In any model of PA, all natural numbers will be finite by this definition. Even if we, from outside of the model, know that many of them aren't. You've been pr

In any model of PA, all natural numbers will be finite by this definition. Even if we, from outside of the model, know that many of them aren't. You've been presented with two different ways to prove this, and a standard textbook. Perhaps read the textbook? On your aside, in PA there is one element that is not the successor of any other. And it likewise has no predecessor. (It is a matter of convention whether you call that special number 0 or 1.) That special number is always the base case of all induction proofs.