The $\Pi^1_2$ Consequences of a Theory

TL;DR

This paper develops a proof-theoretic framework for analyzing $ ext{Pi}^1_2$ theorems using functors in linear orders, extending ordinal analysis beyond traditional ordinal numbers.

Contribution

It introduces a functorial $ ext{Pi}^1_2$ norm and classifies $ ext{Pi}^1_2$-sound theories via new ordinal notions, extending proof theory to higher complexity classes.

Findings

Characterizes $ ext{Pi}^1_2$-soundness ordinals of theories.

Shows admissible ordinals correspond to certain $ ext{Pi}^1_2$-soundness ordinals.

Identifies $ ext{ACA}_0$'s $ ext{Pi}^1_2$-soundness ordinal as $oldsymbol{ ext{omega}}_1^{ck}$.

Abstract

We develop the abstract framework for a proof-theoretic analysis of theories with scope beyond ordinal numbers, resulting in an analog of Ordinal Analysis aimed at the study of theorems of complexity $\Pi^1_2$. This is done by replacing the use of ordinal numbers by particularly uniform, wellfoundedness preserving functors in the category of linear orders. Generalizing the notion of a proof-theoretic ordinal, we define the functorial $\Pi^1_2$ norm of a theory and prove its existence and uniqueness for $\Pi^1_2$-sound theories. From this, we further abstract a definition of the $\Sigma^1_2$- and $\Pi^1_2$-soundness ordinals of a theory; these quantify, respectively, the maximum strength of true $\Sigma^1_2$ theorems and minimum strength of false $\Pi^1_2$ theorems of a given theory. We study these ordinals, developing a proof-theoretic classification theory for recursively enumerable extensions of $\mathsf{ACA}_0$ Using techniques from infinitary and categorical proof theory, generalized recursion theory, constructibility, and forcing, we prove that an admissible ordinal is the $\Pi^1_2$-soundness ordinal of some recursively enumerable extension of $\mathsf{ACA}_0$ if and only if it is not parameter-free $\Sigma^1_1$-reflecting. We show that the $\Sigma^1_2$-soundness ordinal of $\mathsf{ACA}_0$ is $\omega_1^{ck}$ and characterize the $\Sigma^1_2$-soundness ordinals of recursively enumerable, $\Sigma^1_2$-sound extensions of $\Pi^1_1{-}\mathsf{CA}_0$.