It always felt arbitrary to me :

    next(x)={x} would give 

    1={0}
    2={{0}}
    3={{{0}}}

    Kuratowski's encoding gives :
    0=Ø=()
    1={Ø}=(0)
    2={Ø,{Ø}}=(0,1)
    3={Ø,{Ø},{Ø,{Ø}}}=(0,1,2)
    
    The cardinal of N is n and
    every element in N are the predecessors of n.

    Von Neuman's encoding gives :
    0=Ø
    1=0U{0}={Ø,{Ø}}
    2=1U{1}={Ø,{Ø},{Ø,{Ø}}}
   
    Now the cardinal of N is n+1, and n is the maximum
    of the set N defining n.

    Both Von Neuman's and Kuratowski's encoding allows us to define ordered tuples, but I cannot understand how to write the tuples for Von Neuman's in the context of natural numbers.

2 is {Ø,{Ø},{Ø,{Ø}}} with Von Neuman's we can recognize 0 and 1 as the first and second element of the tuple : what is the third one ?