I recently started studying set theory and I have seen this exercise in the textbook. The only example I could think of was the empty set. This is how I think:
A number(for example 1), cannot be a subset of X since a number is not a set. So I need to have sets as the elements of X.
Then, I choose a element like $\{1\}$ for example. But then X must have the subset $\{1\}$, which means it has 1 as an element. This also cannot happen because of what I said before.
And even if choose an element of X as something like $\{\{\{\{1\}\}\}\}$, it will require $\{\{\{1\}\}\}$ to be an element of the set. Which then will require $\{\{1\}\}$ and eventually I come back to what I started with.
Can you show me what is wrong in the way I think of this problem, and can you show some examples of these kinds of sets?
Best Answer
Your argument is right that there must not be any "urelements" hidden anywhere inside the set. But here are some examples: $$\emptyset,\qquad\{\emptyset\},\qquad \{\emptyset,\{\emptyset\}\},\qquad \{\emptyset,\{\emptyset\},\{\emptyset,\{\emptyset\}\}\} $$ and many more.
Note that if $S$ is a set with the desired property, then so is $S\cup\{S\}$.