Meaning of set as an element of another set So I understand that sets can be elements of another sets, for example: S={1,3,5,{12}} In this case, the elements of S are 1, 3, 5 and {12}. But 12 doesn't belong to S. I get that, but what is the meaning of that? If the set only containing 12 is an element of S, why isn't 12 an element of S? Is there any situation where this makes a difference?
 A: Lets first check what $a \subseteq b$ means in Zermelo-Frankael set theory (the standard foundations), shall we?
$$(a \subseteq b) \;\; :\iff \; \forall x \; (x \in a \; \to \; x \in b)$$
Now, we look at $\{12\} \subseteq \mathbf{S}$, does that imply that $12 \in \mathbf{S}$ in the first place? Well, we've to understand that saying $x \subset y$ is not the same as $x \in y$. One implies that every element of $x$ is an element of $y$, and the other implies that $x$ itself is in $y$.
We see that $\{12\} \in \mathbf{S}$ here, and that $1,3,5 \in \mathbf{S}$. We also see that if $a = \{1,3,5, \{12\}\}$ then $a \subseteq \mathbf{S}$. We also see that if $b = \{1,3,5\}$, then $b \subseteq \mathbf{S}$.
Put another way. If $x = \{1,\{1\}\}$, then it is both true that $1 \in x$, and that $\{1\} \subseteq x$, and also, that $\{1\} \in x$. If we say that $x = \{\{1\}\}$, then $\{1\} \subseteq x$ won't be true. Remember that every object in ZF (Zermelo-Frankael set theory) is treated as a set, and so $\{1\}$ is not just a set but a set within a set, which means that $1$ here is also treated as a set. Some other set theories have things called urelements which basically allow for the conception of objects like "elements" which are not sets but maybe elements of sets themselves.
