Philosophy of Mathematics seminar

Hans Robin Solberg, Oxford

Are Large Cardinal Axioms in Bad Company?  

Much ink has been spilled on the bad company problem in the context of neo-logicist, or more generally abstractionist, approaches to the foundations of mathematics. Very roughly, the problem is that abstractionists claim that certain abstraction principles, such as Hume’s Principle, are truths of a certain privileged sort, which motivates their foundational use, yet there are other, schematically similar abstraction principles, such as Frege’s Basic Law V, which are inconsistent with principles the abstractionists are otherwise committed to. The purpose of this talk is to broaden the potential scope of the bad company problem in the context of other axiomatic foundations. To make the comparison somewhat more precise, I will focus on a certain commitment to, and use of, reflection principles in set theory as a way of deriving the existence of various large cardinals. I argue that many of the relevant features of the original bad company problem are present in this case too; enough to raise the question of whether such principles have their very own bad company problem.