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 X in PA if
(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 corresponding numerals substituted for the variables c, d.
Basically, representability can be relative to truth in an interpretation or relative to provability in a theory. Representable functions are 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?