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

[Csaba Henk]

Dear FOM members,

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.)

Thanks for your suggestions!

Regards,
Csaba