[wavemode]

I don't think that's quite it. In many statically-typed languages that we would typically refer to as having "Turing-complete type systems", types cannot be manipulated at runtime, and thus Type is not really a type in the same way that e.g. int or float are types.

It's sort of like having two languages in one. There is a first, interpreted language which manipulates Types and code and produces a program, and then a second language which is the program itself that ultimately gets typechecked and compiled. Typechecking in this case is not (necessarily) undecidable.

This paper is moreso about dependently-typed languages, where the type of one term can depend on the runtime value of another term.