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Monadic Logic of Order over Naturals has no Finite Base

A major result concerning Temporal Logics (TL) is Kamp’s theorem [Kam68, GHR94], which states that the temporal logic over the pair of modalities X until Y and Xsince Y is expressively complete for the first order fragment of monadic logic of order over the natural numbers. We show that there is no finite set of modalities B such that the temporal logic over B and monadic logic of order have the same expressive power over the natural numbers. As a consequence of our proof, we obtain that there is no finite base temporal logic which is expressively complete for the μ-calculus.