Weak Monadic Second Order Theory of Successor is not Element-recurive

Abstract

Let L SIS be the set of formulas expressible in a week monadic second order logic using only the predicates [x =y+1] and [x E z]. Bucci and Elgot [3,4] have shown that the truth of sentences in L SIS (under the standard interpretation < N, successor > with second order variables interpreted as ranging over finite sets) is decidable. We refer to the true sentences in L SIS as WSIS. We shall prove that WSIS is not elementary-recursive in the sense of Kalmar. In fact, we claim a stronger result: