TITLE: The Halting Problem and Church's Thesis
QUESTION [8 upvotes]: In the opening chapters of Hartley Rogers, Jr.'s book Theory of Recursive Functions and Effective Computability, the proofs of the unsolvability of the halting problem and related unsolvability results invoke Church's Thesis. Can Church's Thesis be avoided?
Edit added on September 7 while putting a bounty on the question:
While in no way whatsoever doubting the diagonal argument per se, I am nevertheless in doubt as to how this is meant to work in the undecidability set up as witnessed also by my probes. One $\textit{evasion-possibility}$ that I, given my befuddlements, am not able to exclude a priori is that there $\textit{is}$ a recursive function that tells us whether a given Turing machine halts, but that a diagonalisation of the Turing machine would - pace Church and Turing - provide an effective non-recursive method beyond that recursive function.
I am rather certain that the evasion-possibility does not obtain, but  I want to understand precisely why it does not. Somehow, I would have thought that the answer to this were simple. (The last sentence was not meant to contradict or undermine comments that pointed towards a tediousness of carrying these matters through.)

REPLY [4 votes]: To address your "evasion-possibility" directly: Let's look again at the overall structure of the proof of the unsolvability of the halting problem.  If there were a Turing machine $M$ that could solve the halting problem, then we could construct another Turing machine $M'$ that uses $M$ as a subroutine in a certain way, etc., contradiction.  We can explicitly write down the additional instructions that are needed to convert $M$ into $M'$, since this is a fixed piece of code that is independent of $M$, and so in particular there is no reason (other than laziness) to invoke the Church–Turing thesis to establish the existence of this piece of code.  So if $M$ is a Turing machine then $M'$ must also be a Turing machine.  That's all that we need to prove that there is no Turing machine to solve the halting problem.
This proof doesn't exclude the possibility that there is an effective but non-recursive procedure to solve the halting problem; for that conclusion, we would need to invoke the Church–Turing thesis.  But the proof does exclude the possibility that there is a Turing machine to solve the halting problem for Turing machines, and that is all that we would expect to be able to prove mathematically anyway.