Transfinite Hierarchy of Uncomputability via Higher-Order Oracle Busy Beavers — E8 Intelligence Research

FINDING: Higher-order Busy Beaver functions via oracle Turing machines create a transfinite hierarchy of uncomputability, directly tied to the logical strength of formal systems (arXiv:2507.20321v1). | MATH: Let \(BB(n)\) be the classical Busy Beaver. Define \(BB⁽ᵏ⁾(n)\) as the Busy Beaver for a Turing machine with a \(k\)-th order oracle. The hierarchy satisfies \(BB⁽ᵏ⁾(n) BB⁽ᵏ⁻¹⁾(n)\) for all \(n\) beyond a small threshold. Key result: For any arithmetical formula \(φ\), decidability of \(φ\) is equivalent to the computability of some \(BB⁽ᵏ⁾(n)\) relative to the oracle hierarchy — specifically, \(φ\) is decidable iff its truth is determined by a finite initial segment of the \(BB⁽ᵏ⁾\) sequence for some finite \(k\). This mirrors the arithmetical hierarchy: \(Σ^0_k\)-decidability ↔ \(BB⁽ᵏ⁾\)-computability. | CONNECTION: The hierarchy's growth rates — \(BB(n) ~ nʰʸᵖᵉʳ \), \(BB⁽¹⁾(n) ~ BB(BB(n))\), etc. — exhibit a *self-similar re Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com