axiomatic sets 6 (pairing)

As a set is a gathering of some matters, the elements of a set have no orders.
Given a set having elements $x,y,z$ , then
$\left\{ x,y,z \right\}=\left\{ x,z,y  \right\}=\left\{ y,x,z  \right\}=\left\{ y,z,x  \right\}=\left\{ z,x,y  \right\}=\left\{ z,y,x  \right\} $ .

We want to give an order for elements and
distinguish two sets having same elements of which the orders are not same.

At first we will define a set having only two elements.

\[ \forall x\forall y\exists a\forall w[w\in a\leftrightarrow w=x\vee w=y] \]

Namely, $a=\left\{ x,y  \right\}$  or $\left\{ y,x  \right\}$ . It is called axiom of pairing.

If we write $a=\left\{ x,y  \right\}$ , then it means $x\neq y$ , and
if $x=y$ , then we will write $a=\left\{ x  \right\}$ or $a=\left\{ y  \right\}$ .
Therefore, $\left\{ x,x  \right\}=\left\{ x  \right\}$ .

Given two sets $a=\left\{ x,y  \right\}$ and $b=\left\{ u,v  \right\}$ ,
if $a=b$ , then $x=u$  and $y=v$ , or $x=v$  and $y=u$ by axiom of extensionality.

That is to say $\left\{ x,y  \right\}=\left\{ u,v  \right\}$ and  $\left\{ x,y  \right\}=\left\{ v,u  \right\}$ .

However, if $a=b$ means only the case of $\left\{ x,y  \right\}=\left\{ u,v  \right\}$
and $\left\{ x,y  \right\}\neq \left\{ v,u  \right\}$ ,
then the only conditions of $x=u$ and $y=v$ will be required.

We will define an ordered pair for two elements by
\[ \left\{ \left\{ x  \right\}, \left\{ x,y  \right\} \right\}  \]
and write it $(x,y)$ or $\lt x,y \gt$ . Hence, definitely $\lt x,y \gt\neq \lt y,x \gt$ .

With concrete description, $\lt 0,1 \gt\neq\lt 1,0 \gt$ . Because
$\lt 0,1 \gt=\left\{ \left\{ 0  \right\}, \left\{ 0,1  \right\} \right\}$ ,
$\lt 1,0 \gt=\left\{ \left\{ 1  \right\}, \left\{ 1,0  \right\} \right\}$ ,
although $\left\{ 0,1  \right\} =\left\{ 1,0  \right\}$ .
(and you might know $\left\{ \left\{ 0  \right\}, \left\{ 0,1  \right\} \right\}=\left\{ 1, \left\{ 1,0  \right\} \right\}$ . )

By this definition, an order of two elements in a set will be decided and
two same sets having two different elements become equivalent,
 just only when the order of elements is equivalent.

If you want to set out three elements, you may put
\[  \lt a,b,c \gt=\lt a,\lt b,c \gt \gt=\left\{ \left\{ a  \right\}, \left\{ a,\left\{ \left\{ b  \right\}, \left\{ b,c  \right\} \right\}  \right\} \right\} \]
More elements if you need, then
\[ \lt a_1,a_2,a_3,\cdots,a_n \gt=\lt a_1,\lt a_2,a_3,\cdots,a_n \gt \gt   \]

Axiom of paring means that there is a set such that has only one or two elements.
Using the axiom, we are able to get an order of elements.
For example, an arity of functions will be noted by a set.