From Naive Set Theory by Halmos:
"Let us say that a function f whose domain is the set of strict predecessors of some natural number n (in other words, dom f = n) is an $\omega$-successor function if $f(0) = \omega$ (provided that $n \neq 0$, so that $0 < n$), and $f(m^+) = (f(m))^+$ whenever $m^+ < n$. An easy proof by mathematical induction shows that for each natural number $n$ there exists a unique $\omega$-successor function with domain $n$"(p. 74).
1. What is the unique function with domain $0$? (answered)
I understand (or potentially misunderstanding) that for $n>0$, $0$ is an element in the domain $n$, so $f(0) =\omega$. For $n=0$ though, $0$ itself is the domain and in fact it is empty by definition.
2. Let $S(n,x)$ be "$n$ is a natural number and $x$ belongs to the range of the $\omega$-successor function with domain $n$". What is {$x: S(0,x)$}?
Best Answer
If a function is a collection of pairs, with first coordinate in the domain and second coordinate in the codomain, then the question becomes: What is the unique collection of pairs with no first coordinate? This is of course the empty collection.
If we denote the unique function $\varnothing\ \longrightarrow\ \omega$ by $f$, then $f(0)$ makes no sense, because $0\notin\varnothing$. This is why your excerpt states explicitly "provided that $n\neq0$, so that $0<n$".