Finished Immanuel Kant's Critique of Pure Reason, which was...a pretty dense work. Kant claims that there exist some notions that need not (and perhaps can not) be established based on sensory input. These include, time, space, basic categories, the idea of an object, the idea of causality, and the permanence of material. Even when expose to sensory input, notions in the mind are transcendental since no experience perfectly represents them.
Kant goes on to offer some proofs of his ideas. He talks about the problem of what we might now call false dichotomy, claiming that even saying the world is either finite in time or infinite is a false dichotomy; it is actually neither.
Kant attacks the validity of philosophical proofs of God, and of an eternal soul, but holds with the notion of free will. He talks about the difference between practical knowledge and pure reason.
The book does make me think about my concept of the mind. I think of it like a piece of software, I suppose, and so I agree with Kant that the thinking process has been bootstrapped. It's probably true that some notions are not unthinkable. Perhaps I can not think about non-time, for example. Although I would note that determining what is pure reason -- what intuitions are present without experience -- requires experience of ones own mind, so takes on a kind of empirical nature. So while I understand that axioms are required for thinking, I'm not sure you could ever know for sure you found *the* axioms, even of your own mind.