As far as I know, the definition of Inductive set is given by the Axiom of infinity:
A set $S$ is called inductive if and only if $\emptyset \in S$ and for ecah $x\in S$, $x\cup \{x\}\in S$.
With this definition, I can understand more or less what it is.
Now, let me introduce the definition of an inductive set given by my book. I need a previous definition (J.L. Kelley 141 p. 273).
Nest. A partial ordered class $N$ is called a nest if and only if for all $a,b\in N$, then $aRb$ or $bRa$.
Remark. Kelley introduces nests with the partial order $\subseteq$, so I suppose I must add the condition or $x=y$ to my previous definition. Also note that, a priori, this definition is the same of trichotomy of the partial order $R$.
Now, the definition of inductive set (not stated in Kelley's book):
Inductive class. A class $C$ is inductive if and only if and only if every nest $N\subseteq C$ has an upper bound.
I have two comments about that definition, each of them with one question:
1.- First, I think with this definition $\mathbb N$ is not an inductive set. Because if I set $N=\mathbb N$, then there isn't an upper bound. I'm right? This fact also conditions my second comment.
2.- Even when I were right in 1.- and $\mathbb N$ is not inductive, there is an equivalence between both definitions? If I'm right in 1. the definitions can't be equivalent. But you know, the fact $\mathbb N$ is not an inductive set may be a little problem. You can consider $\mathbb N\cup\{\infty\}$. Has the second definition more important (and unsolvable) problems? If not, can you show the equivalence?
Thanks.