real number system 6

We summarize the properties of real numbers based upon preceding posts.

(1)Real numbers are the set of  all decimals. That is, every real numbers has the form as follow.
\[ C.c_1c_2c_3\cdots, \quad \mbox{or} \quad -C.c_1c_2c_3\cdots \]
where $C$ is any nonnegative integer and $c_i$ is a single digit figure between $0$ and $9$ inclusive.

(In this definition, a finite decimal number always has two forms. For example one is $1.000\cdots $ and $0.999\cdots $. After this, in these cases we shall promise that if $x$ is $0.999\cdots$, we make $x$ be $1$. The same applies to another numbers.)

(2)Real numbers consist of rational numbers and irrational numbers.

(3)Real numbers are a field. On a field we can calculate as prescribed.

(4)Real numbers are ordered by a natural magnitude relationship.

(5)Real numbers are complete.

From (1), (3) and (4), we are not able to get (5). Therefore, we must accept Axiom for completeness or Dedekind cut is satisfied in real numbers. If one is accepted, the other can be proved.

(6)Real numbers are dense. Rational numbers have been already dense in real numbers.

(7)Real numbers are uncountable. Although both $\mathbb{R}$ and $\mathbb{N}$ are infinite sets, $\aleph$ is greater than $\aleph_0$.

(8)There is a one to one correspondence between points of the real number line and real numbers. The one to one function from real numbers to points in the square having a side length $1$ also exists.

Essentially we need (1), (3), (4) and (5). However, do you think all things are very interesting?