This is an exposition of facts about Arithmetic with an approach via mathematical logic. In Section 1 we present Peano Arithmetic, PA, and the complete theory of $\mathbb{N}$, and we show that $\mathbb{N}$ is a prime model of the theory of $\mathbb{N}$. In Section 2 we deal with the Incompleteness Theorems. In Section 3 we deal with non-standard models of arithmetic, in Section 4 we present the Paris-Harrington principle and in Section 5 its independence. The results presented here are quoted from the references listed at the end.