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The hierarchy is built using three fundamental rules of recursion: : The base function is simple incrementation. f0(n)=n+1f sub 0 of n equals n plus 1 Successor Case : For a successor ordinal , the function is defined as the -th iterate of the previous function.
Building a digital calculator for the Fast-Growing Hierarchy is not like building a standard arithmetic calculator. Floating-point numbers fail instantly. Standard BigInt libraries run out of RAM in microseconds.
enum Ordinal Zero, Succ(Box<Ordinal>), Limit(Box<dyn Fn(u64) -> Ordinal>), // fundamental sequence Psi(Box<Ordinal>, Box<Ordinal>), // ψ_α(β) Omega, // ω Veblen(Box<Ordinal>, Box<Ordinal>)
To build a reliable, high-utility FGH tool, developers generally implement a two-tier architecture: an Ordinal Parser Engine and a Recursive Evaluation Simulator.
(α+ωβ)[n]=α+ωβ[n]open paren alpha plus omega raised to the beta power close paren open bracket n close bracket equals alpha plus omega raised to the beta open bracket n close bracket power
Before exploring the tools, it helps to understand the core concepts of FGH. It is a family of functions indexed by ordinals ((f_\alpha: \mathbbN \rightarrow \mathbbN)), defined by three simple rules:
Input: (alpha, n) Stack = [(alpha, n)] While stack not empty: Pop (a, m) if m == 0 → push result else reduce a to a[m-1] …