Consider sets $A$, $B$ and $C$. $A$ is being contained in $C$, that is $A \subseteq C$.
Consider a function $f(x)$, such that $f:C \rightarrow C$.
Set $B$ is inductive if:
Now, what is the intersection of all inductive sets? It seems to me that a question like that is meaningless, since there can only be one inductive set. In other words, there is only one set that contains $A$ and is closed under $f(x)$. Am I correct?
On
No, there can be many inductive sets. For instance, the ordinal $\omega+\omega$ contains the empty set and is closed under successor operation. However the smallest such set is $\omega$ (and this is the intersection of all inductive sets relative to the empty set and the successor operation).
Your intuition for why there’s s unique one is probably just the intuition for why there’s a minimal one.
No, because $B$ needs to include $A$ but it can contain other elements too. Starting from $A$ and applying $f(x)$ to get new elements can lead to one inductive set, but if we start from $B$ that contains elements of $A$ but other elements too, by applying $f(x)$ we may get another inductive set.
Intersection of all these inductive sets should give us the smallest inductive set, the one that starts only from $A$.