Of course, such subsets exist. You can either define such subsets informally yourself, or you can find a total programming language (such as Coq), in which you

Of course, such subsets exist. You can either define such subsets informally yourself, or you can find a total programming language (such as Coq), in which you may only write programs that provably terminate. Even without identifying specific subsets, it's quite possible to predict whether specific programs terminate. For example, it's trivial to write a program that loops forever. Some people misrepresent the halting problem and its corollaries as "we can under no circumstance predict any aspect of the behaviour of any program", which is in fact wrong. What is true is that, for every property we may want to assert about a program, there exists a program (on a Turing machine or in a Turing-complete language, so excluding any of the total programming languages) for which you will fail to tell if it has that property.