I'd be grateful for a pointer to some fundamental background reading; I can only follow enough of your reasoning to be intrigued and committed to learning more :)
I'm trying to write up a blog post about this now. A really good presentation focused on types and programming languages is available in the middle chapters of Practical Foundations for Programming Languages. I originally learned about it from studying Aczel's Antifoundation Axiom from Barwise and Moss' Vicious Circles[0]---in there you can get a much more mathematical foundationalist POV---but Harper's book is more direct.
[0]http://www.amazon.com/Vicious-Circles-Center-Language-Inform...