Second-Order Reflection on Ordinals

Ordinal numbers are natural extensions of the counting numbers. They allow counting ordered collections of objects past infinity. As ordinal numbers grow bigger and bigger, they become harder and harder to describe precisely. This is made precise by the notion of reflection: large enough ordinal numbers cannot be defined by simple formulas. For many classes of formulas C, we can isolate the C-reflecting ordinals: the ordinals which cannot be described by formulas in C. The current project aims at identifying the smallest C-reflecting ordinals for various C and comparing their relative sizes.