Functorial Fast-Growing Hierarchies

TL;DR

This paper explores the categorical extension of fast-growing hierarchies, connecting them to ordinal collapsing functions and providing a new perspective on subsystems of analysis through higher-type wellordering principles.

Contribution

It introduces a functorial approach to fast-growing hierarchies, linking categorical extensions to ordinal collapsing functions and reinterpreting certain subsystems of analysis.

Findings

Categorical extensions of fast-growing hierarchies are isomorphic to ordinal collapsing functions.

Establishes a connection between proof theory and category theory.

Reformulates $ ext{Pi}^1_1$-CA$_0$ as a higher-type wellordering principle.

Abstract

Fast-growing hierarchies are sequences of functions obtained through various processes similar to the ones that yield multiplication from addition, exponentiation from multiplication, etc. We observe that fast-growing hierarchies can be naturally extended to functors on the categories of natural numbers and of linear orders. We show that the categorical extensions of binary fast-growing hierarchies to ordinals are isomorphic to denotation systems given by ordinal collapsing functions, thus establishing a connection between two fundamental concepts in Proof Theory. Using this fact, we obtain a restatement of the subsystem $\Pi^1_1$-CA$_0$ of analysis as a higher-type wellordering principle.