Well, it comes from the fact that apples in a bowl is Exclusively just that, as verified by your Bayesian reasoning. There are no other "chains" of successors (shadow apples? I can't even imagine a good metaphor).

So, now you in fact have that bowl of apples narrowed down to {0, S0, SS0, SSS0, ...} which is isomorphic to the natural numbers, so all other natural number properties will be reflected there.