While reading Munkres' Topology section about integers and reals (Chapter 1, Section 4), he defines the set $\mathbb{Z}_+$ as:
Definition: A subset $A$ of the real numbers is said to be inductive if it contains the number 1, and if for every $x$ in $A$, the number $x+1$ is also in $A$. Let $\cal{A}$ be the collection of all inductive subsets of $\mathbb{R}$. Then the set $\mathbb{Z}_+$ of positive integers is defined by the equation $\bigcap _{A \in \cal{A}} A$
Isn't it possible to define $\mathbb{Z}_+$ as the set defined this way:
$1\in B \wedge (j \in B, n-1 = j) \Rightarrow n\in B$
Then $B = \mathbb{Z}_+$