The Aurellion Function: A Recursive Fast-Growing Hierarchy Beyond Knuth Notation
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866916781147291648 |
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| author | Vodrazka, Daniel |
| author_facet | Vodrazka, Daniel |
| contents | 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 $(f_α)_{α< \varepsilon_0}$, 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 $Γ_0$ due to its ability to majorize all functions $f_α$ for $α< \varepsilon_0$. We also outline possible transfinite extensions indexed by countable ordinals, thus bridging symbolic large-number constructions and ordinal analysis. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_05067 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Aurellion Function: A Recursive Fast-Growing Hierarchy Beyond Knuth Notation Vodrazka, Daniel Logic 03D20 (Primary) 03F15, 03F30 (Secondary) F.4.1 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 $(f_α)_{α< \varepsilon_0}$, 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 $Γ_0$ due to its ability to majorize all functions $f_α$ for $α< \varepsilon_0$. We also outline possible transfinite extensions indexed by countable ordinals, thus bridging symbolic large-number constructions and ordinal analysis. |
| title | The Aurellion Function: A Recursive Fast-Growing Hierarchy Beyond Knuth Notation |
| topic | Logic 03D20 (Primary) 03F15, 03F30 (Secondary) F.4.1 |
| url | https://arxiv.org/abs/2506.05067 |