Intuitively the answer is yes. According to the definition, a set $A$ is infinite if there is no bijection between $A$ and some natural number. 

Now, I don't know the problem of my reasoning. Accordingly the funcion $f:\mathbb{N}\longrightarrow 0$ is a bijection (Because $f\subseteq \mathbb{N}\times 0=\emptyset$ and then it's vacuously true that $\forall a\in \mathbb{N}\forall b\in \emptyset((a,b)\in f \wedge(a,c)\in f\Longrightarrow b=c)$. 

Also, if I wanted to prove that indeed $\mathbb{N}$ is infinite, the only option I see is by making a proof by induction. In such a case then my statement should be false. I don't know...

**Edit**: The approach to prove that there is a biyection is the same way as to proove that it's a function: 

It's injective: It's vacously true that $\forall a\in \mathbb{N}\forall b\in \emptyset((a,b)\in f \wedge(c,b)\in f\Longrightarrow a=c)$.

It's surjective: It's vacuously true that $\forall a\in \emptyset \exists b((a,b)\in f)$.

Now I know my problems are in my understanding of the meaning of "vacously". It would be very nice if you guys could tell me the mistakes in it and also in my definitions.