Logical Theory of the Additive Monoid of Subsets of Natural Integers

Abstract

We consider the logical theory of the monoid of subsets of ℕ endowed solely with addition lifted to sets: no other set theoretical predicate or function, no constant (contrarily to previous work by J̇ez and Okhotin cited below). We prove that the class of true Σ₅ formulas is undecidable and that the whole theory is recursively isomorphic to second-order arithmetic. Also, each ultimately periodic set A (viewed as a predicate X = A) is Π₄ definable and their collection is Σ₆. Though these undecidability results are not surprising, they involve technical difficulties witnessed by the following facts: 1) no elementary predicate or operation on sets (inclusion, union, intersection, complementation, adjunction of 0) is definable, 2) The class of subsemigroups is not definable though that of submonoids is easily definable. To get our results, we code integers by a Π₃ definable class of submonoids and arithmetic operations on ℕ by Δ₅ operations on this class.