[FOM] Plea for literature on recursion theory in V_\omega

[Robert Black]

I haven't actually read it myself, but I think you might find the 
recent Melvin Fitting - _Incompleteness in the Land of Sets_ does 
what you want.

Robert

>I am to write my PhD thesis. I plan to base my "design" on recursion
>theory built up in V_\omega (ie., the universe of hereditarily finite
>sets [as a first-order structure with the language of set theory]).
>
>All textbooks I know of use an arithmetic context for explaining
>recursion theory. I'd like to ask: do you know of some book which uses
>V_\omega for this purpose? Or any book which discusses the well-known
>semantic equivalence (yes, this is a vague term, but you should know
>what I mean...) of <\omega, + , *> and <V_\omega, \epsilon> ?
>
>(I know this is a well explored area, but I don't know how much of this
>remained "folklore"... My focus is on finding nicely worded definitions
>and theorems which I can use as references for my original work.)
>