A few times while studying math I have encountered the notion of a "minimal set". For example, given some set of subsets, what is the "minimal" sigma algebra generated by that set of subsets? Or, in the example I am currently studying: one of the ZFC axioms ensures the existence of an inductive set. Therefore, we define the natural numbers to be the minimal inductive set. I am trying to think of some other examples, but I can't remember off the top of my head. Feel free to comment if you think of some other examples of minimal sets.
Here is my question. I have seen the "minimal" set defined different ways, and I am not sure which statement is definition and which is an implication of the definition.
The first definition I have seen is to call a set $M$ satisfying some property minimal if, for every other set $A$ satisfying the same property as the set $M$, we have $M \subseteq A$: \begin{equation*} M \textrm{ is the minimal set with property } P \iff (\forall A \textrm{ satisfying property P }) M \subseteq A \end{equation*}
The second definition I have seen is to define the minimal set $M$ satisfying some property to be the intersection of all other sets satisfying the same property: \begin{equation*} M \textrm{ is the minimal set in some class } C \iff M=\bigcap C \end{equation*}
The second definition seems more concise, however in the example I am studying right now (defining the set of natural numbers to be the minimal inductive set), I don't know if such a set $C$ (the set of all inductive sets) exists, so I am not even sure if the right hand side of definition 2 even makes any formal logical sense.
Thanks for reading!
The first definition is the more general one (and, as has been said in the comment, can be generalized to arbitrary partial orders besides $\subseteq$). The second definition is not always correct. However, if the class $C$ is nonempty has the property that an arbitrary intersection of members of $C$ is again in $C$, then the second definition is equivalent to the first definition.
Regarding your final problem about a minimal inductive set, note that $C$ need only be a nonempty class, not necessarily a set. In case you worry that $\bigcap $ is only definied for sets, not for (proper) classes of sets: No, the definition $$\tag1\bigcap C:=\left\{\,x\mid \forall c\in C\colon x\in c\,\right\} $$ is perfectly fine and defines a set for any nonempty(!) class $C$ though admittedly $(1)$ uses class builder, not set builder notation. But let $S\in C$ be an arbitrary set and define $$\tag2\bigcap C:=\left\{\,x\in S\mid \forall c\in C\colon x\in c\,\right\}, $$ then the result does not depend on the choice of $S$ (why?) and as $(2)$ is an instance of the Axiom Schema of Comprehension, this shows that $\bigcap C$ is a set. (Then finally, as $C$ is closed under arbitrary intersection, we see that $\bigcap C$ is again an element of $C$ and surely the mnimal element in the sense of the first definition; If $C$ denotes the class of inductoive sets, then the usual formulation of the Axiom of Infnity can be rephrased as simply: $C$ is not empty - which is precisely what we need)