Hazel Brickhill

Reflection Principles and Foundational Acceptance in Set Theory and Proof Theory

Reflection principles within set theory are generally thought of as sharing little but the name with reflection principles in proof theory. Reflection principles in set theory are ontological: they assert that the set theoretic universe as a whole is mirrored in certain small parts (sets). Proof theoretic reflection principles on the other hand are epistemological: they assert that what we can prove from certain axioms is indeed true. I will argue that there are deep connections, both in the form and the motivation of such principles across the two subjects. In particular I develop a notion of foundational acceptance which I argue motivates set theoretic reflection principles. A careful analysis of the process of proof theoretic reflection will show that here too a restricted form of foundational acceptance plays a crucial role.