@Francois: Perhaps I should ask you, what is the reasoning behind your comment. I am assuming (a bad thing to do, I know...) that since SO is axiomatizing the class of sets of ordinals the generalization of the notion of productiveness for SO will be a set, not a proper class. I am not dogmatic on that point. I am willing to learn, if you are willing to teach....

@The User: Thanks. The problems I mentioned in my comments have already been modified. As regards content or tone, is there anything you feel should be modified?

Also, scratch the 'emphasized text' from the question--that was unintentional.

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.

note: '-definable should be -definable 'SOrd' should be SOrd in my question

Actually, it should be "Koepke-Koerwien's system SO". Sorry.

@Prof. Hamkins: Interesting. As regards Koepke's ordinal computability, can one define a notion of productive set in Koepke's system SO (assuming SO not-=L) analogous to the definition of productive set found in ordinary recursion theory? I ask because if one could, in analogy with ordinary recursion theory, such a set would not be able to be generated by one of Koepke's ordinal turing machines and thus a non-constructible set. How would such a non-constructible set be related to 0-sharp?

In fact, what would 0-sharp 'look like' from the perspective of ordinal computability?

(ramified) forcing 'look like' from the perspective of ordinal computability?

@Professor Hamkins: Thanks for the counterexamples--they are very nice! Regarding forcing: since Cohen used forcing to 'create' (would 'create' be the proper term?) nonconstructible sets, could one use forcing to 'create' nonconstructible sets that are in some sense 'simpler' than 0-sharp (I guess for want of a better definition of 'simpler', simpler in this case would mean not implying the consistency of a proper class of inaccessible cardinals)? Also, given Koepke's main theorem: a set S is constructible iff S is ordinal computable from finitely many ordinal parameters, what would

The actual title of the Friedman paper is "The Axiomatization of Set Theory by Extensionality, Separation, and Reducibility". Sorry.

@Andres: Since ZF via Separation and Replacement is infinitely axiomatizable, any finite (or infinite) list of axioms derived from the Separation and Replacement schema are fragments and might be deemed falling short of the mark (I hope that this is not too silly or stupid a remark--if it is, my apologies...). Could, for example, one show that large cardinal axioms could not be construed as instances of replacement or separation (I'm thinking of Harvey Friedman's paper, "The Axiomatization of Set Theory by Extensionality, Separation, and Replacement")? Is this just a silly idea?