Productive and creative sets

Fix a numbering $({\phi }_e{)}_{e\in \mathbb{N}}$ and write $W_e:=\mathrm{dom}({\phi }_e)\mathrm{.}$

A set $P\subseteq \mathbb{N}$ is productive when there exists a partial computable function $p:\mathbb{N}\rightharpoonup \mathbb{N}$ such that for every $e$, $W_e\subseteq P⟹p(e)\downarrow \text{ and }p(e)\in P\setminus W_e\mathrm{.}$ Such a function $p$ is called a productive function for $P$.

A set $C\subseteq \mathbb{N}$ is creative when:

• $C=W_e$ for some $e$, so $C$ is computably enumerable, and
• its complement $\mathbb{N}\setminus C$ is productive.

• Productivity means every c.e. subset of $P$ can be effectively diagonalized against from inside $P$.

• For the fixed acceptable machine coding used later, the standard normalization of productive functions permits the witness to be chosen total and one-one. Whenever a later proof needs that strengthened normal form, it says so explicitly rather than building totality and injectivity into the definition.

• Creativity packages two facts at once: effective enumerability of $C$ and productive richness of its complement.