Say that a *(countable, computable-language)* structure $\mathfrak{A}$ has **computable dimension $\omega$** iff there are infinitely many computable copies of $\mathfrak{A}$ up to computable isomorphism. The simplest example of such a structure is probably the linear order $\mathfrak{O}=(\omega;<)$.

Now $\mathfrak{O}$ - and all "natural" such structures that I'm aware of - satisfy a kind of "productivity" condition, where given a computable sequence of computable copies we can computably produce a new computable copy not computably isomorphic to any of the copies in the sequence. On the other hand, there are more artificial structures with computable dimension $\omega$ for which no infinite set of computable copies exists at all, which of course prevents productivity. (See here for details.)

I'm interested in whether a third extreme behavior can occur. Say that a structure $\mathfrak{A}$ is **listable** iff there is some computable sequence of computable copies of $\mathfrak{A}$ such that every computable copy of $\mathfrak{A}$ is computably isomorphic to one of those copies. Listability clearly contradicts both of the behaviors mentioned in the previous paragraph.

Is there a listable structure with computable dimension $\omega$?

3more comments