Observations on the Halting Problem

When Alan Turing laid the foundation for computation in 1936, he wanted to show what computation can do, and what it cannot do. For the latter, he invented a problem that we now call the “Halting Problem”. In modern terms, it is as follows. In a general-purpose programming language, write a program that reads a text (character string) p representing a program in that same language, and reads another text i representing its input, and outputs true if execution of p with input i terminates, and outputs false if execution of p with input i does not terminate. The choice of programming language does not matter; any general-purpose programming language will do. The problem cannot be solved; there is no such program.