definition of $\mathbb{N}:=\bigcap Ind$ \--- let $A$ a set, $A^+=A \cup \\{A\\}$ \--- let $B$ a set, B is inductive if $\emptyset \in B \wedge \forall A \in B(A^+ \in B )$ \--- let $Ind:=\\{C|C \text{ is inductive }\\}$ is correct this definition: $\mathbb{N}:=\bigcap Ind $, with $\bigcap Ind=\\{x|\forall C \in Ind (x \in C) \\}$ ???? Thanks in advance!!

I don't think your definition is correct. I don't know how to give a meaning to the formula $\bigcap \text{Ind}$ if $\text{Ind}$ is not a set, which it isn't.

A solution is to take inductive set, let's call it $I$, (it exists) and to consider the **set** $\Bbb N:=\\{x\in I\colon \forall y(y \text{ is inductive}\longrightarrow x\in y)\\}$, (it is indeed a set due to the Axiom schema of specification). This set is what one would end up with if one could consider an entity such as $\bigcap \text{Ind}$.