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.