Thinking about the definition of "inductive set", you'll find that there are lots of inductive sets, for example: the set of all real numbers, the set of positive real numbers, the set of integers, the set of rational numbers, and lots more. "Every inductive set" means all of these, not just the ones I listed but also all other sets that satisfy the definition.
Notice that $1$ is in all these sets --- because the definition of "inductive set" says that it as to be there. The (b) clause in the same definition (applied with $1$ as he value of $x$) then ensures that $2$ is in all of the inductive sets. Continuing this way, you can see that $3$, $4$, etc. are also forced to be in every inductive set. On the other hand, $0$ is only in some of the inductive sets, not in all of them (for example, not in the set of positive real numbers). Similarly, $1/2$ is in some but not all of the inductive sets. After thinking about more examples like these, you'll see that the positive integers are in all inductive sets, but all other numbers are in only some, not all, of the inductive sets.
That observation is what Apostol is using to define what he means by positive integer.