axiomatic sets 28 (the number system)

We have seen the system of numbers or the structure of numbers,
where basic elements are the empty set and the axiom of ordered pair.

(1)Natural numbers $\mathbb{N}$ : $\phi,\left\{\phi \right\},\left\{\phi,\left\{\phi \right\} \right\},\cdots$ means $0,1,2,\cdots$ .

(2)Integers $\mathbb{Z}$ : $[(\phi,\phi)],[(\left\{\phi \right\},\phi)],[(\phi,\left\{\phi \right\})],[(\left\{\phi,\left\{\phi \right\} \right\},\phi)],[(\phi,\left\{\phi,\left\{\phi \right\} \right\})],\cdots$ means $0,1,-1,2,-2,\cdots$ .

(3)Rational numbers $\mathbb{Q}$ :
$[([(\phi,\phi)],[(\left\{\phi \right\},\phi)])]=0$ ,
$[([(\left\{\phi \right\},\phi)],[(\left\{\phi \right\},\phi)])]=1$ ,
$[([(\phi,\left\{\phi \right\})],[(\left\{\phi \right\},\phi)])]=-1$ ,
$[([(\left\{\phi,\left\{\phi \right\} \right\},\phi)],[(\left\{\phi \right\},\phi)])]=2$ ,
$[([(\phi,\left\{\phi,\left\{\phi \right\} \right\})],[(\left\{\phi \right\},\phi)])]=-2$ ,  $\cdots$ .

(4)Real numbers $\mathbb{R} $: the set of all "Dedekind cuts" ($\left\{x\in\mathbb{Q} : x\lt a,a\in\mathbb{Q} \right\}$)

We are not conscious for the making of numbers or the system of numbers.
However, have you ever had a little bit of doubts something like why $\frac{-1}{3}=\frac{1}{-3}$ or the definition of irrational numbers is numbers which is not rational numbers, etc, etc ?

The number system will answer some questions and you will see the rigorousness exists in every fields of mathematics.

It makes us very comfortable and gives much confidence.


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In these definitions, the relation of $\mathbb{N}\subset\mathbb{Z}\subset\mathbb{Q}\subset\mathbb{R}$ will require one to one correspondence.