1. ## Von Neumann Numbers

here's the problem:
Consider the following inductively defined function:
s(0):= 0 ; s(S):= s(S) U S
Prove with an appropriate interpretation of 0 and s that the Peano axioms 1-4 hold for your interpretation.
thanks a lot for ur help !!

ps: i think i'm supposed to prove this by induction

2. Originally Posted by yaszine
here's the problem:
Consider the following inductively defined function:
s(0):= 0 ; s(S):= s(S) U S
Prove with an appropriate interpretation of 0 and s that the Peano axioms 1-4 hold for your interpretation.
thanks a lot for ur help !!

ps: i think i'm supposed to prove this by induction
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