I am reading Boolos' famous article "The iterative conception of set" from Benacerraff and Putnam's collection of papers "Philosophy of Mathematics".
At page 493 (2nd edition) Boolos says that: "$\{\emptyset\}$ and $\{\{\emptyset\}\}$ are both pure sets and are formed at stages 1 and 2, repectively".
What sounds weird to me is that in the Von Neumann Universe (henceforth $V$) the set $\{\{\emptyset\}\}$ is formed at stage 3. I think I am missing the reason for this difference and hence I don't know if it has relevant consequences.
What strikes me in particular is that a nice characteristic of $V$ is that every ordinal has a rank equal to itself. That is, obviously, if we stick to Von Neumann definition of ordinal number.
But, using Boolos' stage theory (assuming it is something intelligibly distinguishable from $V$, of which I am not sure and hence this question) and Zermelo's definition of ordinal ($0=\emptyset, 1=\{\emptyset\}, 2=\{\{\emptyset\}\}, 3=\{\{\{\emptyset\}\}\}$, and so on) it would seem that Boolos could keep this nice feature while changing the overall structure.
So, in no particular order whatsoever: is there a difference between $V$ and Boolos' stage theory? If yes, what determines it and what are the consequences? Is the thing about the ordinal being equal to its rank relevant in any way?