TITLE: The difference between the Recursive and the Effective topos.
QUESTION [11 upvotes]: I would like to know which is the real difference between the Recursive topos (in the sense of Mulry) and the Effective topos (in the sense of Hyland). Especially what is related to recursive functions. Do they have the same semantic power?
I will be gratefull with some hints about texts related to this.
Thanks in advance.
PS: I don't give definitions because of their big extension, but I could give them if anybody wants.

REPLY [8 votes]: I am aware that I am answering an old question, but for completeness of MathOverflow, I'd like to point out that G. Rosolini's 1986 thesis "Continuity and effectiveness in topoi" contains (chapter 6) a description of both the effective and the recursive topoi, and a comparison of them by constructing a functor from the effective to the recursive (preserving limits, finite coproducts, images and the natural number object, faithful on the full subcategory of countable objects and full and faithful on the full subcategory of countable separated objects).  According to the author, "the major difference between the effective topos and the recursive topos is that the [natural number object] in [the effective topos] does not generate the topos", and the functor takes an object of the effective topos to the set of maps from the natural numbers object to it, which can be viewed as an object of the recursive topos.