Background: I'm working on a talk that presents Godel's first Incompleteness Theorem from a computability-theoretic perspective. The idea is to show that the first incompleteness theorem follows from the unsolvability of the halting problem. See Scott Aaronson's post for a high-level description: http://www.scottaaronson.com/blog/?p=710
I've almost got the proof down without using the concept of a Godel numbering except for one last step: I need to show that the set of functions definable in Peano Arithmetic is closed under primitive recursion. That will show that all p.r. functions are definable in PA, which lets me complete the proof.
Question: is there a way to prove that PA-definable functions are closed under primitive recursion without using the full machinery of a godel numbering?
The answer presented in the comments was to use Gödel's β function, which does not require Gödel numbering.