# Is infinitary Levy hierarchy well-defined?

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?

• @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? – Thomas Benjamin Aug 3 '14 at 12:03