in "An Intro to Complex Analysis and Geometry, John P. D'Angelo', It has the following:
Definition 3.4
A subset S of R is called inductive if whenever x∈S, then x+1 ∈ S.Definition 3.5
The set of natural number N is the intersection of all inductive subsets of R that contains 1.
I'm a little confused why the intersection of all inductive subsets of R that contains 1 is not the set of integers Z?
surely `... -3, -2, -1, 0, 1 , 2 ... is the one and only set that is the intersection of all inductive subsets of R that contains 1?
The set $\Bbb N=\{1,2,3,\ldots\}$ is an inductive set and $\Bbb N\varsubsetneq\Bbb Z$. Therefore, $\Bbb Z$ cannot be the intersection of all inductive sets.