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

TL;DR

This paper introduces impredicative ordinal analysis concepts like the Bachmann-Howard ordinal and collapsing methods, analyzing their application to $ ext{Pi}^1_1$-comprehension and the extended Kruskal theorem.

Contribution

It provides a detailed exposition of impredicative ordinal analysis techniques and demonstrates their limitations regarding certain combinatorial principles.

Findings

$ ext{Pi}^1_1$-comprehension cannot prove the extended Kruskal theorem

Introduction of collapsing methods for uncountable to countable proof trees

Analysis builds on previous work on Peano arithmetic

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.