The first four Peano axioms are:

1. 0 is a natural number

2. every natural number a has a successor sa

3. no natural number has 0 as its successor

4. a=b iff sa=sb.

Now there seems to be a problem here if I recall Johny's

model of the naturals we start with φ (the null set) being

in N, and

sφ = {φ}

and

sS = S U {S}.

Perhaps you will have better luck showing this construction satisfies

the first four axioms.

RonL