The difference between this exercise and examples on page 91 seems to be in the language used, as here we are missing two essentially used in the examples symbols $0$ and $\mathbb{S}$ . That is, the purpose of the exercise seems to be to show that those two symbols do not expand the set of what can be defined in $\mathbb{N}$ , in particular, we show that using our restricted language we can still define both $\{0\}$ and the successor of any natural number.

(a) $\forall v_{2}v_{1}\cdot v_{2}=v_{1}$ ($m=0$ iff for every $n$ , $m\cdot n=m$ ).

(b) $\forall v_{2}v_{1}\cdot v_{2}=v_{2}$ ($m=1$ iff for every $n$ , $m\cdot n=n$ ).

(c) $\exists v_{3}(v_{2}=v_{1}+v_{3}\wedge\forall v_{4}v_{3}\cdot v_{4}=v_{4})$ ($n=m+1$ iff for some $k$ , $n=m+k$ and $k=1$ ).

(d) $\exists v_{3}(v_{2}=v_{1}+v_{3}\wedge\exists v_{4}v_{3}\cdot v_{4}\neq v_{3})$ ($n>m$ iff for some $k$ , $n=m+k$ and $k\neq0$ ).