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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 ...
Download QR code; Print/export Download as PDF; Printable version; In other projects ... Countable set, Uncountable set; Power set; Relative to a topology
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 ]
A set A ⊆ R has strong measure zero if and only if A + M ≠ R for every meagre set M ⊆ R. [5] This result establishes a connection to the notion of strongly meagre set, defined as follows: A set M ⊆ R is strongly meagre if and only if A + M ≠ R for every set A ⊆ R of Lebesgue measure zero.
One of the earliest results in set theory, published by Cantor in 1874, was the existence of different sizes, or cardinalities, of infinite sets. [2] An infinite set is called countable if there is a function that gives a one-to-one correspondence between and the natural numbers, and is uncountable if there is no such correspondence function.
The Cantor–Bendixson theorem states that closed sets of a Polish space X have the perfect set property in a particularly strong form: any closed subset of X can be written uniquely as the disjoint union of a perfect set and a countable set. In particular, every uncountable Polish space has the perfect set property, and can be written as the ...
It is possible for a Sierpiński set to be a subgroup under addition. For this one modifies the construction above by choosing a real number x β that is not in any of the countable number of sets of the form ( S α + X )/ n for α < β , where n is a positive integer and X is an integral linear combination of the numbers x α for α < β .
Silver later gave a fine-structure-free proof using his machines [1] and finally Magidor gave an even simpler proof. The converse of Jensen's covering theorem is also true: if 0 # exists then the countable set of all cardinals less than ℵ ω {\displaystyle \aleph _{\omega }} cannot be covered by a constructible set of cardinality less than ...