Fast Growing - Hierarchy Calculator !!top!!

Beyond Infinity: The Quest for a Fast-Growing Hierarchy Calculator

Part 4: Implementation Challenges (And Why Most “Calculators” Are Fakes)

Search online for “FGH calculator,” and you’ll find toy scripts that handle ( f_\alpha(n) ) for ( \alpha < \omega^2 ) and ( n < 5 ). A full-featured one is a beast.

Features (of a good calculator)

• Support for ordinals up to ( \Gamma_0 ) or ( \psi(\Omega_\omega) )
• Step-by-step expansion of ( \alpha[n] ) down to 0
• Iteration count display
• Output in normal form or approximated as ( g_\textnumber )

2. Choices that matter for a calculator

• Base function f0: choose n+1 (standard) or something else (e.g., n+2) — affects constants but not asymptotic hierarchy.
• Ordinal notation system: decide how far you want to support (natural numbers, ω, ω^ω, ε0, Γ0, etc.).
• Fundamental sequences: must provide a computable sequence (λ[n]) for each countable limit ordinal λ in your notation.
• Evaluation domain: inputs (n) must be small; values explode quickly — treat outputs as big integers or use growth descriptors.
• Representation of outputs: exact integers become impossible quickly; instead provide:
  • Exact values for small α,n
  • Logarithmic scales (height, number of digits)
  • Ordinal notation for function growth (e.g., "f_ω(3) ≈ f_ω-1^(3)(3)")
  • Descriptive bounds (tower-of-exponentials, hyperoperators)

1. Googology (The study of large numbers)

Communities like the Googology Wiki use FGH calculators to verify the growth rates of new functions. If you invent a function G(n), you feed it into an FGH calculator to see if it matches ( f_ω^2(n) ) or ( f_Γ_0(n) ). fast growing hierarchy calculator

1. Ordinal Input Modes

• Cantor Normal Form (CNF): ω^ω, ω^2 + ω, ε_0, etc.
• Veblen / φ notation: φ(1,0) for ε₀, φ(2,0) for ζ₀, φ(1,0,0) for Γ₀.
• Symbolic builder: Clickable buttons to construct ordinals (e.g., +, ω^, φ).

4. Competitive Code Golf

Extreme coders compete to write the shortest program that approximates large FGH values using the fewest bytes. Beyond Infinity: The Quest for a Fast-Growing Hierarchy

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7. Implementation outline

• Languages: Python (with BigInt), optionally Rust for performance.
• Key modules:
  • Ordinal parser/CNF module
  • Fundamental sequence generator
  • Evaluator with memoization and limits
  • Representation/pretty-printer (Knuth arrows, Conway notation)
  • CLI / Web UI with inputs: ordinal string, n, max-steps, output mode (exact/symbolic/approx)
• Pseudocode sketch (core evaluator):

function eval(ordinal α, int n, limits):
  if α == 0: return n+1
  if α is successor β+1:
    return iterate(eval(β, ·), n, n, limits)
  if α is limit:
    λn = fundamental_sequence(α, n)
    return eval(λn, n, limits)

• iterate(f, count, x): perform count-fold composition with fast strategies and abort if exceeds limits; return symbolic form if aborted.