FOM: Reply to Davis

[Kanovei]

Joseph Shoenfield <jrs at math.duke.edu> writes: 
**
We all know
that in developing recursion theory, it is best to use functions as
the basic notions, and define recursive sets in terms of recursive
functions.   Can this be done in set theory and would the resulting
axiom system be useful?                                         
**

Recursion theory enjoys once and for all fixed 
underlying domain of simple nature. 
On the contrary any set-theoretic system needs 
to explain how to gather objects, already constructed, 
at limit steps. Either one just gathers them in a set 
or one employs constructions like "the sobobject classifier", 
whichever considered better. 
At the first look it seems more optimistic to look 
for a function-theoretic setup in terms of NBG 
(with functions defined on all sets). 

V.Kanovei