I'm wondering about the urelements in $\mathrm{NFU}$.
- How many of them are there?
- Can the urelements be collected into a set?
- Are the urelements, whether they form a set on their own or not, contained in the universal set?
The set theory NFU has a universal set, shown below.
$$ \{ x : x = x \} \; \text{is the universal set because the well-formed formula $x=x$ is stratifiable} $$
So, $x=x$ is always true, and therefore all sets at least are present in $\{x : x=x\}$, but I'm not sure whether the urelements are also included or not.
The set $ \{ x : \forall a \mathop. a \not\in x \} $ seems like it would contain the urelements and $\varnothing$.