$\Pi^1_1$-Comprehension as a Well-Ordering Principle

TL;DR

This paper establishes an equivalence between $ ext{Pi}^1_1$-comprehension and the existence of well-founded fixed points for dilators, linking a logical principle to a structural property of linear orders.

Contribution

It proves that $ ext{Pi}^1_1$-comprehension is equivalent to all dilators having well-founded Bachmann-Howard fixed points, confirming a conjecture by Rathjen and Montalbán.

Findings

$ ext{Pi}^1_1$-comprehension characterized by fixed points of dilators

Established equivalence between logical and order-theoretic principles

Confirmed a conjecture in proof theory

Abstract

A dilator is a particularly uniform transformation $X\mapsto T_X$ of linear orders that preserves well-foundedness. We say that $X$ is a Bachmann-Howard fixed point of $T$ if there is an almost order preserving collapsing function $\vartheta:T_X\rightarrow X$ (precise definition to follow). In the present paper we show that $\Pi^1_1$-comprehension is equivalent to the assertion that every dilator has a well-founded Bachmann-Howard fixed point. This proves a conjecture of M. Rathjen and A. Montalb\'an.