axiomatic sets (ZFC) 2

Axiomatic set theory was in large part found in 1908-1922 by E. Zermelo and A. Fraenkel. 

It is much different from the set theory which we study in beginner class.
We will start from;

(1)undefined terms
(2)axioms
(3)notations and symbols which are restricted

For example, if a set $A$  and a set $B$ are sets, then the couple $\left\{ A,B \right\}$ is a set. This is "axiom of pairing" . We will formally express;
\[  \forall x\forall y\exists z\forall w[w\in z \leftrightarrow w=x\vee w=y]  \]

When a set $A$ and a set $B$  are sets, we have definitely believed $A \cup B$  is a set. However, the axiomatic system requires the condition.

Why was the axiomatic set theory needed?

The classical set theory, we know as "the naive set theory", is very intuitive and easy to understand, but it had been fraught with some contradictions and problems which are not solved.

axiomatic sets (ZFC)

In axiomatic set theory, all objects is a set.
Although it is very natural, it is not easy to understand.

For example, $\mathbb{N}$  is "natural number" and must be a set in the theory.

The definition is as follow;

You must know a empty set $\phi$ .

$\exists y \forall x\neg [x\in y]$ .

$y$ means $\phi$ .

We represent $\phi=0$ and $\left\{ \phi \right\}=1$ .

Next, $2=1+\left\{ 1  \right\}=\left\{\phi, \left\{ \phi \right\}  \right\}$ ,

$\cdots \cdots \cdots$ , and

$n+1=1+\left\{ n  \right\}$ .

In addition, axiom of infinity is defined;

$\exists x[0\in x\wedge \forall y(y\in x\rightarrow (y\cup \left\{y \right\})\in x)]$ .

How do you feel this.