# Give severeal examples for a set X such that each element of X is a subset of X.

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?

• In the usual set theory formalism, a number is a set. – WoolierThanThou Sep 26 '19 at 16:41
• I think in the book I was studying it was not the case that a number is a set. Or at least nothing like that was mentioned so far. It was Cantor's set theory. – zzlawlzz Sep 26 '19 at 16:46
• Well, if a number isn't a set, then what is it? The reason for the formalism is that you can encode the mathematical structures you'd like to consider using only sets. – WoolierThanThou Sep 26 '19 at 16:47
• Your analysis is working its way toward what's known as the Axiom of Foundation: $\forall y \exists x \in y ~(x \cap y = \emptyset).$ – Robert Shore Sep 26 '19 at 17:01

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 $$S\cup\{S\}$$.
• @RüzgarAyan A transitive set not reachable by the method I described is $\{ \{\{\{\{\emptyset\}\}\}\}, \{\{\{\emptyset\}\}\}, \{\{\emptyset\}\},\{\emptyset\},\emptyset\}$, but that too has nothing but emptyset at its "bones" - necessarily by the OP argument – Hagen von Eitzen Sep 26 '19 at 16:49