So according to the above explanation, any set can be inductive as long as it contains the positive integers? It means the set of rational numbers is inductive only and only because it contains all the positive integers. Also, in the definiton, that "$x$" has to be an integer or it can be any number in the given set?
What is the use of the inductive property of rational numbers? I know for the set of positive integers, PMI is the best example. But, what about rational numbers?
Please help!
No, it says that any inductive set must include the positive integers. It does not say that any set that includes the positive integers is inductive. For example, $\{\frac 12\} \cup \Bbb N$ is not inductive because $\frac 12+1$ is not a member of the set. Yes, the rationals are inductive because they are closed under addition of $1$. Similarly, the integers are inductive. The set of naturals union the naturals plus $\frac 13$ is also inductive.