# Aritm\'etica

## TL;DR

This paper provides an exposition of arithmetic through mathematical logic, covering Peano Arithmetic, models of natural numbers, incompleteness theorems, non-standard models, and independence results like the Paris-Harrington principle.

## Contribution

It offers a comprehensive logical perspective on arithmetic, including foundational theories, models, and independence results, with references to key results in the field.

## Key findings

- N is a prime model of its theory
- Incompleteness Theorems are discussed
- Paris-Harrington principle's independence is shown

## Abstract

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.