# A Question Regarding Productive Sets in the Koepke-Koerwien System SO (Sets of Ordinals)

In their paper "The Theory of Sets of Ordinals" (arXiv), Koepke and Koerwien propose a theory SO axiomatizing the class of sets of ordinals in a model of ZFC and show that SO and ZFC are bi-interpretable. They also define *-recursiveness over SO which generalizes the ordinary recursive functions from omega to Ord, and show that the *-recursive sets (i.e. the *-definable sets) form the smallest inner model of SO; SOrd*, and that SOrd*=L. Considering that SOrd* is the smallest inner model of SO and that *-recursion generalizes the ordinary recursive functions from omega to Ord, it seems reasonable to assume that one can define productive sets of ordinals in nearly the same manner as in ordinary recursion theory, where one is defining productive sets of finite ordinals. Of course (in analogy with ordinary recursion theory), the productive sets of ordinals will not be *-definable, and therefore will not be constructible. Are there aguments against the existence of productive sets of ordinals, and, more importantly, in what way are productive sets of ordinals (if they exist) related to 0-sharp?

• note: '-definable should be -definable 'SOrd' should be SOrd in my question Jun 10, 2013 at 14:40
• It should also be noted that 'SOrd' should have an asterisk at the upper right-hand corner and ' -definable' should have a centered asterisk before the '-definable' as in the Koepke-Koerwien paper. Jun 10, 2013 at 14:46
• Also, scratch the 'emphasized text' from the question--that was unintentional. Jun 10, 2013 at 14:48
• @Thomas You can still modify your question. Jun 10, 2013 at 14:59
• I'm not sure what you mean by a productive set of ordinals. When I try to generalize the idea from recursion theory, I arrive at a productive class of ordinals. Could you clarify what you mean? Jun 10, 2013 at 19:35