Every finite subset and its complement in $\mathbb{N}$ can be defined in $\mathfrak{N}_{S}$ (Exercise 4 of Section 3.1), and, hence, in $\mathfrak{N}_{A}$ . Therefore, by using disjunctions and conjunctions we can modify any finite number of points of a subset, and it is sufficient to show that every periodic set of natural numbers is definable in $\mathfrak{N}_{A}$ . But every periodic set $D$ with period $p$ is determined by memberships of the first $p$ natural numbers. Hence, if $D\cap\{0,\ldots,p−1\}=\{d_{1},\ldots,d_{k}\}$ , then $D$ is defined by $\exists x(v_{1}=d_{1}+\underbrace{x+\ldots+x}_{p}\vee\ldots\vee v_{1}=d_{k}+\underbrace{x+\ldots+x}_{p})$ .