A local normal form theorem for infinitary logic with unary quantifiers

摘要

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-Fraisse 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.