So I have a question:
Let: Allow set B = {x: x $\notin$ x}. Then, B $\in$ B $\iff$ B $\notin$ B ? Does this apply for the zero set?
Because I'm a bit confused. The definition is a zero set is always in the zero set. But then if $\emptyset$ $\in$ $\emptyset$, then $\emptyset$ $\in$ $\emptyset$ $\iff$ $\emptyset$ $\notin$ $\emptyset$.
Is there a mistake I'm making? Something doesn't feel right here. Sorry if it's a confusing question. (I don't know if this has been posted before?)
Also:
Let C = {$\emptyset$}. Is there an element? Because if there's an element, what is that element? Is it the empty set? If it's the empty set, then there's no set there is there? It's nothing in the set in there. But there's a set in there. (Please explain this also). Can somebody also explain if I placed the logic of Russell's Paradox in this set with empty set?
Thank you for the help. (Thank you everybody) ありがとう 皆さん
By definition no set is an element of $\varnothing$, in particular itself. So it is not true that $\varnothing\in\varnothing\iff\varnothing\notin\varnothing$.
For the second part you're confusing "being empty" and "non-existing". If my wallet is empty, it doesn't mean that I don't have a wallet. The empty set is just a set which has no elements, it doesn't mean that it doesn't exist. $\{\varnothing\}$ has a single element, and that element is the empty set. It doesn't mean that there are no elements there, since sets can be elements of other sets as well.
In any case, the Russell paradox comes to show that not every "property" defines a set. In particular the set $\{x\mid x\notin x\}$ cannot exist. That's all that it shows. It says nothing about $\varnothing$ or otherwise.