There are two categories of "self-contained":

1. Logically self-contained, which just means that the treatment doesn't assume outside knowledge. Difficulty level is not bounded from above.

2. Beginner-friendly, which means the presentation is tailored to people without a strong background in the subject.

Penrose falls squarely in Category 1. An intelligent reader can probably push the definitions around enough to follow the presentation, but that's not the same as understanding what's actually going on.

That's correct, but it is difficult enough to effectively not be self-contained, I think. Being able to apprehend the concepts at the pace and brevity at which Penrose introduces them would require significant prior training in thinking mathematically, or a quite unusually agile mind.

I don't actually remember, it's been a while. I do remember that it's not completely explained - he does skip steps (and that's not just me, I read that in a review of the book). I also remember getting confused and frustrated. I bought the book with high hopes and put more than a little bit of effort into it, but eventually I gave it away to someone who knew more math.

I suppose I could go take it out of the library and try again.