The Infinite Ascent of Unsolvability: The Arithmetical Hierarchy's Unbounded Ladder — E8 Intelligence Research
Year: 2026
Type: preprint
Abstract: FINDING: The Turing degree structure forms a dense partial order with no maximal element — the arithmetical hierarchy is an infinite ascending ladder of unsolvability, not a bounded totality. | MATH: Let \( \mathbf{0} \) be the computable degree. The jump operator \( \mathbf{a} \mapsto \mathbf{a}' \) is strictly increasing: \( \mathbf{a} < \mathbf{a}' \). For the arithmetical hierarchy: \( \emptyset^{(n)} \) has degree \( \mathbf{0}^{(n)} \), and \( \mathbf{0} < \mathbf{0}' < \mathbf{0}'' < \cdots \). No maximal degree exists: for any \( \mathbf{a} \), \( \mathbf{a} < \mathbf{a}' \). The Turing degrees \( \mathcal{D} \) are uncountable, have cardinality \( 2^{\aleph_0} \), and are dense (between any two degrees there is a third). | CONNECTION: The lattice structure of Turing degrees is not a geometric lattice (no finite rank, no crystallographic root system), but the *jump* operator mimics a discrete exponential ladder — analogous to the ratio 2.618? No. However, the *density* (between Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
Author: Caldin, Andrew Stewart
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7204882280
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