I'm not really sure I completely understand where the conflict is here. Even in the "machine model" the church-turing thesis "fails spectacularly" once you sta

I'm not really sure I completely understand where the conflict is here. Even in the "machine model" the church-turing thesis "fails spectacularly" once you start adding in unreasonable things like halting oracles. TM+Halting oracle > TM. It's just that the TM+halting oracle doesn't exist in our universe, even though you can define it mathematically. When building a language, it's similarly easy to introduce "unreasonable" things that either require huge complexity to translate to the machine representation or might even be completely unrealizable in the real world! Even ignoring higher types, compiling a C++ program requires a potentially unbounded amount of work because templates are a turing-complete functional programming language. As stated in the article, just determining if a C++ program is valid is undecidable! Nothing stops you from designing these languages, but you can't say that Church-Turing fails until you consider also the systems required to actually execute these languages in the real world. If you can't, then your programming language is perhaps an interesting mathematical abstraction, but just as useful ultimately as that TM+halting oracle model of computation.