The Fickleness of Recursively Enumerable Turing Degrees — E8 Intelligence Research

FINDING: The recursively enumerable (r.e.) Turing degrees form a rich, non-distributive upper semilattice with intricate join/meet properties, including "fickleness" — a measure of how degrees fail to have meets — revealing deep structural pathology in the lattice of computability. | MATH: The r.e. degrees (R, ≤_T) under Turing reducibility form an upper semilattice (join = ⊕, the Turing join: a⊕b = {2n: n∈a} ∪ {2n+1: n∈b}). Meets (greatest lower bounds) do not always exist. Key constants: 0 (computable degree), 0′ (Turing jump, the halting problem degree). Fickleness: a degree a is "fickle" if it has no meet with some other degree, and the "fickleness" of a degree measures the complexity of the set of degrees that fail to meet it. The lattice is dense (between any two r.e. degrees there is another), but not distributive — the distributive law fails: a ∧ (b ∨ c) ≠ (a ∧ b) ∨ (a ∧ c) in general. | CONNECTION: The r.e. degrees form a *lattice* — a poset with join and meet — which is a dis Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com