axiomatic sets 15 (the set of all Natural numbers)

We got all natural numbers.
\[ 0=\left\{ \right\}, \quad n+1=n\cup\left\{n\right\}\quad (n\ge 0)   \]
You are able to know any natural numbers which you desire, increasing one by one.

However, it is not a set or collection of all natural numbers.

In the set $\mathbb{N}=\left\{0,1,2,\cdots  \right\}$ ,
'･･･' does not mean any natural number.

No matter how a large number, it does not mean '･･･'.

Hence, the axiom is needed.
\[ \exists x \forall y[y\in x\rightarrow (y\cup\left\{ y \right\})\in x]   \]
It is called an axiom of infinity.

There can be exists a set whose elements are infinite.

Let $y$ be $0$ , the set $x$ is the set of all natural numbers.