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

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 $\vartheta :T_X\to X$ (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. Montalbán.

This version has been accepted for publication in Advances in Mathematics

References in corpus (2)

• Derivatives of normal functions in reverse mathematics
• A Categorical Construction of Bachmann-Howard Fixed Points

Cited by in corpus (5)

• Derivatives of normal functions in reverse mathematics
• A Categorical Construction of Bachmann-Howard Fixed Points
• How strong are single fixed points of normal functions?
• Set-theoretic reflection is equivalent to induction over well-founded classes
• Reverse mathematics of a uniform Kruskal-Friedman theorem