# Epsilon induction

I recently came upon this technique called epsilon induction, and was searching for some proof using the same. But I saw no such proof. Does someone know of any proof using this technique?

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Exercise for you: prove that every set has a well-defined rank. Next, show that epsilon induction and well-foundedness are equivalent. –  Zhen Lin May 1 '12 at 17:05
You can use epsilon-induction to prove that every set belongs to the von Neumann hierarchy. For ordinals $\xi$ define $V_\eta$ in the usual way: $V_0=0, V_{\eta+1}=\wp(V_\eta)$, and $V_\eta=\bigcup_{\xi<\eta}V_\xi$ for limit $\eta$. Let $x$ be any set, and suppose that $\forall y\in x\,\exists\xi\in\mathbf{ON}\,(y\in V_\xi)$. For $y\in x$ let $\xi(y)$ be the least ordinal such that $y\in V_{\xi(y)}$, and let $\eta=\sup\{\xi(y):y\in x\}$. (We can do this using Replacement.) Then $x\subseteq V_\eta$, so $x\in V_{\eta+1}$. It now follows immediately by epsilon-induction that every set is in some $V_\xi$.