From 'Chaos and Fractals; the New Frontiers of Science' the textbook:
The Cantor Set
[Figure 2.4] The Cantor Set was represented by vertical lines whose base points are exactly at all the different points belonging to the set.
Greg Cantor (1845 - 1918) was a German mathematician at the University of Halle where he carried out his fundamental work in 'Set Theory'. First published in 1883, his theories soon emerged as an example of certain "exceptional" sets (meaning, a perfect, nowhere dense subset). The Cantor Set plays a deep role in chaotic dynamical systems, and is hidden as a kind of framework behind many fractals, like the Julia Set.
The basic Cantor set is an infinite set of points in the unit interval [0,1]. It can be interpreted as a set of certain numbers, for example, 0,1,1/3, 2/3, 1/9, 2/9, 7/9, 8/9, 1/27, 2/27,...assuming we knew what they were plotting these and all other points would not make enough of a picture. Rather than plotting just points, we plot vertical lines of all the same length, whose base points are exactly at all the different points belonging to the Cantor Set. Then we see the distribution of these points better. (see Figure 2.4)
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CONSTRUCTION OF THE CANTOR SET
Start with the interval [0,1]. Now take away the (open) interval (1/3, 2/3), i.e; remove the middle third from [0,1], but not the numbers 1/3 and 2/3. This leaves two intervals [0,1/3] and [2/3, 1] of length 1/3 each, 
and completes a basic construction step. Now we repeat, look at the remaining intervals [0,1/3] and [2/3, 1] and remove their middle thirds, which leaves four intervals of a length 1/9. Continue on in this way. There is a feedback process in which a sequence of (closed) intervals is generated - 1 interval after the 1st step, 2 after the 2nd step, 3rd after the third step, 8 after the 4th step, etc. (i.e., 2n intervals of length 1/3n after the nth step). Figure 2.5 here visualizes this construction.
The Cantor Set is the set of points which remain if we carry out only the removal steps infinitely often. For instance, if a point - say x - is in the Cantor set we can guarantee that no matter how often we carry out the removal process, the point xwill not be taken out. Obviously 0, 1, 1/3, 2/3, 1/9, 2/9, 7/9, 8/9, 1/27, 2/27,... are examples of such points because they are the end points of the intervals which are created in the steps; and therefore, they must remain. All these points are related to powers of 3, or rather to powers of 1/3. It is tempting to draw the conclusion that any point in the Cantor set is of this kind (i.e., an end point of one of the small intervals generated in the process) but this is categorically wrong.
If the Cantor set were just the end points of the intervals of the generating process, we easily enumerate them. But the Cantor set is known to be uncountable. There is no way to enumerate the points in the Cantor set. Thus, there must be many more points which are not end points. To give an example, we can characterize a Cantor set by using triadic numbers.
CHARACTERIZATION OF THE CANTOR SET:
Triadic numbers are number which are represented with respect to base 3. This means only using the digits 0, 1, and 2. There are a few examples in this table.
TRIADIC CONVERSION: Conversion of four decimal numbers into the triadic representation.
The Decimal System and its Representation
When we write 0.32573, we mean: 3 * 10-1 + 2 * 10-2 + 5 * 10-3 + 7 * 10-4 + 3 * 10-5
So any number x in [0,1] can be written as x = a1 * 10-1 + a2 * 10-2 + a3 * 10-3..., where the a1,a2, a3 are numbers from {0,1,2,...,9}, the decimal digits. This is called the decimal expansion of x, and may be infinite (e.g. x = 1/3) or finite (e.g. x = 1/4). When we say the expansion is finite we actually mean that it ends with infinitely consecutive zeros, i.e. 0.20000...describes a limit of 2, because the only numbers following it are zeros.
Digital computers depend on binary expansions of numbers. In computers 10 as base is replaced by 2. For example 0.11001 is 1 * 2-1 + 1 * 2-2 + 0 * 2-3 + 0 * 2-4 + 1 * 2-5. There is a bit of ambiguity in these representations. We can write 2/10 as 0.19 = 0.1999... or 0.2(000...) or in base 2 the number 1/4 can be represented as 0.001 = 0.00111...or 0.01 = 0.01000....
Now we can completely describe the Cantor set by representing numbers from [0,1] in their triadic expansion, i.e., we switch to expansions of x with respect to base 3 as shown in this equation: x = a1 * 3-1 + a2 * 3-2 + a3 * 3-3 + a4 * 3-4 + ... Thus, here are the a1, a 2, a3...numbers from { 0,1,2 }.
[Figure 2.9]: Visualization of binary expansions by a 2-branch tree. In contrast to real trees, we draw address trees with the root at the top. Any number in the interval [0,1] at the bottom can be reached from the root of the tree by following branches. Writing down the labels of these branches (0 for left, 1 for right) in a sequence will yield a binary expansion of the chosen real number. The tree has obvious self-similarity: any 2 branches at any node are a reduced copy of the whole tree.
[Figure 2.10]: A 3-branch tree visualizes the triadic expansion of numbers from the unit interval. The first main branch 
covers all numbers between 0 and 1/3. Following the branches all the way to the interval and keeping note of the labels 0, 1, and 2 for choosing the left, middle or right branches will produce a triadic expansion of the number in the interval which is approached in this process.
CANTOR'S DIAGONALIZATION ARGUMENT
Suppose that the infinity of decimal numbers between zero and one is the same as the infinity of counting numbers. Then all the decimal numbers can be denumerated in a list.
1
d1 = 0.d11d12d13d14...
2 d2 = 0.d21d22d23d24...
3 d3 = 0.d31d32d33d34...
4 d4 = 0.d41d42d43d44...
continuing few iterations...in between 4 and n...
n dn = 0.dn1dn2dn3dn4...
Consider the decimal number x = 0.x1x2x3x4x5...where x1is any digit other than d11; x2 is different from
d22; x3 is not equal to d33;x4 is not d44;and so on. Now xis a decimal number, and x is less than one, so it must be in our list somewhere.
x can't be first, since x's first digit differs drom d1's first digit. x can't be second in the list, because x and d2 have different hundredths place digits. In general, x is not equal to dn, since there nthe digits are not the same.
x is nowhere to be found in the list, we have exhibited a decimal number that should be in the list but isn't. No matter how we try to list the decimal numbers, at least one will be left out. Therefore, 'listing' the decimal numbers is impossible, so the infinity of decimal numbers is greater than the infinity of counting numbers.
So there are infinitely many counting numbers and infinitely many fractions. But even though there are infinitely many fractions between each pair of consecutive counting numbers, the arrows in the diagram (shown here) show how to match the red counting numbers with the blue fractions. This work proves the infinities are the same. There are just as many counting numbers as there are fractions.
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