@TerryTao: Would the qualifier ⁷"algorithmic" explain why the Paris-Harrington theorem (for example) and other similar theorems wouldn't be the sort of examples you are looking for? Since for $PA$ the set of all true wffs and the set of all false wffs are productive sets, and so is the set of unprovable wffs of $PA$ (the set of provable wffs of $PA$ is c.e. and I assume there are no false wffs in the set of provable wffs), would you say that the productive functions of the productive sets capture completely the notion of diagonalization?

Could you list the paper you are referring to, please?

You're very welcome!

@ZuhairAl-Johar: also take a look at the Stanford Encyclopedia of Philosophy entry , "Alternative Axiomatic Set Theories" It should be helpful as well.

@ArvidSamuelsson: So you would (in effect) be saying that $\emptyset$ at $V_{0}$ would not be a set?

@ArvidSamuelsson: so does the Axiom of Foundation as you (properly) define it hold for every stage in the cumulative hierarchy?

Thanks, however, for pointing these out. Very helpful.

Check the context in which the implication occurs. Also as regards Foundation (since Coskey shows that Foundation holds for $V_0$ and $V_1$), how does his sloppiness cause problems for his proof that Foundation holds at each stage of the hierarchy?