Reading through Hodges' "A Shorter Model Theory", he gives the following symbolism (pgs. 23-25) for the first-order language constructed in the normal way with only finitely many formulas conjoined/disjoined together and only finitely many quantifiers in a row: $L_{\omega \omega}$. The first subscript is, from what I can gather, meant as a strict upperbound for conjunction/disjunction, while the second is meant as a strict upperbound on quantification (I know that doesn't quite make sense--hence the question). The notation is also meant to generalize in the obvious way to $L_{\alpha \beta}$.
Why is Hodges using the ordinal $\omega$ rather than, say, the cardinal $\aleph_0$? Is the ordering he has in mind simply from left to right? If so, how can this be meaningful given that, for example, his disjunction/conjunction operations are over sets of formulas (and thus seem to be without an ordering)?
As far as I know, the only reasons for writing $\omega$ rather than $\aleph_0$ in this context are (1) tradition and (2) avoiding second-level subscripts (for a while). In particular, if conjunctions, disjunctions, and quantifier blocks are given as sets rather than sequences, then the $\omega$ here and the $\alpha$ and $\beta$ in similar contexts should be understood as strict upper bounds for cardinalities. No orderings should be involved.