Let the successor function be s(n) = the next power of 2 larger than n.

So, let's check that rules 1,2,5 hold:

(1) we have a first element (rule 1 is true)

(2) Each of these elements has a successor element in the set (s(1)=2, s(2)=4, s(4) = 8, etc, is a long chain, but also 3 is in the set and s(3) = 4. and s(5) = 8.

(5) induction holds.

But rule (3) is violated since s(2) = s(3) but 2 != 3.

And rule (4) is violated since 3 is not the successor of any element.

If you imagine N with a successor function of: s(n) = the next power of 2 larger than n, then we satisfy (1) and (2) but not (3) or (4). Unfortunately, (5) also doesn't seem to hold in this setting. (As pointed-out in the comments)