There are few definitions in mathematical logic that don't vary between textbooks. I see two options.What does A(c,d) represent X mean?

(1) A represents X if iff for all .

(2) A represents Xin PAif

(a) for all , and

(b) for all .

Here is the standard model of natural numbers, PA is Peano arithmetic, and if is a pair of natural numbers, then denotes A with correspondingnumeralssubstituted for the variables c, d.

Basically, representability can be relative to truth in an interpretation or relative to provability in a theory. Representablefunctionsare usually defined relative to provability in PA. See "Computability and logic" by Boolos and Jeffrey, chapter "Representability of Recursive Functions."

Under option (1) above this is true by definition.also why does (m,n) element X iff A(m,n) interpreted in the natural way as a statement in number theory is true?