The R.E. Degrees: Joins, Missing Meets, and Fickleness — E8 Intelligence Research

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Authors

Publication Details

Journal
Zenodo (CERN European Organization for Nuclear Research)
Published
2026-09-18
DOI
https://doi.org/10.5281/zenodo.22823736
Primary Topic
Computability, Logic, AI Algorithms
Type
preprint
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preprint

The R.E. Degrees: Joins, Missing Meets, and Fickleness — E8 Intelligence Research

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

The R.E. Degrees: Joins, Missing Meets, and Fickleness — E8 Intelligence Research

Andrew Stewart Caldin
preprint en

Abstract

FINDING: The recursively enumerable (r.e.) Turing degrees form a rich algebraic structure — an upper semilattice with a join operation, but not a full lattice (meet fails generally); recent work explores "fickleness" and bounding lattices within this degree structure. | MATH: The r.e. degrees \(\mathcal{R}\) under \(\leq_T\) have join \(a \vee b\) (least upper bound) but meet \(a \wedge b\) (greatest lower bound) exists only for special pairs (e.g., when one is low or when they are contiguous). Key constants: none numeric, but structural invariants — the least degree \(\mathbf{0}\), the jump \(\mathbf{0}'\), and the density theorem (between any two r.e. degrees there is another). Lattice-theoretic: \(\mathcal{R}\) is not a lattice; it is an upper semilattice with zero \(\mathbf{0}\) and no maximal element. | CONNECTION: The lattice structure here is **not** geometric — it is a **distributive-free, non-modular** algebraic lattice. However, the *join* operation mirrors the **supremum** i 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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The R.E. Degrees: Joins, Missing Meets, and Fickleness — E8 Intelligence Research — Andrew Stewart Caldin · Zenodo (CERN European Organization for Nuclear Research) (2026) | TGRS Research Map | TGRS