Foundation of the natural numbers

At last I found a more in depth explanation of the natural numbers. In this idea numbers are labels assigned to sets and adding 1 to a number represents a map between two sets. If we would analyze the mathematics of a hypothetical alien civilization we would see that they have natural numbers too, they just used different labels in their definition.

Mathematically it roughly works like the following:

Let A be the collection of all sets.

Let f be a map
 f:A->A
 a: |-> a U {a}

Assign label '0' to Emptyset
If a has label 'x' then assign label 'x+1' to f[a]

Label a f[a]  a U {a} 
--------------------- 
0 Empty 

1  f[Empty] Empty U {Empty}
    = {Empty}
    = {0}

2 {0} f[{0}]  {0} U {{0}}
    = {0, {0}}
    = {0, 1}

3 {0,1} f[{0,1}] {0,1} U {{0,1}}
    ={0,1,{0,1}}
    ={0,1,2} 

More later. ( Confirmed as an article in the NT Wiki. )