above summation argument shows, the Cantor set is uncountable but has Lebesgue measure 0. Since the Cantor set is the complement of a union of open sets, it itself is a closed subset of the reals, and therefore a complete metric space. Since it is also totally bounded, the Heine–Borel theorem says that it must be compact.
changed index "N" to "L" to avoid confusion with index "n" (used in case 2 and 3) with screen readers, cf. en:User_talk:Jochen_Burghardt#Need small change to Georg Cantor's first set theory article: 09:52, 9 November 2015: 332 × 66 (18 KB) Jochen Burghardt: duplicated recent changes of corresponding .pdf file: 09:44, 24 September 2015: 332 × ...
the set of natural numbers, irrespective of including or excluding zero, the set of all integers, any infinite subset of the integers, such as the set of all square numbers or the set of all prime numbers, the set of all rational numbers, the set of all constructible numbers (in the geometric sense), the set of all algebraic numbers,
Download as PDF; Printable version; In other projects ... This is a topic category for the topic Georg Cantor ... Cantor set; Cantor space;