Being a member of a set in itself

Question

Hello to all math lovers. I was studying Russell's paradox in set theory and came across something ambiguous that I couldn't justify no matter how hard I tried. The strange thing was: sets that are members of themselves and sets that are not members of themselves. This statement threw me thousands of meters away from the paradox itself. What does Russell mean by this? We know that when something is a part of a set, we say it is a member of that set. But how can a set be a part of itself or not? What unique characteristic makes a set a member of itself or not? I don't understand the concept of being a member of a set in itself. I would appreciate your guidance. Thank you, a math enthusiast :)

Answer 1

The first step that might help is to start thinking about sets that contain other sets. For example, the set $\{\{1, 2\}, \{3, 4\}\}$ has two elements - the set $\{1, 2\}$ and the set $\{3, 4\}$.

After that, you can start thinking about more complicated nesting of sets. For example, we can start with the empty set $\varnothing = \{\}$ and then look at the set that contains just the empty set $\{\varnothing\}$. Then let's make a set that contains those two sets, $\{\varnothing, \{\varnothing\}\}$. And we can keep going, putting those three sets in a set together to get $\{\varnothing, \{\varnothing\}, \{\varnothing, \{\varnothing\}\}\}$. You can even take that to infinitely many sets, giving you something that looks like: $\{\varnothing, \{\varnothing\}, \{\varnothing, \{\varnothing\}\}, \ldots\}$.

So let's consider a set $U$. And this is going to be a special set, because $U$ is a set that contains every set. Name a set, it's an element of $U$. The empty set? That's in $U$. The set of natural numbers? It's in there. The set containing the set containing the empty set? Sure. So is $U$ an element of $U$? Well, $U$ is a set, and every set is an element of $U$, so therefore $U$ must be an element, and hence $U$ is a set that contains itself.