IndexIntroductionAnalysis of the topicReferencesIntroductionThis document will be a summary of my findings in answering the questions: "how large can a set with zero 'length' be?". In this article I will explain the facts regarding the Cantor set. The Cantor set is the best example to answer this question since it is considered to be zero in length. Say no to plagiarism. Get a tailor-made essay on "Why Violent Video Games Shouldn't Be Banned"? Get an Original Essay Topic 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 eliminating the open middle third, from the interval [0,1], leaving the line segments. The open middle third of the remaining line segments is erased, and this process is repeated infinitely. At each iteration of this process, s of the initial length of the line segment (at that given step) will remain. The total length of the line segments at the nth iteration will then be: Ln = n, and the number of line segments at this point will be: Nn = 2n. From this we can also understand that the open intervals that will be removed by this process at the nth iteration will be ++ . . . + .Since the Cantor set is the set of points not removed by the previous 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 towards. the length removed will then be the geometric progression: = + + + + .. = () = 1. It is easy to understand that the proportion left is 1 – 1 = 0, suggesting that the Cantor set cannot contain any interval other than zero length. The sum of the removed intervals is then equal to the length of the original interval. At each step of the Cantor set the measure of the set is , so we can find that the Cantor set has a Lebesgue measure equal to n at step n. Since the construction of the Cantor set is an infinite process, we can see how this measure tends to 0, . Therefore, the entire Cantor set has a total measure of 0. However, something should remain since the removal process leaves behind the endpoints of the open intervals. Furthermore, subsequent steps will not remove these endpoints, or indeed any other endpoints. The points removed are always the interior points of the open range selected to be removed. The Cantor set is therefore non-empty and contains an innumerable number of elements, however the extrema of the set are countable. An example of endpoints that will not be removed are and , which are the endpoints of the first step of removal. Within the Cantor set there are more elements than endpoints that are not removed. A common example of this is that contained in the range [0]. It is easy to say that there will be infinitely more numbers like this example between any two of the closed intervals in the Cantor set. From above it is easy to see that the Cantor set contains all points in the line segments not canceled out by this infinite process in the interval [0,1]. Since the construction process is infinite, the Cantor set is considered 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] that have at least one ternary expansion containing only the digits 0 and 2, this is a result of how the ternary expansion is written. As it is written in base three, the fraction will be equal to the decimal 0.1 (also 0.0222..), it is therefore equal to 0.2 and equal to 0.01. In the first step of building the set, we removed all the real numbers whose