axiomatic sets 14 (a natural number)

Natural number is an ordinal.

You must know well,
the first=1,
the second=1+1=2,
the third=2+1=3,
the fourth=3+1=4,
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In addition, '0' is also a natural number.

In axiomatic set theory, those must be sets.

We have already defined
\[ 0=\phi=\left\{ \right\} . \]

As, if 'n' is a natural number,　then 'n+1' is also a natural number and an ordinal,
we will define
\[ n+1=n\cup\left\{ n\right\}=\left\{ n, \left\{ n\right\} \right\}  . \]

That is to say,
$1=0\cup\left\{ 0\right\}=\phi\cup\left\{ 0\right\}=\left\{ 0\right\}=\left\{ \phi \right\}=\left\{ \left\{ \right\}\right\}$ ,
$2=1\cup\left\{ 1\right\}=\left\{ 0\right\}\cup\left\{ \left\{ 0\right\} \right\}=\left\{ 0,\left\{  0\right\}\right\}$ ,
$3=2\cup\left\{ 2\right\}=\left\{ 0,\left\{  0\right\},\left\{ 0,\left\{  0\right\}\right\}  \right\}$ ,
$4=3\cup\left\{ 3\right\}=\left\{ 0,\left\{  0\right\},\left\{ 0,\left\{  0\right\}\right\},\left\{ 0,\left\{  0\right\},\left\{ 0,\left\{  0\right\}\right\}   \right\}  \right\}$ ,
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You will understand
$1=\left\{0 \right\}$ ,
$2=\left\{0,1 \right\}$ ,
$3=\left\{0,1,2  \right\}$ ,
$4=\left\{0,1,2,3  \right\}$ ,
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We are able to get all natural numbers.