# Section 3.2: 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
Show that in the structure $(\mathbb{N};+)$ the following relations are definable:
(a) Ordering, $\{m,n|m .
(b) Zero, $\{0\}$ .
(c) Successor, $\{m,n|n=S(m)\}$ .
This exercise shows that the theory of $(\mathbb{N};+)$ is, in fact, coincides with the theory of $\mathfrak{N}_{A}=(\mathbb{N};0,S,<,+)$ .
(a) $\exists x(x+x\neq x\wedge v_{1}+x=v_{2})$ .
(b) $v_{1}+v_{1}=v_{1}$ .
(c) $\exists x(x+x\neq x\wedge\forall y\forall z(y+y\neq y\rightarrow z+z\neq z\rightarrow y+z\neq x)\wedge v_{1}+x=v_{2})$ .