axiomatic sets 35 (ordinal numbers)

You may remember the structure of natural number $\mathbb{N}$ ,
\[ 0=\left\{ \right\},1=\left\{0\right\},2=\left\{0,1\right\},3=\left\{0,1,2\right\},\cdots , \]
based on axiom of infinity ($n+1=n\cup\left\{n\right\}$) .

We will also define
\[ \mathbb{w}=\left\{0,1,2,\cdots \right\}    \]
as all natural numbers $n$ .

If we go to the next number from $\mathbb{w}$, then
\[ \mathbb{w}+1=\left\{0,1,2,\cdots,\mathbb{w}  \right\},  \]
Furthermore,
\[ \mathbb{w}+2=\left\{0,1,2,\cdots,\mathbb{w},\mathbb{w}+1  \right\} , \]
and $\cdots\cdots\cdots$

Then,
\[ \mathbb{w}+\mathbb{w}=\left\{0,1,2,\cdots,\mathbb{w},\mathbb{w}+1,\mathbb{w}+2,\cdots  \right\}  \]
and we naturally put
\[ 2\mathbb{w}=\mathbb{w}+\mathbb{w} .   \]

Further and furthermore,
\[ 2\mathbb{w},3\mathbb{w},\cdots, \mathbb{w}\mathbb{w} \]
\[\mathbb{w}^2=\mathbb{w}\mathbb{w}=\mathbb{w}+\mathbb{w}+\cdots  \]
\[ \mathbb{w}^3=\mathbb{w}^2+\mathbb{w}^2+\cdots  \]

(you must also remember that $3^2=3\times 3=3+3+3,4^2=4\times 4=4+4+4+4$ and $3^3=3^2\times 3$)

Although we can not write anymore,
we are formally able to get $\mathbb{w}^{\mathbb{w}},\mathbb{w}^{\mathbb{w}^{\mathbb{w}}}\cdots$ .

These are called an Ordinal number.
It is the expansion of the property of Natural number.

When you find out eternal infinite repetitions of an infinite number,
could you feel the depth of real number $\mathbb{R}$ ?