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

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 $(f_{\alpha }{)}_{\alpha <{\epsilon }_0}$, 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 $\alpha <{\epsilon }_0$. We also outline possible transfinite extensions indexed by countable ordinals, thus bridging symbolic large-number constructions and ordinal analysis.

arXiv comment6 pages, 0 figures. v1, 5 June 2025. Keywords: Large numbers, fast-growing functions, proof theory, computability, Knuth notation, ordinal analysis, Peano Arithmetic