A Categorical Construction of Bachmann-Howard Fixed Points

Abstract

Peter Aczel has given a categorical construction for fixed points of normal functors, i.e. dilators which preserve initial segments. For a general dilator $X\mapsto T_X$ we cannot expect to obtain a well-founded fixed point, as the order type of $T_X$ may always exceed the order type of $X$. In the present paper we show how to construct a Bachmann-Howard fixed point of $T$, i.e. an order $\mathrm{BH}(T)$ with an "almost" order preserving collapse $\vartheta :T_{\mathrm{BH}(T)}\to \mathrm{BH}(T)$. Building on previous work, we show that ${\Pi }_1^1$-comprehension is equivalent to the assertion that $\mathrm{BH}(T)$ is well-founded for any dilator $T$.

This version has been accepted for publication in the Bulletin of the London Mathematical Society

References in corpus (2)

• ${\Pi }_1^1$-Comprehension as a Well-Ordering Principle
• Computable Aspects of the Bachmann-Howard Principle

Cited by in corpus (3)

• ${\Pi }_1^1$-Comprehension as a Well-Ordering Principle
• Derivatives of normal functions in reverse mathematics
• How strong are single fixed points of normal functions?