I guess I could have been more precise and said that the lambda calculus (rather than Scheme/Lisp) is not the foundation of computing. It seems like there are people who still think this; see my recent response here:
https://news.ycombinator.com/item?id=11412392
You could say Lisp and Scheme are proof of that. To actually be bootstrapped, they had to add all this other stuff like vectors and hash tables. I don't know the details of how well those are axiomatized. Paul Graham's Arc tried to a little further down, i.e. unifying functions and macros, defining numbers in terms of lists a la Peano arithmetic, etc., but I'm not sure how far that effort went.
I mentioned all my experience with Lisp... doing SICP 19 years ago, and then coming back to it. As I said, I think it's outstanding research, but if you are trying to build an entire computing universe out of it, that's folly. Good luck. It's just not powerful enough -- once you add all the stuff you actually need, you're not far from the complexity of C.