Why is the class of all sets a stage? I want to prove that the class of all sets $\mathbb{S}=\{x \mid x=x \}$ is a stage (p. 15) (and then that it is a limit thus that it is the successor of another stage).
One way to do it is to proof that $$\mathbb{S} = acc(H(\mathbb{S}))$$ 
where $H(S)$ is the history (p. 15) of a class $S$ and
$$ acc(A) := \{x \mid \exists y \in A; \   x \in y \lor x \subseteq y \}.$$
I'm trying to figure out what the history of $\mathbb{S}$. Any hints on that? Is that even a good approach to proof that $\mathbb{S}$ is a stage?
 A: Let $\mathbb{H}$ be the class of all stages that are sets. We show that $\mathbb{H}$ is a history and $\text{acc}(\mathbb{H})=\mathbb{S}$, which shows that $\mathbb{S}$ is a stage. 
By Lemma 2.9 (c), every stage that is a set is hereditary and transitive. So to show that $\mathbb{H}$ is a history it suffices that for every stage $S$ sthat is a set, $S=\text{acc}(\mathbb{H}\cap S)$. But this follows directly from Lemma 2.9 (d). So $\mathbb{H}$ is a history.
We are now ready to show $\mathbb{S}=\text{acc}(\mathbb{H})$. Since $\text{acc}(\mathbb{H})$ is a class, we have trivially that $\text{acc}(\mathbb{H})\subseteq\mathbb{S}$. So let $a$ be any set. By the axiom of creation, there is a stage $S$ with $a\in S$. Hence $a\in\text{acc}(\mathbb{H})$ and  since $a$ was arbitrary, we have $\mathbb{S}=\text{acc}(\mathbb{H})$.
A: As Achim Blumensath already says on his p.11, $\mathbb{S}$, the class of all sets is not a set. So a fortiori it is not a stage (as a stage is a special kind of set). What you say you want to prove is indeed not provable in his (or any) version of Scott-Potter set theory!
[That's wrong: "or any" overshoots, as I was forgetting about the earlier version of Potter as @Michael Greinecker points out. But it is perhaps worth leaving the answer in place, as it at least will serve to point up a difference between Potter's smooth later version of Scott-Potter -- which I'd thought of as the "best buy" -- and Blumensath.] 
