The Aurellion Function: A Recursive Fast-Growing Hierarchy Beyond Knuth Notation

TL;DR

The paper introduces the Aurellion Function, a new recursive hierarchy based on Knuth's notation, which grows faster than classical hierarchies and surpasses Peano Arithmetic's provability limits, with potential for transfinite extension.

Contribution

It defines the Aurellion Function, analyzes its growth relative to known hierarchies, and explores its provability and transfinite extensions, advancing the understanding of large-number hierarchies.

Findings

Aurellion Function dominates all provably total functions in PA.

It is situated near the proof-theoretic ordinal $oldsymbol{ extGamma_0}$.

Potential for transfinite extensions indexed by countable ordinals.

Abstract

We introduce the Aurellion Function, a novel recursively defined fast-growing hierarchy based on Knuth's up-arrow notation, defined by $A_1 = 10 \uparrow\uparrow\uparrow 10$, $A_{n+1} = 10 \uparrow^{A_n} 10$, where the number of arrows in the operation increases superexponentially with $n$. We analyze its growth rate relative to classical hierarchies such as the fast-growing hierarchy $(f_\alpha)_{\alpha < \varepsilon_0}$, and discuss its provability status in formal arithmetic. We provide formal bounds showing $A_n$ dominates all functions provably total in Peano Arithmetic, situating the Aurellion Function near the proof-theoretic ordinal $\Gamma_0$ due to its ability to majorize all functions $f_\alpha$ for $\alpha < \varepsilon_0$. We also outline possible transfinite extensions indexed by countable ordinals, thus bridging symbolic large-number constructions and ordinal analysis.