let $A$ be any inductive set, then $\{C \in P(A)|C \text{ is inductive set} \}$ is a set? ... and $\mathbb{N}$...?

let $A$ be any inductive set, then $\{C \in P(A)|C \text{ is inductive
set} \}$ is a set? ... and $\mathbb{N}$...?

let $A$ be any inductive set, then $\{C \in P(A)|C \text{ is inductive
set} \}$ is a set?
if $\{C \in P(A)|C \text{ is inductive set} \}$ is a set I can defined
$\mathbb{N}:=\bigcap\{C \in P(A)|C \text{ is inductive set} \}$??
Thanks in advance!!
P.S.=$P(A)$ is power of (A)