[UnderProtest]

You're hitting the nail on its flattest head here. The metaphors are deliberately broken. The ways in which they break are meant to show aspects of the system.

Theorems have characteristics that aren't obvious to non-mathematicians. Hofstatder's goal was to convey some of these characteristics. A metaphor that says "if a theorem were a record and a mathematical system were a record player, here is what would happen to the record player when it played a self-referential theorem record" is an excellent way to show how theorems are different than anything we normally deal with.

A more modern book would have been able to rely on the metaphor of a program "crashing" which would have been sad and inept. There are a thousand ways a program can crash and very few of them reflect any aspects of Gödelian incompleteness. They're just poorly made, like a record with a big hole in the track that breaks the record needle.

There are essays in Hofstadter's collection "Metamagical Themas" that expand significantly on his idea of what metaphors and analogies actually are and what he thinks they're capable of doing. Reading those makes it very clear why he'd use broken metaphors in GEBEGB.

I think GEBEGB is most useful in exposing people who don't yet have any formal training in mathematical theory to some deeper concepts and inspiring them to find the subtle edges of these systems. I read it at about 12 and it changed my life by exposing subtle fripperies of thought. If I'd read it at 20, as the author of this article did, I too would probably have experienced these little amusements as lack of rigor. Instead I experienced them as mysteries, and now as an adult I experience them as very dry jokes and thought prompts.

[erikb]

> The metaphors are deliberately broken

Also adding to this. A main point of the book is also that there is no logical system that can be complete, so someone trying to get the real big picture needs to be able to live outside logic comfortably as well.

A little bit like Zen thinking. You cannot make a clapping sound with one hand. But you have to accept that the question can exist in a space with zero answers.