(related to Proposition: Uniqueness of Natural Zero)
We will show that the natural number zero \(0\) is unique, i.e. there can be only one such number, for which
\[n=n+0\text{ for all }n\in\mathbb N\quad\quad ( * )\]
Suppose, \(0^{\ast}\) is any (other) natural number, for which
\[n=n+0^{\ast}\text{ for all }n\in\mathbb N\quad\quad ( * * )\]
Applying the commutativity law for adding natural numbers, we get
\[\begin{array}{rcll} 0^{\ast}&=&0^{\ast}+0&\text{ by }( * )\\ &=&0+0^{\ast}&\text{ by commutativity of adding natural numbers}\\ &=&0&\text{ by }( * * ) \end{array} \]
Thus, both zeros are equal.
