von Neumann Integer

The von Neumann integers represent natural numbers and are constructed using the set notation. They are defined recursively as follows:

  0 := {} (the empty set)
  N+1 := N  {N}

The smallest von Neumann integer zero is defined as the empty set. The von Neumann integer N is defined as the finite set with N elements which are the von Neumann integers 0 to N-1.

0 := {}
1 := 0  {0} = {0} = {{}}
2 := 1  {1} = {0,1} = {{},{{}}}
3 := 2  {2} = {0,1,2} = {{},{{}},{{},{{}}}}

N+1 := N  {N} = {0,1,2,...,N}

The set of von Neumann integers is infinite though every its element N is finite.

The von Neumann integer grows exponentially, for N>0 it has 5*2^N-1-1 characters (braces and commas).

Check the von Neumann integer generator or the von Neumann integer test tool.