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In Peano arithmetic, Addition is defined recursively.

## Definition

Given an arbitrary $a \in \mathbb{N}$, we will define $a+b$ recursively as follows: $a + 0 = a$ and $a+b' = (a+b)'$, for all $b \in \mathbb{N}$.

## Properties

Addition on the Natural Numbers has two important properties: commutativity and associativity. Also, multiplication is distributive over addition.