axiomatic sets 16 (Peano's axioms)

By an axiom of infinity, the collection of all natural numbers was gotten.

By Peano's axioms, all natural numbers $\mathbb{N}$  has been defined.

$w$ which satisfies the following conditions (1)-(5) is $\mathbb{N}$ ;

(1)$0\in w$
(2)if any $n\in w$ , then $n\cup\left\{ n \right\}\in w$
(3)for each $n\in w$ , $n\cup\left\{ n \right\}\ne 0$
(4)if $x$  is a subset of $w$ such that $0\in x$  and if $n\in x$ , then $n\cup\left\{ n \right\}\in x$ ,
   then $x=w$
(5)if $n,m\in w$ and $n\cup\left\{ n \right\}=m\cup\left\{ m \right\}$ ,  then $n=m$

Please remember that
\[ n\cup\left\{ n \right\}=n+1 .   \]
By using preceding axioms, these can be proved.

$n\cup\left\{ n \right\}$  is called a successor set of $n$ .

Please note that (4) means the principle of "mathematical induction".
It may seem same as  an axiom of infinity.

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.