Identity Element of Natural Number Addition is Zero

Theorem
The identity element for the natural numbers under addition is zero ($0$).

Proof
The Natural Numbers are a Naturally Ordered Semigroup.

We also have that naturally ordered semigroups are unique up to isomorphism.

A naturally ordered semigroup has an element zero which behaves as an identity element for addition.

This is therefore defined as being the zero ($0$) of the natural numbers.