A crevice on the Crane Beach: Finite-degree predicates

First-order logic (FO) over words is shown to be equiexpressive with FO equipped with a restricted set of numerical predicates, namely the order, a binary predicate MSB
₀
, and the finite-degree predicates: FO[A
_RB
] = FO[≤, MSB
₀
, F
_IN
]. The Crane Beach Property (CBP), introduced more than a decade ago, is true of a logic if all the expressible languages admitting a neutral letter are regular. Although it is known that FO[A
_RB
] does not have the CBP, it is shown here that the (strong form of the) CBP holds for both FO[≤, F
_IN
] and FO[≤, MSB
₀
]. Thus FO[≤, F
_IN
] exhibits a form of locality and the CBP, and can still express a wide variety of languages, while being one simple predicate away from the expressive power of FO[A
_RB
]. The counting ability of FO[≤, F
_IN
] is studied as an application.