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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 ...
From Wikipedia, the free encyclopedia. Redirect page. Redirect to: Uncountable set ...
When considered as a set, the elements of are the countable ordinals (including finite ordinals), [1] of which there are uncountably many. Like any ordinal number (in von Neumann's approach ), ω 1 {\displaystyle \omega _{1}} is a well-ordered set , with set membership serving as the order relation.
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:
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 ]
About Wikipedia; Contact us; Contribute Help; ... 1 uncountable set. 2 comments. 2 countable set. 2 comments. 3 The quadratic formula to solve x.