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This article gives a thorough overview of what is known about first-order logic with counting quantifiers and with arithmetic predicates. As a main theorem we show that Presburger arithmetic is closed under unary counting quantifiers. Precisely, this means that for every first-order formula ϕ(y,z) over the signature {x,z) which expresses over the structure 〈&U2115;,x is interpreted exactly by the number of possible interpretations of the variable y for which the formula ϕ(y,z) is satisfied. Applying this theorem, we obtain an easy proof of Ruhl's result that reachability (and similarly, connectivity) in finite graphs is not expressible in first-order logic with unary counting quantifiers and addition. Furthermore, the above result on Presburger arithmetic helps to show the failure of a particular version of the Crane Beach conjecture.