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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?

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    I have never heard the term "stage" in the context of set theory. Nor the term "history".2012-10-28
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    I was trying to avoid all the definitions in my question. Do you think it is okay to give references when aksing a question which contains "special" definitions?2012-10-28
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    @joachim When the definitions are special, it is necessary. Sounds like [Scott-Potter set theory](http://en.wikipedia.org/wiki/Scott%E2%80%93Potter_set_theory).2012-10-28
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    I guess that "all the world's a stage"2012-10-28
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    On p.19 there is an exercise to prove that "$\mathbb{S}$ is a stage with history $H (\mathbb{S}) = \{ S : S \text{ is a stage} \}$."2012-10-28
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    Is this the same $\mathbb{S}$? It is the union of all stages not the class of all sets, isn't it?2012-10-28
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    So... all those stages are the von Neumann hierarchy? And essentially you want to show that all the sets are generated by this hierarchy.2012-10-28
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    joachim: I sense your last comment was directed at me. I think once you have to so-called "Axiom of Creation" (also p.19) there isn't a difference. The author mentions that without this axiom one cannot prove that every set is in a stage (or, rather, one cannot prove that stages exist), and it would seem that the desired result may not hold.2012-10-28
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    @Asaf: In my reading, Yes. It seems to be a way to get the von Neumann hierarchy without introducing ordinals, but with the expense of requiring additional axioms.2012-10-28

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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 $\mathbb{S}=\text{acc}(\mathbb{H})$.

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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.]

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    p. 19 bottom: $\mathbb{S}$ is the only stage that is a proper class2012-10-28
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    @PeterSmith In Potter's previous book *Sets: An Introduction*, he extends the cumulative hierarchy beyond the class of all sets.2012-10-28
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    I stand corrected! Compare Potter's canonical *Set Theory and Its Philosophy*. He talks of levels rather than stages, but with the same definition (I think). And in his axiomatization now, for every level there is a later one -- which rules out the universe as counting as a level. [The surplus in the early version was a sop to category theorists, who didn't appreciate it, so it is not there in the later version. If I recall!]2012-10-28