(related to Corollary: Existence of Natural Zero (Neutral Element of Addition of Natural Numbers))
According to the definition of natural numbers, the number \(0\) has been identified with the empty set \(\emptyset\), which exists according to the axiom of empty set. Therefore, the natural number \(0\) also exists.
As a corollary to the definition of adding natural numbers "\( + \)" it also follows that \(0\) is neutral with respect to this operation, i.e. \[x + 0= x\] for all \(x\in\mathbb N\).
It remains to be shown that also the equation \(0+x=x\) holds for all \(x\in\mathbb N\). It follows immediately from the commutativity of adding natural numbers.