I'm reading Godel incompleteness theorem from A mathematical introduction to logic by Enderton.
GÖDEL INCOMPLETENESS THEOREM (1931)
If A ⊆ Th N and #A is recursive( the set of Godel number of sentences in A), then Cn A(the set of sentences true in A) is not a complete theory.Thus there is no complete recursive axiomatization of Th N.
PROOF . Since A ⊆ Th N, we have CnA ⊆ Th N. If Cn A is a complete theory, then equality holds. But if Cn A is a complete theory, then #Cn A is recursive (item 21 of the preceding section). And by the above corollary, #Th N is not recursive.
Q1. what it miss is that we need to show that the set of Godel numbers of axioms of Peano arithmetic without induction axiom is recursive. how is that proven? is it because any finite st of natural numbers is decidable and hence recursive? or does it have another proof/reason
Q2. The text provides lots of representable relations in PA,like the set of Godel numbers of variables, terms, formulas,..etc. later on it states that "Any recursive function is representable in PA. then later on it proves fix point lemma which only uses an specific function and by the help of fix point lemma it proves the incompleteness theorem.
What I don't get is if all those representable functions and relations are necessary in the proof of Godel theorem? if so what do I miss? If not why bother mentioning all of those?
Thanks for your help.