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.