In the standard construction of natural numbers in axiomatic set theory (ZFC), zero is defined as being the empty set.
However, if we consider, for instance, the function $f:\mathbb N\rightarrow \mathbb N$ defined by $f(n)=n+1$, we have $f(0)=1$, but $f(\emptyset)=\emptyset$, because the image of the empty set is always empty.
Is this contradictory? What am I missing here?