Section 2.1: Problem 2 Solution

Working problems is a crucial part of learning mathematics. No one can learn... merely by poring over the definitions, theorems, and examples that are worked out in the text. One must work part of it out for oneself. To provide that opportunity is the purpose of the exercises.
James R. Munkres
With the same language as in the preceding exercise, translate back into good English the wff $\forall x(Nx\rightarrow Ix\rightarrow\neg\forall y(Ny\rightarrow Iy\rightarrow\neg x .
First, let us translate this construction appearing twice: $\forall x(Nx\rightarrow Ix\rightarrow\alpha)$ $=\forall x(Nx\rightarrow(Ix\rightarrow\alpha))$ $\sim\forall x(\neg Nx\vee(Ix\rightarrow\alpha))$ $\sim\forall x(\neg Nx\vee\neg Ix\vee\alpha)$ $\sim\forall x(\neg(Nx\wedge Ix)\vee\alpha)$ $\sim\forall x(Nx\wedge Ix\rightarrow\alpha)$ . “For every interesting number $x$ , $\alpha$ .” So, we have “For every interesting number $x$ , it is not true that (for every interesting number $y$ , it is not true that $x )”. We can further paraphrase the second part (in the parentheses), namely, $\neg\forall y(Ny\wedge Iy\rightarrow\neg\alpha)$ , as $\exists y\neg(Ny\wedge Iy\rightarrow\neg\alpha)$ , or $\exists y(Ny\wedge Iy\wedge\alpha)$ . So, we have “For every interesting number $x$ , there exists an interesting number $y$ such that $x ”. Simply put, “For every interesting number, there is a larger interesting number” or “There is no largest interesting number”.