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](https://mathoverflow.net/questions/377687/does-productive-dimension-omega-for-computable-structures) 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.



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> Is there a listable structure with computable dimension $\omega$?
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Is there a "listable" structure of computable dimension $\omega$?
Yes. [Hirschfeldt and Khoussainov](https://www.jstor.org/stable/4147757) built such a structure. See the start of section 3, on page 1208. In fact, their listing is injective (into equivalence classes modulo computable isomorphism). Interestingly, they also consider idea of a productive structure, although they call it "effectively infinite computable dimension"; see page 1200.


