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)ω whose formulas involve arbitrary unary quantifiers but finite quantifier rank. We use a local Ehrenfeucht-Fraisse type game similar to the one in [9]. A consequence is that every sentence of L∞ω(Qu)ω 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. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)