Identity Element of Natural Number Addition is Zero

Theorem

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

$\forall n \in \N: 0 + n = n$

Proof

Take the definition of natural numbers $\N$ as a naturally ordered semigroup $\struct {\N, +, \le}$.

The result follows from Zero is Identity in Naturally Ordered Semigroup.

$\blacksquare$

Sources

• 1982: Alan G. Hamilton: Numbers, Sets and Axioms ... (previous) ... (next): $\S 1$: Numbers: $1.1$ Natural Numbers and Integers: Examples $1.1 \ \text {(b)}$