Successor-invariant first-order logic on finite structures

We consider successor-invariant first-order logic (FO+ succ)inv, consisting of sentences Φ involving an “auxiliary” binary relation S such that (A, S1) |= Φ ⇐⇒ (A, S2) |= Φ for all finite structures A and successor relations S1, S2 on A. A successorinvariant sentence Φ has a well-defined semantics on finite structures A with no given successor relation: one simply evaluates Φ on (A, S) for an arbitrary choice of successor relation S. In this article, we prove that (FO + succ)inv is more expressive on finite structures than first-order logic without a successor relation. This extends similar results for order-invariant logic [7] and epsilon-invariant logic [10]. §