#MathsInMinutes Day 29: Cantor Sets
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#MathsInMinutes Day 29: Cantor Sets
Cantor Sets are the earliest appearance of the objects known as fractals. The diagonal argument developed by Georg Cantor shows that certain intervals of the real number line are uncountable sets. But do all uncountable sets contain such line intervals? Cantor showed that it was possible to construct an uncountable set that contains no line intervals. Cantor sets are infinitely intricate; they have structure on finer and finer scales.
One example is called the middle third Cantor set. It is obtained by starting with an interval and removing the middle thirds from all the intervals remaining at each stage. At the nth stage of building, it has 2n intervals, each of the length 1/(3n), and a total length of (2/3)n. As n tends towards infinity, so does the number of points within it, while the length of the set shrinks towards zero. It takes a bit more work to show that there really is something left at the infinite limit of this subdivision and to prove that the set is uncountable, but it can be done.
Constructing the Cantor set:
Begin with the closed unit interval, the real numbers between 0 and 1 including the end-points, and remove the middle third, leaving two closed intervals of length 1/3, including their end-points. Now remove the middle third of each of these intervals, so we have four (22) closed intervals, each of length 1/9 (1/32). Repeat the process ad infinitum.
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