 In what follows, we shall sometimes use the variables \(m\), \(n\), and \(o\) to range over the natural numbers. In other words, we’ll use formulas of the form \(\forall n(\ldots n\ldots)\) to abbreviate formulas of the form \(\forall x(Nx \to \ldots x\ldots)\), and use formulas of the form \(\exists n(\ldots n\ldots)\) to abbreviate formulas of the form \(\exists x(Nx \amp \ldots x\ldots)\).