As suggested on page 192, we use the fact that a subset $A$ is definable in $\mathfrak{N}_{S}$ iff there is a quantifier-free formula $\phi$ that defines it. In fact, $\phi$ can be further assumed to be in its disjunctive normal form. Each atomic formula is of the form $\mathbb{S}^{m}t=\mathbb{S}^{n}u$ , where $t$ and $u$ are either $v_{1}$ or $0$ . If $t=u$ , then, if $m=n$ , the atomic formula is always satisfied, otherwise, using axioms S2 and S4, the atomic formula is never satisfied. Thus, if $t=u$ , the atomic formula defines either $\emptyset$ or $\mathbb{N}$ . If $t\neq u$ , then, without loss of generality, it is $\mathbb{S}^{m}v_{1}=\mathbb{S}^{n}0$ . If $m>n$ , then, using axioms S2 and S1, we conclude that the atomic formula is never satisfied. If $m\le n$ , then, using axiom S2, $A_{S}\vDash\mathbb{S}^{m}v_{1}=\mathbb{S}^{n}0\leftrightarrow v_{1}=\mathbb{S}^{n-m}0$ . In the first case, the atomic formula defines the empty set, and in the second case, the atomic formula defines a singleton. Overall, an atomic formula can define a subset containing 0, 1 or all points only (and every such subset can be defined in the standard model of $\mathfrak{N}_{S}$ , using the fact that for every $n\in\mathbb{N}$ , $\{n\}$ is defined by $v_{1}=\mathbb{S}^{n}0$ ). Therefore, the negation of an atomic formula can define any subset containing 0, all or all but one points and only such subsets of $\mathbb{N}$ . Each component of $\phi$ (the conjunction of atomic formulas or their negations) defines the intersection of the subsets defined by the atomic formulas or their negations, i.e. a subset containing 0, 1, all or all but a finite number of points (and any such subset can be defined). Now, $\phi$ , as the disjunction of its components, defines a subset that is the union of the subsets defined by the components, i.e. a subset containing a finite number of points or all points but a finite number of them, and any such subset can be defined by a formula.