> a basically-similar argument in Kleene's 1967 textbook.

This textbook is an astonishing piece of work of an astonishing mathematical genius. The amount of all kinds of weird (and useful? I'm not much of a logician myself) logical stuff in it is just mind-boggling, just as its writing style. You can keep come back to this book for years and find something new in it every time.

Barendregt's book on lambda calculus is another example of such a book.