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first-order and higher-order logic, model theory, set theory, proof theory, computability theory, formal languages, definability, interplay of syntax and semantics, constructive logic, intuitionism, philosophical logic, modal logic, completeness, Gödel incompleteness, decidability, undecidability, theories of truth, truth revision, consistency.

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Syntactical statements in PA about primitive recursive functions [closed]

How does one generally show syntactical statements in PA about primitive recursive functions or relations? For example something like: Let $A$ be a prim. rec. relation such that $n\in A$ for every …