I said: So it may be consistent and satisfiable to ask for a Y-program to compute halting for all X-programs. At least it has not yet been proven impossible.
You said: Alas, it has been proven. RAM computers are equivalent to Turing machines. If X and Y are Turing-complete languages then all X-programs can be converted to Turing machines, your proposed Y-program can also be converted into a Turing machine. Put another way, unless your language X is too weak to be useful, there should exist a deterministic program to convert any program written in Y into a program written in X with identical behavior (with regard to halting). This should be evident by the fact that the computers we use day-in and day-out execute one binary machine language. Any X-program and Y-program reduce to the same assembly language.
I say: You cannot have read my papers carefully. I spent a lot of time explaining, with examples, why programs with subjective specifications cannot be translated from one TM-equivalent language to another TM-equivalent language.
You said: This would disprove the Church-Turing thesis.
I say: Again, you cannot have read the paper carefully. There is a section in each of www.cs.utoronto.ca/~hehner/OSS.pdf and www.cs.utoronto.ca/~hehner/EGT+.pdf titled "Church-Turing Thesis", explaining that it applies to all objective specifications, which are the only specifications considered by Church and Turing, but not to subjective specifications.
You said: Regarding Cantor, indeed, there are many ways to compare the "sizes" of two sets. Cardinality is one such way. It can have applications, for instance Cantor's original paper used it to prove that there are infinitely many transcendental numbers. It is also used in real analysis, in Lebesgue integration, and in measure theory.
I say: Again, you cannot have read the paper carefully. Almost every time I mention applications in the paper, I say "applications outside mathematics". If you consider mathematics to be an application of mathematics, then all mathematics has applications.
You said: the "Program Analogy" section of the paper contains what I think is a flawed argument.
I say: Yes, I point out the flaw in the paper in the paragraph that starts with the sentence "Like the original Cantor argument, this program-analogy version is informal, and the informality may hide serious errors.". Did you miss it?
You will "win" this debate by throwing misunderstandings at me faster than I can reply. And it's disheartening because the replies are already in the papers, if you read them carefully.
You point out quite correctly that if halting can be solved, then we can use it to settle certain open mathematical questions. But this does not tell us whether halting can be solved. (And if it can, it may not be a practical way to settle those questions.) I have many papers on the halting problem on my website at www.cs.utoronto.ca/~hehner/halting.html. The most recent is www.cs.utoronto.ca/~hehner/EGT+.pdf published in SN Computer Science v.1 p.308, 2020 September. Its final paragraph says:
Let X and Y be two programming languages, or two computers, or two locations. It is inconsistent to ask for an X-program to compute halting for all X-programs due to a twisted self-reference. Twisted self-reference is characteristic of subjective specifications. So it may be consistent and satisfiable to ask for a Y-program to compute halting for all X-programs. At least it has not yet been proven impossible.
If you want to know what a twisted self-reference is, or what a subjective specification is, you have to read the paper. This conclusion is not what you claimed it is. The Cantor paper you are unable to find is on my website. Its name is "the Size of a Set", and it is at www.cs.utoronto.ca/~hehner/SetSize.pdf. Its conclusion says:
It is popularly believed that Cantor's diagonal argument proves that there are more reals than integers. In fact, it proves only that there is no onto function from the integers to the reals; by itself it says nothing about the sizes of sets. Set size measurement and comparison, like all mathematics, should be chosen to fit the needs of an application domain. For all application domains that I know of, Cantor's countability relation is not the most useful way to compare set sizes.
I have never claimed, as you said, that there's anything wrong with Cantor's diagonal argument. My paper on Goedel, published in Beauty is our Business, Springer-Verlag silver series, New York, p.163-172, 1990, is at www.cs.utoronto.ca/~hehner/God.pdf. Nowhere in that paper is there any suggestion that "Goedel was wrong", as you claimed I said.
Mathematical results should never be accepted or rejected by whether you "trust this guy". You should read the work, think about it, and decide for yourself whether the results are right or wrong.
I may be notorious, but not for the work I've done, rather for the misinformation about me that is repeated by people who don't check. I have written and published papers that say halting for programs in one formal system (language) cannot be solved in that same system, but can be solved in a different formal system. I would be delighted if you find an error in that work, but no-one has done so yet. I have never said that Goedel was wrong; I have written a paper explaining Goedel's results that other authors have used in textbooks and translations. I have never said Cantor's proof is wrong, but I have suggested there are other measures of set size than the one he proposed.