I state my conclusion first that there are two kinds of infinite of potency of countable set and property of continuity.


Potency means the number of elements of infinite set, which is an extended concept of the number of finite set.

[Potency of countable set]

When primitively counting the number of elements of set, we chant or breathe 1, 2, 3, ---. Therefore, the element of finite set is one-to-one correspondent to a set of natural numbers {1, 2, 3, ---,n}.
Among infinite sets, a set equivalent (, or one-to-one correspondent) to a set of natural numbers is defined as a countable set. The number of these elements is defined as potency of countable set and represented by a sign (aleph zero).
Examples of countable set
(1) set of natural even numbers
A set of natural even numbers {2, 4, 6, 8, ---} is allowed to correspond to a set {1, 2, 3, 4, ---}. Therefore, a set of even numbers is a countable set. In the same way, a set of odd numbers is a countable set.
{set of even numbers} U {set of odd numbers} = {set of natural numbers} then
However, the following subtraction =0 does not hold, upon subtracting from either.
(2) set of positive rational numbers
Since rational numbers are represented by p/q (where p and q is natural numbers), p and q are successfully allowed to become one-to-one correspondent to natural numbers in the order of numerator p and denominator q.

Therefore, a set of rational numbers is a countable set. This leads to
However, the following division =1 does not hold, upon dividing either member by .

[Potency of continuity]

In the case of a set of real number [X: 0<X<1]
First, let A=5 and B=6, for example, 0.56 is to be represented by 0.AB.
If a countable set, then a set of real number has to be X1, X2, X3, X4,--- by arranging them in a row.
Suppose the n'th real number among them is Xn=0. An1An2An3An4---,and arrange them in ascending order from X1,
Focusing our attention on the diagonal figures and assuming Bn=1 for Ann as an even number and Bn=2 for Ann as an odd number.
Suppose a number of Y: Y=0. B1B2B3B4----- then, Y is not equal to any Xn.
In other words, there is at least one element in real number such as cannot be arranged if done as a countable set.
Thus, real number is not countable. Real number is Potency of continuity and represented by a sign (aleph).
This proof method is referred to as Diagonal method.
If naturally fairly taken in, you cannot apply this line of argument to rational number either
For example, a rational number 1/7=0.142857142857142857----- will be a repeating decimal by a unit of 142857.
Generally, if a rational number p/q is an infinite decimal having its decimal part repeated infinitely, that decimal never fails to repeat (I recommend you to prove this because it is easy).
Then, if we create another number Y such as Y=0. 1211221211121211122222--- 1 and 2 recurs randomly so failing to be a rational number.

[Relationship between and ]

2 (2 to the =)

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