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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]
Thus, the Cantor set is a homogeneous space in the sense that for any two points and in the Cantor set , there exists a homeomorphism : with () =. An explicit construction of h {\displaystyle h} can be described more easily if we see the Cantor set as a product space of countably many copies of the discrete space { 0 , 1 } {\displaystyle \{0,1\}} .
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 ...
Any finite natural number can be used in at least two ways: as an ordinal and as a cardinal. Cardinal numbers specify the size of sets (e.g., a bag of five marbles), whereas ordinal numbers specify the order of a member within an ordered set [9] (e.g., "the third man from the left" or "the twenty-seventh day of January").
Such sets are now called uncountable sets, and the size of infinite sets is treated by the theory of cardinal numbers, which Cantor began. 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.
Georg Ferdinand Ludwig Philipp Cantor (/ ˈ k æ n t ɔːr / KAN-tor; German: [ˈɡeːɔʁk ˈfɛʁdinant ˈluːtvɪç ˈfiːlɪp ˈkantoːɐ̯]; 3 March [O.S. 19 February] 1845 – 6 January 1918 [1]) was a mathematician who played a pivotal role in the creation of set theory, which has become a fundamental theory in mathematics.
All subsets of a set S (all possible choices of its elements) form the power set P(S). Georg Cantor proved that the power set is always larger than the set, i.e., |P(S)| > |S|. A special case of Cantor's theorem is that the set of all real numbers R cannot be enumerated by natural numbers, that is, R is uncountable: |R| > |N|.
The set is itself an ordinal number larger than all countable ones, so it is an uncountable set. Therefore, ℵ 1 {\displaystyle \aleph _{1}} is distinct from ℵ 0 {\displaystyle \aleph _{0}} .