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 ( Q u ) ω whose formulas involve arbitrary unary quantifiers but finite quantifier rank. We use a local Ehrenfeucht‐Fraïssé type game similar to the one in [9]. A consequence is that every sentence of L ∞ ω ( Q u ) ω of quantifier rank n is equivalent to an infinite Boolean combination of sentences of the form (∃ ≥ i y ) ψ ( y ), where ψ ( y ) has counting quantifiers restricted to the (2 n –1 – 1)‐neighborhood of y . (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)