On page.238 of Enderton's "A Mathematical Introduction to Logic", Church's Theorem is stated (The set of Gödel numbers of valid sentences (in the language of R) is not recursive.)
My question is basic. Does it thereby follow that the set of valid sentences of R is not recursive. Why?
I suppose my initial question can be revised: If we show the set of Gödel numbers of valid sentences (in the language of R) is not recursive what relation does this bear to the question about the valid sentences in the language R (why should we care about the Godel numbers, surely it is the valid sentences that we care about). Why does the function mapping elements of the language R to Gödel numbers show us something about the valid sentences in R?
Also, if only sets of numbers can be recursive, and not sets of sentences of R, then why does Enderton write the following sentence after stating Church's theorem, "The set of Gödel numbers of valid wffs is not recursive either, lest the set of valid sentences be recursive." Here he clearly speaks of the set of valid sentences (of R) as being recursive.
This question might be, and therefore my answer will have to be, slightly pedantic. The issue is that in this context the adjective 'recursive' is a property of sets of integers. So, when speaking with complete formality, it's not clear what is meant by saying a set of sentences is recursive or not recursive.
Of course, there is a really nice one-to-one correspondences between sentences and integers: the Gödel numbering. So when we 'step back' to look at the big picture, when we say 'the set of Gödel numbers of valid sentences is not recursive' what we are really saying is 'there is no computational procedure which can perfectly decide whether a given sentence is valid or not'.
So if we dispense with the formality, sure, you can think of it that way.