axiomatic sets 26 (real numbers 2)

We can not give a construction of real numbers from rational numbers.
Because it's proof is very long and a little bit tedious for blogs.
We will see a brief outline of the construction.

Let us make a subset $\alpha$ in rational numbers $\mathbb{Q}$ as follow.

(1)$\alpha\ne \phi,\mathbb{Q}$ .
(2)$\alpha$ is, for a number $a$, $\left\{x\in\mathbb{Q} : x\lt a \right\}$ .
(3)There is not a maximum number in $\alpha$ .

You must remember that all rational numbers is completely ordered "$\lt$" .
Therefore, if $y\lt a$ ,then $y\in \alpha$ and if $y\ge a$ ,then $y\notin\alpha$ .

As rational numbers are dense, if $y\lt a$ , then there must exists a $z$ such that $y\lt z\lt a$ .
Thus, $\alpha$ has not a maximum number.

By the number $a$ , rational numbers $\mathbb{Q}$ are separated to two sets $\alpha$ and $\alpha^c$ .
(You may prefer to think the both-sides open interval $\alpha=(-\infty, a)$ in the line of $\mathbb{Q}$ .
 Please note that $\alpha$ does not have a point $a$ . )

It was called the 'Dedekind cut' $\alpha$  in the preceding post.

When we give many rational numbers to $a$ , there can be many 'Dedekind cuts' $\alpha$ .

We will express the set of all 'Dedekind cuts'   $\mathcal{C}$ .
That is, $\mathcal{C}$ has the all intervals in which the left-side is $-\infty$
and the right-side is a rational numbers.

It is an astonishing fact that the element of the set $\mathcal{C}$ is a real number.

(We have to recognize the term "all" is much useful. )

axiomatic sets 25 (real numbers)

We have constructed rational numbers by using the ordered pairs of two integers.

Unfortunately irrational numbers cannot be explicitly constructed from integers by such the way.
But we know there are many irrational numbers. For example, $\sqrt{2},\sqrt{3},\pi,e,\cdots$ .

We are enough in living using rational numbers. Almost people will be satisfied by the number "3.14" as $\pi$ .
No one requires the numbers which has infinite digits.

However, in rigorous thought as mathematics numbers which can not be represented by rational numbers are needed.
Area of circle must be $\pi r^2$ , not $3.14r^2$ , and
the answer of the equation $x^2-2=0$  must be $\pm\sqrt{2}$ , not $\pm 1.4142$ .  

Irrational numbers may be defined by numbers which cannot be represented by a rational number
(that is, one ordered pair of two integers. )

As you know well, real numbers are made by rational numbers and irrational numbers.
In other words, real numbers are made by rational numbers and numbers which are not rational numbers.

Such a definition will give us much frustration. I will also agree with it.
We have already known complex numbers or quaternion and these are neither rational
nor irrational numbers.

We want to know how to make real numbers by rational numbers and irrational numbers.

Thus, what is real number was one of much important issues around the end of 19century.

The answer was given by R.Dedekind or G.Cantor, etc.
(In the preceding posts, we have seen the properties of real numbers.)