[FOM] A question for Recursion Theorists

[steve newberry]

I begin by freely confessing that I am SADLY deficient in my study of
Recursion Theory, and am now trying to make up for it by plodding through
Hartley Roger's great monograph, THEORY OF RECURSIVE FUNCTIONS AND EFFECTIVE
COMPUTABILITY.  The book is copyright 1967, and most of the foundation work
was done in the thirties and forties.  Presumably, there have been a lot of
new theorems, methods, and proof techniques developed in the interim.

On p. 31, Rogers states, "It is not inaccurate to say that our theory is, in
large part, a 'theory of diagonalization'."

Does anyone know WHY this is so?  Are there no other ways to prove the 
theorems?
Are none of the theorems capable of proof within the methods of Proof Theory?
Or the methods collected under the rubric of "Predicativity"? Or ANYTHING other
than the deadly repetition of one or another versions of the "Diagonal Proof"?

Or is it the case that no one feels that answers of such questions have any 
relevance to Foundations?

I'm NOT trying to be wise guy, I really want to know.

Please, somebody, reply,

Steve Newberry