axiomatic sets 4 (empty set)

First axiom of ZFC is the existence of empty set.

It is stated by following logical formula;
\[ \exists x\forall y[y\notin x]  \]

Using A familiar expression,
\[ \phi=\left\{ \right\}=\left\{x ; \forall y\notin x\right\},  \]
or,
\[  \forall y\notin \phi \]

This axiom means three things. One is a set(e.g. empty set) always exists in any spaces,
two is there is a set which has no elements,
and three is any sets has empty set as a element

Namely, it is always true that if $x\in\phi$ , then any $y$ has the member $x$ .

As the number of elements of empty set is zero, $0$ can be used instead of $\phi$ .
Please note that $0$  is a set in the theory.

This axiom is proved by another axioms.
However, it is one of ZFC-axioms.

axiomatic sets 3 (ZFC "a class")

In "naive set thory" (initiated by G. Cantor and R. Dedekind in the 1870s),
a set has been given the following conditions.

A set is a group of some matters which are usually called elements or members and satisfies ;
(1)It is possible to determine whether an element (in the space) should be included in the set or not,
(2)It is possible to determine whether two elements in the set are identical or not.

However, such weak conditions of a set are fraught with some paradoxes and contradictions.

Zermelo-Fraenkel set theory (referred to as the ZFC including axiom of Choice ) begins
with putting some axioms in order to avoid such problems.
Let us have a look at these axioms.

Before beginning with the axioms, we have to define  "a class".
In the naive set theory, any gathering of some matters is a set
whereas in the ZFC, it would be merely be a class rather than a set.
A class consists of sets and proper classes.
Although a proper class is a gathering of some matters, it is not a set.

Only a gathering of elements which is adapted to ZFC-axioms is deemed as a set.
A proper class is a gathering of elements, but it is not a set.

Please do remember that it is the great difference between the ZFC and naive set theory.