Type: Process Essays
Sample donated: Natasha Burke
Last updated: December 31, 2019
How large can a setwith zero ‘length’ be? This paper will be a summary of my findings in answering thequestions, “how large can a set with zero ‘length’ be?”.
Throughout this paperI will be explaining facts regarding the Cantor set. The Cantor set is the bestexample to answer this question as it is regarded as having length zero. TheCantor set was discovered in 1874 by Henry John Stephen Smith and it was laterintroduced by Gregor Cantor in 1883. The Cantor ternary set is the most commonmodern construction of this set.
TheCantor ternary set is constructed by deleting the open middle third, (,) from the interval 0,1, leaving the line segments and . The open middle third of the remaining line segments aredeleted and this process is repeated infinitely. At each iteration of thisprocess, s of the initial length of the line segment (at that given step)will be remaining.
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 to be: Nn = 2n. From this we can also work out that the openintervals which will be removed by this process at the nth iterationwill be + + . . . + .
Asthe Cantor set is the set of points not removed by the above process it is easyto work out the total length removed, and from above it is easy to see that atthe nth iteration the length removed is tending towards 1. The totallength removed will therefore be the geometric progression: = + + + + .. = () = 1.
Itis easy to work out the proportion left is 1 – 1 = 0, suggesting the Cantor setcannot contain any interval of non-zero length. The sum of the removedintervals is therefore equal to the length of the original interval. Ateach step of the Cantor set the measure of the set is , so we can find that the Cantor set has Lebesgue measure of n atstep n. Since the Cantor set’s construction is an infinite process, we can seeas this measure tends to 0, . Therefore, the whole Cantor set itself has a total measure of 0.
Thereshould, however, be something left as the removal process leaves behind the endpoints of the open intervals. Further steps will also not remove theseendpoints, or in fact any other endpoint. The points removed are always theinternal points of the open interval selected to be removed. The Cantor set istherefore non-empty and contains an uncountable number of elements, however theendpoints in the set are countable. An example of end points that will not beremoved are and , which are the endpoints fromthe first step of removal.
Within the Cantor set there are more elements otherthan the endpoints which are also not removed. A common example of this is which is contained in theinterval 0. It is easy to tell that there will be infinitely many othernumbers like this example between any two of the closed intervals in the Cantorset. Fromabove it is easy to see that the Cantor set contains all the points in the linesegments not deleted by this infinite process in the interval 0,1. As theconstruction process is infinite, the Cantor set is regarded to be an infiniteset, i.e. it has an infinite number of elements. The Cantor set contains allthe real numbers in the closed interval 0,1 which have at least one ternaryexpansion containing only the digits 0 and 2, this is the result of how theternary expansion is written.
As it is written in base three, the fraction will beequal to the decimal 0.1 (also 0.0222..), is therefore equal to 0.2and equal to 0.
01. Inthe first step of the construction of the set, we removed all the real numberswhose ternary decimal representation contain a 1 in the first decimal place,except for 0.1 itself (this is and we have found out it iscontained in the Cantor set). Choosing to represent as 0.222.. this removes allthe ternary decimals that have a 1 in the second decimal place. The third stageremoves those with a 1 in the third decimal place and so on.
After all thenumbers have been removed the numbers that are left, i.e. the Cantor set, are thoseconsisting of ternary decimal representations consisting entirely of 0’s and2’s. Itis then possible to map every 2 in any number in the Cantor set to a 1, if wedo this it will give the full set of numbers in the interval 0,1 in binaryand therefore mapping the whole of the interval 0,1. This means that there isa mapping which has its image as the whole of the interval 0,1, meaning thatthere is a surjection from the Cantor set to all the real numbers in theinterval 0,1. Since the real numbers are uncountable, the Cantor set mustalso be uncountable.
The Cantor set must therefore contain as many points asthe set it is made from and it contains no intervals. Thecompliment of the Cantor set is made up of the points which are not containedin the Cantor set, i.e. the points which are removed from the interval 0,1during the construction of the Cantor set.
From above we worked out that thetotal length removed was equal to 1, which means the compliment of the Cantorset must equal 1 as it is defined precisely as that. An example of a number inthe compliment is the number . Like the Cantor set itself, there is an uncountable number ofelements in the compliment.
At each step of the cantor set, n, there are nnumber of open intervals in the compliment. Between any two endpoints of theCantor set it is obvious to point out that there is an entire interval in thecompliment, i.e. the open intervals removed from 0,1 to form the Cantor set. TheCantor ternary set, talked about above, and in fact the general Cantor set areexamples of fractal sets. A fractal set is a set which is constructed by thesame repeated pattern at every scale. The ternary Cantor set evidentially canbe classed as a fractal set, the pattern demonstrated in the following picture.
The Cantor set split at every step by removing the same fraction of the patternat every step and the number of closed intervals doubles as you move to thenext stage of construction. Thefractal dimension of the Cantor set is . Theabove idea of construction by the ternary method can be generalised to anyother length of removal to form another form of the general Cantor set. Thepattern of forming a generalised Cantor set follows the same constructionpatterns as above also. Another interesting fact about the Cantor set is thatthere can exist “Cantor dust”.
The difference between the two is that Cantordust is the multi-dimensional version of a Cantor set. The dust is formed bytaking the finite cartesian product of the Cantor set with itself, this makesit a Cantor space. The Cantor dust, like the Cantor set, also has a measure of0.