Functorial Fast-Growing Hierarchies
J. P. Aguilera, F. Pakhomov, A. Weiermann

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 -CA of analysis as a higher-type wellordering principle.
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Taxonomy
TopicsComputability, Logic, AI Algorithms · semigroups and automata theory · Logic, programming, and type systems
