No...?

The above basically says that P(k) is "is within the successor chain of 0". Note that the base case is k = 0, not k = o. Anyway, the point is, since such a property is possible (including only the objects that are some n-successor of 0), the axiom of induction implies that numbers that follow 0 are the only numbers.

ETA: Reading your parent post again, the problem is it's impossible to have an o that has every property 0 does. As a demonstration, Z(k) := (k = 0) is a valid property. It's true only of 0. R(k, 0) is similarly a property that is only true of 0, or SS..0.