Their Math1a uses Apostol as its textbook (http://www.math.caltech.edu/~2013-14/1term/ma001a/#text).

This is the classic proofs based text, in other schools what's commonly used for "honors calculus", including MIT but not with such a name. I suppose in theory a really mathematically adept student could start with it, but that's not what I remember any school I've looked into, including CalTech, requiring before you start it.

The link you cited is just an attack on a particular contention; it's entirely unrelated to my examination of it, except second hand reports of it (some of the phases quoted are familiar) probably in part prompted mine.