When, if ever, are subsets elements?

Question

A very basic question on terminology that I have not seen addressed despite some searching: are any sets (such as subsets) considered elements? My understanding is that the empty set is a subset of all sets and it is not considered an element; is that because it is empty or because it is a set?

Answer 1

There are sets that have the emptyset as element, for example $\{\emptyset\}$ or $\{\emptyset,\text{dog},\text{cat}\}$. And $\{\emptyset,\{\emptyset\}\}$ has both the empty set and my first example as element. Hoever, $\{\{\emptyset\}\}$ does not have the empty set as element; instead it has a single element and that has the empty set as element. We can build up arbitrarily convoluted nestings like this.