The set of primitive recursive functions is defined as the smallest subset $F\subseteq \cup _{k\in \mathbb{N}} \{f:\mathbb{N}^k \rightarrow \mathbb{N}\}$, satisfying the properties
1) $F$ contains the constant zero function: $0:\mathbb{N}^0 \rightarrow \mathbb{N}$, the successor function and the projection functions
2) $F$ is closed under composition by functions
3) $F$ is closed under primitive recursion
My question is: Why can one even assume, that a set with properties 1)-3) exists, so that one can take the smallest such set ? As well, I was told, that one can construct $F$ by using hull operators, "from the bottom" and "from the top", but I'm not quite sure, about the meaning of that.