How does the "Cogito" prove anything? You constantly fail to actually prove any of your assertions, other than just reasserting them in different ways.

By the way, concerning that dialogue, I think that this is just another case of someone just saying that he can do something without actually doing it. I don't believe that this person, or any person, can really go up as many levels as he pleases. He can say that he's going up as many levels as you can imagine (and I would say that that's a finite number, as well). If not, then tell me, what insights do you extract from being aware (of being aware)^9 that you couldn't extract from being aware (of being aware)^7?

B. The Principle of Mathematical Induction is an _assumption_ you must make about the natural numbers. It can't be proven from other, simpler axioms (though it is equivalent to the Well Ordering Principle).
But I digress. My point is, that the above dialogue is just a silly game of definitions, with too many implicit assumptions and not enough clear terminology to be useful. I again ask you to tell me of the startling conclusions you've come to by thinking (about thinking)^10 that were impossible to arrive at by thinking (about thinking)^7.