Table of contentsIntroductionTopic analysisReferencesIntroductionThis article will be a summary of my findings in answering the questions “how big can a set be with zero “length”? » Throughout this article, I will explain the facts about the Cantor set. The Cantor set is the best example to answer this question because it is considered to have zero length. Say no to plagiarism. Get a tailor-made essay on “Why Violent Video Games Should Not Be Banned”?Get the original essayTopic AnalysisThe Cantor set was discovered in 1874 by Henry John Stephen Smith and was later introduced by Gregor Cantor in 1883. The Cantor ternary set is the most common modern construction of this set. The Cantor ternary set is constructed by removing the open middle third, from the interval [0,1], leaving the line segments. The open middle third of the remaining line segments is removed and this process is repeated infinitely. In each iteration of this process, s will remain of the initial length of the line segment (at that given step). The total length of the line segments at the nth iteration will therefore be: Ln = n, and the number of line segments at this point are: Nn = 2n. From this we can also infer that the open intervals that will be removed by this process on the nth iteration will be + + . . . +.As the Cantor set is the set of points not removed by the above process, it is easy to calculate the total length removed, and from above it is easy to see that at the nth iteration , the removed length tends to. the removed length will therefore be the geometric progression: = + + + + .. = () = 1. It is easy to determine that the remaining proportion is 1 – 1 = 0, which suggests that the Cantor set cannot contain no interval of non-zero values. length. The sum of the deleted intervals is therefore equal to the length of the original interval. At each stage of the Cantor set, the measure of the set is , so we can see that the Cantor set has a Lebesgue measure of n at stage n. Since the construction of the Cantor set is an infinite process, we can see that this measure tends to 0, . Therefore, the Cantor set itself has a total measure of 0. Something should, however, remain when the deletion process leaves behind the endpoints of the open intervals. The following steps will also not remove these endpoints, or indeed any other endpoints. The deleted points are always the internal points of the open interval selected to be deleted. The Cantor set is therefore not empty and contains an uncountable number of elements, however the ends of the set are countable. An example of endpoints that will not be deleted are and , which are the endpoints of the first deletion step. In the Cantor set, there are more items other than endpoints that are also not removed. A common example of this is that which is contained in the interval [0]. It is easy to say that there will be an infinity of other numbers like this example between two closed intervals of the Cantor set. From above it is easy to see that the Cantor set contains all the points of the line segments not removed by this infinite process in the interval [0,1]. Since the construction process is infinite, the Cantor set is considered to be an infinite set, that is, it has an infinite number of elements. The Cantor set contains all real numbers in the closed interval [0,1] which have at least one ternary expansion containing only the digits 0 and 2, this is the result of how the