A set S is denumerable if and only if it is equivalent to the set of natural numbers N . A denumerable set S has cardinal number ℵ0 and write S =ℵ0. If a set is finite or denumerable, it is countable, otherwise the set is uncountable.
The symbol ℵ0 is the first infinite cardinal
number. Other infinite cardinal numbers are associated with uncountable sets.
The interval ) 0,1 ( is an example of an uncountable set. The cardinal number of ) 0,1 ( is defined to be c (which stands for continuum).
In the years 1871-1884 Georg Cantor invented the theory of infinite sets.
In the process Cantor constructed a set which is called a "Cantor" set.
To construct the Cantor set, take a line and remove the middle third.
There are two line segments left. Take the remaining two pieces and remove their middle thirds. Repeat this process infinite number of times.
The resulting collection of points is called a "Cantor" set. Indeed repeatedly removing the middle third of every piece, we could also keep removing any other fixed percentage (other than 0 % and 100 %) from the middle.
The resulting sets are all homeomorphic to the Cantor set, i.e. these sets are topologically the same.
The Cantor set is an unusual object. The deletion process produces an infinite set of points. On the other hand these points are uncountable, also it has no interior point.
Let's focus on the ternary representations of the decimals between 0 and 1. Since, in base three, 1/3 is equivalent to 0.1, and 2/3 is equivalent to 0.2.
We see that in the first stage of the construction (when we removed the middle third of the unit interval) we actually removed all of the real numbers whose ternary decimal representation have a 1 in the first decimal place, except for 0.1 itself. (Also, 0.1 is equivalent to 0.0222.. in base three, so if we choose this representation we are removing all the ternary decimals with 1 in the first decimal place.In the same way, the second stage of the construction removes all those 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. (By noticing that 1/9 is equivalent to 0.01 and 2/9 is equivalent to 0.02 in base three.
Thus, after every thing has been removed, the numbers that are left – that is, the numbers making up the Cantor set – are precisely those whose ternary decimal representations consist entirely of 0’s and 2’s.
Then the Cantor middle third set 1/ 3 C is precisely the set of points in the interval I having a ternary expansion.
Since in base four expansion, 4 / 1 is equivalent to 1. 0 and 4 / 3 is equivalent 3. 0 . We see that in the first stage of construction (when we removed the middle half of the unit interval) we actually remove all elements ] x ∈[0,1 such that 3 0.1 < x < 0. , that is we remove all of the real numbers whose four decimal representation is 1 and 2 in first decimal place, except for 0.1 itself. (Also 0.1 is equivalent to 0.0333 . . . in base four, so we choose the representation in which we are removing all the four
decimals with 1 and 2 in the first decimal place). In the same way, the second stage of the construction removes all those fourth decimals that have a 1 and 2 in the second decimal place. The third stage removes those with a 1 and 2 in the third decimal place, and so on. (By noticing that 1/16 is equivalent to 0.01 and 3/16 is equivalent to 0.03 in base four expansion).finally all numbers left, making up the Cantor middle half set 1 / 2 C are precisely those whose four decimal representations which consist entirely of 0's and 3's.
Thus 1 / 2 C is the set of points, x , in the unit interval such that there is a base four expansion of x that uses only zeros and threes.
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