I am trying to solve next task: Proove that each natural number is definable in structure $(\mathbb{N},+)$.
Here is $\mathbb{N} = \{1,2,3,4,\ldots\}$.
Formule $\phi_{1}(x) = (\forall y)(\forall z)\neg( x = y + z)$ define $\{1\}$. At the other side, formule $\varphi(x) = (\exists y)(x = y + 1)$ define set $\mathbb{N}\setminus\{1\}$. Next, formule $\psi(x) = (\exists y)(x = y + y)$ define all even numbers. Of course, $2$ is even number, but I need formule which define only $\{2\}$.
So, my problem is how to define $\{2\}, \{3\}, \ldots$.
One more question: If I proove, relation $\leq$ is definable in $(\mathbb{N},+)$, may I use this relation to construct formule which define $\{k\}, k \in \mathbb{N}$. For example: $\phi_{2}(x) = (\forall y)(1 \leq y \wedge y \leq x) \Rightarrow (1 = y \vee y = x)$ define $\{2\}$.
Your formula $\varphi_1(x)$ correctly defines $1$ in the structure $(\mathbb{N}_{>0},+)$.
Now in general, if a formula $\varphi(x)$ defines an element in a structure $A$, you can proceed with definability in $A$ as if you had a constant symbol $c$ naming that element. Indeed, given a formula $\psi(y,c)$ using the constant symbol $c$, this is equivalent to $\exists x\, (\varphi(x)\land \psi(y,x))$, and also to $\forall x\, (\varphi(x)\to \psi(y,x))$, since the element named by $c$ is the unique element satisfying $\varphi(x)$.
In your case, you can define the natural number $n$ by $$y = \underbrace{1+\dots+1}_{n\text{ times}},$$ which we can rewrite as $$\exists x\, (\varphi_1(x)\land y = \underbrace{x+\dots+x}_{n\text{ times}}).$$
An alternative approach is to note that the usual order $x<y$ on $\mathbb{N}_{>0}$ is definable by $\varphi_<(x,y)\colon \exists z\, (x+z = y)$.
Now we can define $1$ as the least element: $$\lnot\exists y\, \varphi_<(y,x),$$ and we can define $n>1$ as the unique element that has exactly $(n-1)$ elements less than it: $$\exists y_1,\dots,y_{n-1} \left(\bigwedge_{i < j} y_i\neq y_j\land \bigwedge_{i} \varphi_{<}(y_i,x)\land \forall z\, (\varphi_<(z,x)\rightarrow \bigvee_{i} z = y_i)\right).$$