9
The Proof
•Idea: recall that we can express EVEN in MSO(<).

Let S(x,y) =  :(9 z.x<z Æ z<y) Æ x < y
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•We can’t say 9 P in FO.  Instead, we will use a vocabulary s that allows to assert the existence of all 2n sets P.
9 P.(
    8 x.y.(S(x,y) ) (P(x)Æ: P(y) Ç : P(x)ÆP(y))
 Æ P(min)
 Æ : P(max))
EVEN=