In analogy with the terminology for sets, say that a *(countable, computable language)* structure $\mathfrak{A}$ is **productive** if there is a computable way to properly expand any computable list of computable isomorphism types of computable copies of $\mathfrak{A}$. That is, $\mathfrak{A}$ is productive iff there is some partial computable function $F$ such that for all $a,b$:

If $W_a=\overline{W_b}$, and every element of $W_a$ is an index for a computable copy of $\mathfrak{A}$, then $F(a,b)$ is defined and is an index for a computable copy of $\mathfrak{A}$ not computably isomorphic to any of the copies with indices in $W_a$.

*(The "$W_a=\overline{W_b}$"-bit just says that $W_a$ is in fact a computable, not just c.e., set of names for copies of $\mathfrak{A}$, and we're giving this set to $F$ as a computable set rather than a c.e. set.)*

Recall that the *computable dimension* of a structure is the number of computable copies it has up to isomorphism. Obviously any productive structure must have a computable copy (take $W_a=\emptyset$) and must have computable dimension $\omega$ (iterate $F$ appropriately). However the converse is not clear to me. My question is:

**Is every computable structure with computable dimension $\omega$ productive?**

All the "natural" examples I can think of are easily seen to be productive, but I don't see any generally-applicable principle at work here. There are various results in the literature of similar "flavor" such as Montalban's work on copy/diagonalize games but none that I'm aware of seem directly applicable.

*My suspicion is that the answer to this question is "fragile" in the sense that there is a computable structure with infinite computable dimension which is non-productive, but that every structure is either computably categorical on a cone or "productive on a cone" in the appropriate sense; this is motivated by (general perversity and) the combination of Goncharov's theorem that there are computable structures of computable dimension strictly between $1$ and $\omega$, and McCoy's theorem that every structure is either computably categorical on a cone or has computable dimension $\omega$ on a cone.*

classicallyisomorphic, since they're copies of $\mathfrak{A}$. I don't know what "computably non-isomorphic" would mean, it sounds stronger than merely "not computably isomorphic." $\endgroup$ – Noah Schweber Nov 29 '20 at 6:51