In Boolos, Burgess, and Jeffrey (2007, 5th edition), the following statement is found on p. 224:
"[Consider] that the set of sentences that are provable and the set of sentences that are disprovable from any recursive set of axioms is semirecursive, and that all recursive sets are definable by ∃-rudimentary formulas. It follows that there are formulas PrvT (x) and DisprvT (x) of forms ∃y PrfT (x, y) and ∃y DisprfT (x, y) respectively, with Prf and Disprf rudimentary, such that A is a theorem of T if and only if the sentence PrvT ( A ) is correct or true in the standard interpretation."
My question is this: Why does it follow from the first sentence in the quotation that Prf and Disprf are rudimentary? Actually, if you could just explain why Prf and Disprf must be rudimentary, that would suffice for my purposes... Thanks in advance.