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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!

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    $\begingroup$ The two are equivalent (assuming closure under intersections), so you can treat either as the definition as you prefer. $\endgroup$ – Qudit Mar 15 '15 at 20:45
  • $\begingroup$ Thanks for your comment. I will try to prove that these two are equivalent. Typically, however, which statement is usually treated as the primary definition? $\endgroup$ – Mathemanic Mar 15 '15 at 20:47
  • $\begingroup$ The first definition is generalizable to orders other than $\subseteq$, so it's better. $\endgroup$ – Git Gud Mar 15 '15 at 20:47
  • $\begingroup$ @EthanAlvaree I'm not sure. I've definitely seen both as well. $\endgroup$ – Qudit Mar 15 '15 at 20:49
  • $\begingroup$ @GitGud, I'm not sure what you mean by generalizable to orders other than $\subseteq$. Can you please explain more? $\endgroup$ – Mathemanic Mar 15 '15 at 20:50
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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)

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  • $\begingroup$ Yes, I was worried about $\bigcap C$ being defined in set theory when $C$ is not a set. I think I understand, but please tell me if this is right. So $\bigcap C$ in both definitions $(1)$ and $(2)$ are equal to each other? But definition $(2)$ is preferred because it represents how we actually use the ZFC axioms to ensure existence of the minimal inductive set, correct? For instance, your set $S$ is the set whose existence is provided by the axiom of infinity, correct? I think I understand, but please let me know if I got this right. Thanks for your great answer! $\endgroup$ – Mathemanic Mar 15 '15 at 21:15
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Given a set $S$ of sets you can consider the containment relation between sets. This relation is what is called a partial order, and makes $S$ into a poset. A minimal element $s\in S$ is an element that is not properly contained by another element in $S$. Of course when $S$ is closed under arbitrary intersections the minimal element of $S$ is the intersection of all the elements of $S$.

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