How large can a set

with zero ‘length’ be?

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. 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 and . 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 1. 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:

The

following websites are those where I acquired information in order to complete

this research project.

·

https://en.wikipedia.org/wiki/Cantor_set

·

https://www.missouriwestern.edu/orgs/momaa/ChrisShaver-CantorSetPaper4.pdf

·

http://web.math.ucsb.edu/~padraic/ucsb_2013_14/mathcs103_s2014/mathcs103_s2014_zhixing_presentation.pdf

·

http://platonicrealms.com/encyclopedia/Cantor-set