Impredicativity and Trees with Gap Condition: A Second Course on Ordinal Analysis

Abstract

These lecture notes introduce central notions of impredicative ordinal analysis, such as the Bachmann-Howard ordinal and the method of collapsing, which transforms uncountable proof trees into countable ones. Specifically, we analyze parameter-free $\Pi^1_1$-comprehension and show that it cannot prove the extended Kruskal theorem due to Harvey Friedman (not even for two labels). In terms of prerequisites, we build on a previous lecture on the ordinal analysis of Peano arithmetic. The present material is intended for 12 lectures and 6 exercise sessions of 90 minutes each.