axiomatic sets 5 (extensionality)

Second axiom of ZFC is the extensionality of a set.
By this axiom, we are able to distinguish sets by means of members.

It is stated by following formula;
\[ \forall x\forall y[\forall z[z\in x\leftrightarrow z\in y]\leftrightarrow x=y]   \]

Using a familiar expression,
\[ x=y\leftrightarrow \forall z[z\in x\leftrightarrow x\in y]  \]

This axiom means three things.
one is a set (except for empty set) has elements,
two is the difference of two sets is only dependent on each elements, and
three is a set can be extended by adding distinct elements.

If $x\neq y$ are two sets, then there are some elements (or a element)
that belong to one but not the other.
For example, if
$x=\left\{ x;1\leq x\leq \sqrt{2}, x=\mbox{ real number}  \right\}$ , and
$y=\left\{ y;1\leq y\lt \sqrt{2}, y=\mbox{ real number}  \right\}$ , then $x\neq y$ .
If
$x=\left\{ x;1\leq x\leq \sqrt{2}, x=\mbox{ rational number} \right\}$ , and
$y=\left\{ y;1\leq y\lt \sqrt{2}, y=\mbox{ rational number} \right\}$ , then $x= y$ .

The axiom of extensionality requires that a set will be only identified by the elements.
When we deal with a set, we have to observe the elements of a set carefully.

By contraries if the elements are not defined or are defined ambiguously,
 the set can not exist.
(However, you will see the gathering of some matters not a set,
 although the elements are defined with no doubt. )

As the elements create a set, empty set is always one in any spaces.
There do not exist two different empty sets.

It can be possible that empty set becomes a element.
\[ \left\{ \phi \right\} =\left\{ \left\{ \right\}\right\}  \]

We will define this set $\left\{\phi\right\}$ as $1$ ;
\[ \left\{\phi\right\}=1   \]
because in the theory $1$ also has to be a set.

By this definition, you must understand that  $\phi$  is not equivalent to $\left\{\phi\right\} $ .

In a addition, we will always be able to use the number $0$ and $1$ in any spaces.