Adding urelements to the Von Neumann hierarchy A very simple question: suppose we would like to generate something like the usual Von Neumann hierarchy, but where we also add in some set of $n \in \Bbb N$ different urelements. Let's call these urelements $x_1, x_2, ..., x_n$ and say that $X$ is the set of all urelements.
With the usual Von Neumann heirarchy, we start with $V_0 = \{ \}$, and then for successor ordinals we have $V_{n+1} = P(V_n)$. However, if we naively do that here, starting with $V_0 = X$ instead, we don't naturally have $V_n \subset V_{n+1}$. We could perhaps modify this to say that $V_{n+1} = V_n \cup P(V_n)$. This gives the following:
$$
V_0 = X\\
V_1 = V_0 \cup P(V_0) \\
V_2 = V_1 \cup P(V_1) \\
... \\
V_{\alpha+1} = V_\alpha \cup P(V_\alpha) \\
... \\
V_{\lambda} = \bigcup V_{\alpha < \lambda}
$$
with $\alpha$ a successor ordinal and $\lambda$ a limit ordinal.
My only question: is this the correct, standard way of doing this? I would appreciate any references on what the equivalent of $V$ would be with a small set of urelements and how this is usually done.
 A: Yes, this is the correct, standard way of doing this. See Jech (either the beginning of chap 4 of the AC book or the end of chap 15 in the big book).
A: Your method does not actually build “urelements” at all. $X$ will eventually occur in the ordinary hierarchy at some stage $V_\alpha$, assuming we’re working in ZF in the metatheory. Your method can only work if every element of $X$ already lies outside the ordinary cumulative hierarchy, which cannot be the case in ZF (unless $n = 0$).
Let us call your version of the hierarchy $V’$. Then it is easy to show by ordinal induction that $V_\beta \subseteq V’_\beta \subseteq V_{\alpha + \beta}$ for all $\beta$, thus proving that the two hierarchies coincide.
In order to actually produce a model of ZFA (ZF with atoms/urelements), we need to define the hierarchy a little differently. Let $atom(x) = (0, x)$, and let $set(S) = (1, S)$. The idea is that everything in our universe $A$ will be either of the form $atom(x)$ for some $x \in A$, or will be of the form $set(S)$ for some set $S \subseteq A$. And for $a, b \in A$, we have $a \in_A b$ is defined as $\exists S (a \in S \land b = set(S))$. This ensures that we have no overlap between our urelements and our sets. The element $set(\{atom(x) \mid x \in X\})$ is the set of atoms.
Then we define $A_\alpha = \{set(S) \mid S \in \bigcup\limits_{\beta < \alpha} P(A_\beta)\} \cup \{atom(x) \mid x \in X\}$. For convenience, let $Atoms = \{atom(x) \mid x \in X\}$. Then we have $A_0 = Atoms$, $A_{\alpha + 1} = \{set(S) \mid S \subseteq A_\alpha\} \cup Atoms$, and $A_\beta = \bigcup\limits_{\kappa < \beta} A_\kappa$ for $\beta$ a limit ordinal.
We can then set $A = \bigcup\limits_{\alpha} A_\alpha$.
