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.
Best Answer
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).