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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 ...
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 .
In mathematical logic and philosophy, Skolem's paradox is the apparent contradiction that a countable model of first-order set theory could contain an uncountable set. The paradox arises from part of the Löwenheim–Skolem theorem ; Thoralf Skolem was the first to discuss the seemingly contradictory aspects of the theorem, and to discover the ...
A set of polygons in an Euler diagram This set equals the one depicted above since both have the very same elements.. In mathematics, a set is a collection of different [1] things; [2] [3] [4] these things are called elements or members of the set and are typically mathematical objects of any kind: numbers, symbols, points in space, lines, other geometrical shapes, variables, or even other ...
From Wikipedia, the free encyclopedia. Redirect page. Redirect to: Uncountable set ...
More generally, let S be an infinite set of positive integers, such as the set of even positive numbers or the set of primes, if A is a subset of S, and if the proportion of elements of S below n that are in A (out of all elements of S below n) tends to 1 as n tends to infinity, then it can be said that almost all elements of S are in A. Examples:
The Cantor set is an example of an uncountable null set. It is uncountable because it contains all real numbers between 0 and 1 whose ternary form decimal expansion can be written using only 0’s and 2’s, and it is null because it is constructed by beginning with the closed interval of real numbers from 0 to 1 and multiplying the length by 2 ...
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 ]