Calculator Link: Fast Growing Hierarchy

This piece covers the mathematical foundations, the engineering challenges of building such a calculator, and provides a working code implementation for the computable levels of the hierarchy.

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Limit λ:

• Finite ordinals: 0, 1, 2, …
• ( \omega ) (first limit ordinal)
• Addition: ( \omega + 2 )
• Multiplication: ( \omega \cdot 3 )
• Exponentiation: ( \omega^\omega ), ( \omega^\omega^\omega ) (epsilon numbers)
• Veblen hierarchy and beyond (Γ₀, the Feferman–Schütte ordinal, etc.)

• $f_0(n) = n + 1$
• $f_1(n) = f_0^n(n) = n + 1 + 1 \dots = 2n$
• $f_2(n) = f_1^n(n) = 2 \cdot 2 \cdot \dots \cdot n = n \cdot 2^n$
• $f_3(n)$: Iterates exponentiation. This is roughly equivalent to tower of exponents of height $n$. This is "tetrational" growth.
• $f_\omega(n)$: Diagonalization. It selects $f_n(n)$.

  Fast-Growing Hierarchy Calculator — Detailed Guide

  This guide explains fast-growing hierarchies (FGHs), how to compute values at small ordinals, practical strategies for a calculator implementation, algorithms and data structures, performance considerations, and examples. It assumes familiarity with ordinals up to ε0 and basic recursion theory; if not, the worked examples will still illustrate concrete cases. fast growing hierarchy calculator

  print(fgh(2, 3)) # Output: 24 print(fgh('w', 2)) # Output: fgh(2,2) = 8 Finite ordinals: 0, 1, 2, … ( \omega