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The well-known Levy hierarchy of formulas consist of two $\omega$-sequences of sets of formulas of different complexity $\langle\langle \Sigma_n:n\in \omega\rangle,\langle \Pi_n:n\in \omega\rangle\rangle$.

Question: Is it possible to define an "ordinal valued Levy hierarchy" $\langle\langle \Sigma_{\beta}:\beta\in \alpha\rangle,\langle \Pi_{\beta}:\beta\in \alpha\rangle\rangle$ for a given ordinal number $\alpha$ in an infinitary logic appropriately?

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It is well-known that the Levy hierarchy can be extended to set theories in infinitary languages (see K. Gloede's paper of the same title, "Set Theory in Infinitary Languages", Lecture Notes in Mathematics, Volume 499, pp 311-362, in particular pp 322-323, which specifically talks about the Levy hierarchy in such languages). Perhaps this will help you to answer your question.

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  • $\begingroup$ @Haim: since the n in $\Sigma_n$, $\Pi_n$, and $\Delta_n$ is just the n'th iteration of blocks of $\exists$ and $\forall$ quantifiers in the Levy hierarchy, when one has a set theory (i.e. ZFC) over the infinitary language $\mathcal{L}_{\kappa\lambda}$, $\kappa$ $\le$ $\lambda$, n can be an infinite ordinal. Does this answer your question? $\endgroup$ – Thomas Benjamin Aug 3 '14 at 12:03

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