User blog comment:P進大好きbot/A Generic System Analogous to Kleene's O/@comment-37233444-20191111033045/@comment-35470197-20191111071139

No, there is no assumption of the finiteness. A partial function \(\mathbb{N} \to \mathbb{N}\) is a map in the sense of set theory whose domain is a subset of \(\mathbb{N}\) and whose values are in \(\mathbb{N}\).