axiomatic sets 23 (types of numbers)

There are some kinds of numbers. Let us list these samples up.

The empty set means the number "0", that is, $\phi=\left\{ \right\}=0$ .
We will see that "0" is $0_N$  in natural numbers.

The natural number "1" is the set whose member is only the empty set, $1_N=\left\{\phi\right\}$ .
The natural number "2" is the set whose members are natural numbers "0" and "1",
$2_N=\left\{0_N,1_N \right\}=\left\{\phi,\left\{\phi\right\} \right\}$ .

The integer "0" is the ordered pair class $0_Z=[(0_N,0_N)]$ ,
but it's class has only one element $(0_N,0_N)$ .

The non negative integer "1" is $[(1,0)]$ ,where "1" is the natural number $1_N$, and "[ ]" means a equivalent class. Therefore,
$$1_Z=[(1_N,0_N)]=[(\left\{\phi\right\},\phi)]=[(\left\{\left\{ \right\}\right\},\left\{ \right\})]$$
As the equivalent relation is for two integers $(a_N,b_N),(c_N,d_N)$ ,
$a_N+d_N=b_N+c_N$  must be satisfied,
you have to note that $(1_N,0_N)\sim (2_N,1_N)\sim (3_N,2_N)\sim\cdots$ .
Those are in $1_Z$ ,and we use every elements as $1_Z$ .

The non negative integer "2" is also the ordered pair class $[(2_N,0_N)]=2_Z$ ,
The negative integer "-2" is $[(0_N,2_N)]=-2_Z$ .

The rational number "0" is also the ordered pair class $[(0,a)]$ ,where $a$  is an arbitrary integer, but not zero, and "[ ]" means a equivalent class.
$$0_Q=[(0_Z,a_Z)]=[([(0_N,0_N)],[(a_N,0_N)])] .$$
The equivalent relation on rational numbers is for two rational nummbers $(p_Z,q_Z),(r_Z,s_Z)$
(and $q_Z,s_Z\ne 0)$, $p_Zs_Z=q_Zr_Z$ must be satisfied.

The non negative rational number "1" is $[(1,1)]$ ,where "1" is the integer number "$1_Z$".
Therefore,
$$1_Q=[(1_Z,1_Z)]=[([(\left\{\phi\right\},\phi)],[(\left\{\phi\right\},\phi)])] ,$$
and for $x\in 1_Q$ ,
$$x\sim (1_Z,1_Z)\sim (-1_Z,-1_Z)\sim (2_Z,2_Z)\sim (-2_Z,-2_Z)\sim (3_Z,3_Z)\sim\cdots$$
By same way, $2_Q=[(2_Z,1_Z)]=\left\{(2_Z,1_Z),(-2_Z,-1_Z),(4_Z,2_Z),(-4_Z,-2_Z),(6_Z,3_Z),\cdots \right\}$ .

You will find these kinds of numbers are constructed by $\phi$  and axioms of sets.
Please try to make some types of numbers.