A local normal form theorem for infinitary logic with unary quantifiers

We prove a local normal form theorem of the Gaifman type for the infinitary logic L∞ω(Qu) whose formulas involve arbitrary unary quantifiers but finite quantifier rank. We use a local EhrenfeuchtFräıssé type game similar to the one in [KL04]. A consequence is that every sentence of L∞ω(Qu) of quantifier rank n is equivalent to an infinite Boolean combination of sentences of the form (∃≥iy)ψ(y) where ψ(y) has counting quantifiers restricted to the (2n−1 − 1)-neighborhood of y.