A Categorical Construction of Bachmann-Howard Fixed Points

TL;DR

This paper constructs Bachmann-Howard fixed points for dilators using categorical methods and establishes their connection to $oldsymbol{ ext{Pi}}^1_1$-comprehension, linking fixed point well-foundedness to a key logical principle.

Contribution

It introduces a method to construct Bachmann-Howard fixed points for general dilators and relates their well-foundedness to $oldsymbol{ ext{Pi}}^1_1$-comprehension.

Findings

Construction of Bachmann-Howard fixed points for dilators.

Equivalence of well-foundedness of these fixed points to $oldsymbol{ ext{Pi}}^1_1$-comprehension.

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 $\operatorname{BH}(T)$ with an "almost" order preserving collapse $\vartheta:T_{\operatorname{BH}(T)}\rightarrow\operatorname{BH}(T)$. Building on previous work, we show that $\Pi^1_1$-comprehension is equivalent to the assertion that $\operatorname{BH}(T)$ is well-founded for any dilator $T$.