Thread: Help with first order peanos arithemtic

Let A(c,d) be a first order peano arithmetic formula containing no free variables occurence other than c and d. LLet X be a subset of N^2 ( N is the set of natural numbers)

What does A(c,d) represent X mean?

also why does (m,n) element X iff A(m,n) interpreted in the natural way as a statement in number theory is true?

Thanks

What does A(c,d) represent X mean?

There are few definitions in mathematical logic that don't vary between textbooks. I see two options.

(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."

also why does (m,n) element X iff A(m,n) interpreted in the natural way as a statement in number theory is true?

Under option (1) above this is true by definition.