Cantor dust: a ►fractal set of ►real numbers with paradoxical properties.

Here is a recipe for producing Cantor dust: Take a range of real numbers such as the range from 0 to 1. Now cut out 80% of all numbers in the middle of this range, that is, all numbers between 0.1 and 0.9. Then cut out the middle 80% from each of the remaining 10% at the beginning and the end of the original range. These should be all numbers from 0.01 to 0.09 and from 0.91 to 0.99. The remaining four number ranges should be those from 0 to 0.01, from 0.09 to 0.10, from 0.90 to 0.91, and from 0.99 to 1. Now, remove the middle 80% from each of these ranges once again, and so forth.

Repeat these 80%-cuts an infinite number of times. When you are finished you will have thereby generated Cantor dust. It is the set of all remaining numbers after an infinite number of cuts as described above:

0th cut: 0-----------------------------------------------1
1st cut: 0----------0.1                              0.9----------1
2nd cut:   0---0.01     0.09---0.1                                       0.9---0.91     0.99---1
3rd cut: 0-0.001  0.009-0.01  0.09-0.091  0.099-0.1    0.9-0.901  0.909-0.91  0.99-0.991  0.999-1
4th cut: ..................................................................................................................

Now go ahead and count the remaining numbers. To your astonishment, you will notice that just as many numbers remain as were within the original range with which you started out. And this is so despite your having removed numbers an infinite number of times. How is this possible?

The Frustrated Cutter

The first cut removes all numbers whose first position after the decimal point takes a value other than 0 or 9. Similarly, the second cut removes from the set of remaining numbers all numbers whose second position after the decimal point takes a value other than 0 or 9. This process continues infinitely. After an infinite number of steps only those numbers will remain that consist entirely of the digits 0 and 9 - such as 0.9090909, 0.00900999, etc. This also applies to the last number of our range, the number 1, for this number can also be written as 0.99999999.*

A number consisting of only two digits, however, is nothing but a binary number (see number systems). To illustrate this, let us replace in all of our remaining numbers the digit 9 by the digit 1, which obviously will not affect the overall cardinality of our set. In this way, we obtain all binary numbers between 0 and 1, such as 0.1010101, 0.00100111, and so forth. Since it does not matter with regard to numerical value whether we represent it as a binary or a decimal number, we have hereby generated precisely our original number range between 0 and 1.

Of course, this paradox is just another way of representing the fact that an infinite set can have just as many elements as one of its subsets, which you will encounter frequently in this encyclopedia (see ►Countability). Here's a trick question: What happens if we once again cut out 90% of numbers each step in an infinite series of steps, only this time cutting from the end of the number range rather than from the center?**

* Proof: 9 = 9.999. - 0.999. = (10-1) ∙ 0.999. =  9 ∙ 0.999. = 9 ∙ 1, therefore 0.9999. = 1. Accordingly, we also have 0.1 = 0.09999. etc.

** The first cut removes all numbers greater than 0.1. The second removes all numbers greater than 0.01, and the third all numbers greater than 0.001. With each step the digit 1 is shifted one position further to the right until eventually, after an infinite number of steps, only the digit 0 remains. Thus, even though we have hereby removed just as many numbers as before, the result is quite different.

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