The Cantor Set: How It Works

Introduction

This paper will be a summary of my findings in answering the questions, “how large can a set with zero ‘length’ be?”. Throughout this paper I will be explaining facts regarding the Cantor set. The Cantor set is the best example to answer this question as it is regarded as having length zero.

Topic analysis

The Cantor set was discovered in 1874 by Henry John Stephen Smith and it 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 deleting the open middle third,  from the interval [0,1], leaving the line segments. The open middle third of the remaining line segments are deleted 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 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 open intervals which will be removed by this process at the nth iteration will be + + . . . + .

As the Cantor set is the set of points not removed by the above process it is easy to work out the total length removed, and from above it is easy to see that at the nth iteration the length removed is tending towards.

The total length removed will therefore be the geometric progression: = + + + + .. = () = 1. It is easy to work out the proportion left is 1 – 1 = 0, suggesting the Cantor set cannot contain any interval of non-zero length. The sum of the removed intervals is therefore 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 Lebesgue measure of n at step n. Since the Cantor set’s construction is an infinite process, we can see as this measure tends to 0, . Therefore, the whole Cantor set itself has a total measure of 0.

There should, however, be something left as the removal process leaves behind the end points of the open intervals. Further steps will also not remove these endpoints, or in fact any other endpoint. The points removed are always the internal points of the open interval selected to be removed. The Cantor set is therefore non-empty and contains an uncountable number of elements, however the endpoints in the set are countable. An example of end points that will not be removed are and , which are the endpoints from the first step of removal. Within the Cantor set there are more elements other than the endpoints which are also not removed. A common example of this is which is contained in the interval [0]. It is easy to tell that there will be infinitely many other 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 the points in the line segments not deleted by this infinite process in the interval [0,1]. As the construction process is infinite, the Cantor set is regarded to be an infinite set, i.e. it has an infinite number of elements. The Cantor set contains all the 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 ternary expansion is written. As it is written in base three, the fraction will be equal to the decimal 0.1 (also 0.0222..), is therefore equal to 0.2 and equal to 0.01.

In the first step of the construction of the set, we removed all the real numbers whose ternary decimal representation contain a 1 in the first decimal place, except for 0.1 itself (this is and we have found out it is contained in the Cantor set). Choosing to represent as 0.222.. this removes all the ternary decimals that have a 1 in the second decimal place. The third stage removes those with a 1 in the third decimal place and so on. After all the numbers have been removed the numbers that are left, i.e. the Cantor set, are those consisting of ternary decimal representations consisting entirely of 0’s and 2’s.

It is then possible to map every 2 in any number in the Cantor set to a 1, if we do this it will give the full set of numbers in the interval [0,1] in binary and therefore mapping the whole of the interval [0,1]. This means that there is a mapping which has its image as the whole of the interval [0,1], meaning that there is a surjection from the Cantor set to all the real numbers in the interval [0,1]. Since the real numbers are uncountable, the Cantor set must also be uncountable. The Cantor set must therefore contain as many points as the set it is made from and it contains no intervals. The compliment of the Cantor set is made up of the points which are not contained in the Cantor set, i.e. the points which are removed from the interval [0,1] during the construction of the Cantor set.

From above we worked out that the total length removed was equal to 1, which means the compliment of the Cantor set must equal 1 as it is defined precisely as that. An example of a number in the compliment is the number . Like the Cantor set itself, there is an uncountable number of elements in the compliment. At each step of the cantor set, n, there are n number of open intervals in the compliment. Between any two endpoints of the Cantor set it is obvious to point out that there is an entire interval in the compliment, i.e. the open intervals removed from [0,1] to form the Cantor set.

The Cantor ternary set, talked about above, and in fact the general Cantor set are examples of fractal sets. A fractal set is a set which is constructed by the same repeated pattern at every scale. The ternary Cantor set evidentially can be 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 pattern at every step and the number of closed intervals doubles as you move to the next stage of construction. The fractal dimension of the Cantor set is.

The above idea of construction by the ternary method can be generalised to any other length of removal to form another form of the general Cantor set. The pattern of forming a generalised Cantor set follows the same construction patterns as above also. Another interesting fact about the Cantor set is that there can exist “Cantor dust”. The difference between the two is that Cantor dust is the multi-dimensional version of a Cantor set. The dust is formed by taking the finite cartesian product of the Cantor set with itself, this makes it a Cantor space. The Cantor dust, like the Cantor set, also has a measure of 0.

References

1. https://en.wikipedia.org/wiki/Cantor_set
2. https://www.missouriwestern.edu/orgs/momaa/ChrisShaver-CantorSetPaper4.pdf
3. http://web.math.ucsb.edu/~padraic/ucsb_2013_14/mathcs103_s2014/mathcs103_s2014_zhixing_presentation.pdf
4. http://platonicrealms.com/encyclopedia/Cantor-set