In Apostol's «Calculus I», on page 22 there is the following definition:
A set of real numbers is called an inductive set if it has the following two properties:
(a) The number 1 is in the set
(b) For every x in the set, the number x + 1 is also in the set.
Next there is a definition of positive integers:
A real number is called a positive integer if it belongs to every inductive set.
So my question is what is meant by «every inductive set». I don't quite understand this definition.
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.
The statement "every inductive set" means the set of all sets that are inductive, i.e. $S:=\{ A \subseteq \mathbb{R} : A \text{ is inductive}\}$. Note that $S$ is nonempty since $\mathbb{R}$ is clearly inductive. Hence $\cap S$ makes sense and we use this to define $\mathbb{N}:= \cap S$.
Note that this is a "top down" approach to constucting $\mathbb{N}$. Apostol is assuming the existence of $\mathbb{R}$ and using it to construct $\mathbb{N}$ because this is easy and useful for the study of real analysis. There is a separate approach which I would call "bottom up" which comes from a more set theoretic standpoint. Assuming the existence of an inductive set (inductive in a slightly different meaning in this case) being one of the axioms of ZF. This gives us immediately $\mathbb{N}$ and we then work tediously to construct $\mathbb{Z}$ then $\mathbb{Q}$, then $\mathbb{R}$. These constructions are good to know, but don't have much to do with elementary real analysis, so the top down approach is usually taken when studying real analysis.
You have a definition of an inductive set, which in this context - it seems - talks about subsets of $\Bbb R$.
If $x$ is such number that whenever $A$ is an inductive set, $x\in A$ -- then we say that $x$ is a positive integer.
