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

[*Potency*]

*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 X_{1}, X_{2}, X_{3}, X_{4},--- by arranging them in a row.

Suppose the n'th real number among them is X_{n}=0. A_{n1}A_{n2}A_{n3}A_{n4}---,and arrange them in ascending order from X1,

w_{1}OD`_{11}`_{12}`_{13}`_{14}---

w_{2}OD`_{21}`_{22}`_{23}`_{24}---

w_{3}OD`_{31}`_{32}`_{33}`_{34}---

w_{4}OD`_{41}`_{42}`_{43}`_{44}---

·

·

·

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. B_{1}B_{2}B_{3}B_{4}----- 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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