I think I see now. So the author wasn't saying that construction of values is what defines the type. Rather it is a necessary step before you can start talking about sets of values. (And you can talk about a type without knowing how to construct it, so that makes types different from sets?)
I think I see what the question is now. A type is defined by picking how its values are introduced and/or used. It doesn’t mean that those rules have to be actually productive. In fact there are many kinds of uninhabited types (the empty type being the simplest, but in languages with rich type systems there are many others).
