The notion of Gödel Encoding is extremely important from a philosophy of mathematics standpoint, but seems to be rarely noted explicitly. Take the following claims for example (which are implied by the inner-machinations of proofs such as Gödel's Second Incompleteness Theorem):
Claim #1: A consistent axiomatic system which includes Peano arithmetic can Gödel Encode ANY axiomatic system.
Claim #2: There does not exist any property of any consistent formal system which is not representable within Peano arithmetic.
Are these claims widely accepted as true? If so, it seems like this implies a perspective where the structure generated by Peano arithmetic contains all possible structures, and what formal systems do is merely select a subset of this structure to discuss with a particular language. If not, how can one hold the affirmative of something like Gödel's Second Incompleteness Theorem?
And finally, if lambda calculus includes Peano arithmetic (it does) and a universal turing machine includes Peano arithmetic (it does), then does this imply an affirmative proof of the Church-Turing Thesis?