Ordinal Notation Choice, Not Ordinal Size, Determines Fast-Growing Hierarchy Speed — E8 Intelligence Research

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Authors

Publication Details

Journal
Zenodo (CERN European Organization for Nuclear Research)
Published
2026-09-16
DOI
https://doi.org/10.5281/zenodo.22786265
Primary Topic
Computability, Logic, AI Algorithms
Type
preprint
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preprint

Ordinal Notation Choice, Not Ordinal Size, Determines Fast-Growing Hierarchy Speed — E8 Intelligence Research

Andrew Stewart Caldin
Zenodo (CERN European Organization for Nuclear Research)
Computability, Logic, AI Algorithms
preprint

Ordinal Notation Choice, Not Ordinal Size, Determines Fast-Growing Hierarchy Speed — E8 Intelligence Research

Andrew Stewart Caldin
preprint en

Abstract

FINDING: Fast-growing hierarchies (FGH) are indexed by transfinite ordinals; diagonalization at limit ordinals creates self-referential "slots" that accelerate growth, but the hierarchy's rate depends on the ordinal notation system chosen, not just the ordinal itself. | MATH: FGH defined as \(f_0(n)=n+1\), \(f_{\alpha+1}(n)=f_\alpha^n(n)\), \(f_\lambda(n)=f_{\lambda[n]}(n)\) for limit \(\lambda\) with fundamental sequence \(\lambda[n]\). Key ordinals: \(\omega\), \(\varepsilon_0 = \sup\{\omega, \omega^\omega, \omega^{\omega^\omega}, \dots\}\), \(\Gamma_0\) (Veblen fixed point), and ordinal collapsing functions (OCFs) like \(\theta\) (Buchholz) reaching \(\Pi^1_1\text{-CA}_0\). Growth rates: \(f_\omega(n) \sim n \uparrow\uparrow n\), \(f_{\varepsilon_0}(n)\) outgrows all primitive recursive functions. | CONNECTION: The ordinal indexing mirrors the golden-ratio-like self-similarity: \(\varepsilon_0 = \phi(\omega)\) where \(\phi\) is the fixed-point operator, analogous to \(\phi = 1+\frac Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

Zenodo (CERN European Organization for Nuclear Research)
Computability, Logic, AI Algorithms
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